1To my daughter Han**in**.

ACKNOWLEDGEMENTSThe work presented **in** this **the**sis was performed at **the** Laboratory **of** MIA "Ma**the**matics,Computer Science and Applications" - UFR **of** Ma**the**matics, Computer Science andApplications, Department **of** Ma**the**matics and Computer Science at **the** Faculty **of**Sciences Rabat - University **of** Mohammed V-Agdal.First **of** all, thanks to ALLAH who awarded me with patienace, courage and selfconsitancy **in** order to successfully carry out this research work. I would like to warmlythank my worthy **of** respect supervisor, Mr. Mohamed El MARRAKI, Pr**of**essor **of**Higher Education **in** **the** Faculty **of** Sciences Rabat, for his support, his availability,patience, close collaboration and k**in**d assistance that allowed me to complete this **the**sis.I express my pr**of**ound gratitude to Mr. Driss ABOUTAJDINE, Pr**of**essor **of** HigherEducation at **the** Faculty **of** Sciences Rabat, for **the** honor that he gave me to evaluatethis work, on one hand, and to chair **the** **the**sis committee, on **the** o**the**r hand. Therefore,he f**in**ds **the** expression **of** my deep gratitude here.I wish to thank Mr. El Mamoun SOUIDI, Pr**of**essor **of** Higher Education at **the**Faculty **of** Sciences Rabat, and Mr. Abdelmalek AZIZI, pr**of**essor **of** Higher Educationat **the** Faculty **of** Sciences - University **of** Mohammed I - Oujda, for hav**in**g accepted **the**burden to be **the** penal secretaries **of** **the**sis committee and for **the**ir valuable commentsthat helped me to improve this manuscript.I am also quite grateful to Pr**of**essor Hussa**in** BEN-AZZA, University **of** Mollay Ismal,ENSAM **in** Meknes, for giv**in**g me an honor to accept **the** evaluation burden **of** my **the**sisbe**in**g **the**sis exam**in**er and for his k**in**d help on **spann ing**

4my wife who has been very supportive dur**in**g **the** period **of** my research. At last but notleast, I rema**in** unable to conclude without mention**in**g **the** k**in**d assistance and thank**in**gall **the** members **of** our research group "Laboratory **of** Ma**the**matics, Computer Scienceand Applications" at **the** Faculty **of** sciences Rabat. My thanks **of** course to all my dears, Icame **in**to contact dur**in**g **the**se four years and who helped me directly or **in**directly dur**in**g**the** course **of** this **the**sis.

ContentsIntroduction 15I Prelim**in**aries 191 Def**in**itions and Properties 211.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.1 Vertex Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.1.2 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.1 Prelim**in**aries and notations on maps . . . . . . . . . . . . . . . . . 301.2.2 Faces and Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . 321.2.3 Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3 Trees and Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.3.1 Properties **of** Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.3.2 Spann**in**g Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4 Distance **in** **trees** and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Spann**in**g **trees** 412.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2 Basic concepts and research background . . . . . . . . . . . . . . . . . . . 412.3 Spann**in**g Trees and Enumeration . . . . . . . . . . . . . . . . . . . . . . . 452.3.1 Enumeration **of** Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 452.3.2 Spann**in**g **trees** **in** graphs . . . . . . . . . . . . . . . . . . . . . . . . 452.4 Matrices associated to a graph . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Degree Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.3 Incidence Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4.4 Laplacian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.5 Notation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

6 CONTENTSII Theoretical Part 533 How to count **the** **number** **of** **spann ing**

CONTENTS 76 Count**in**g **the** **number** **of** **spann ing**

8 CONTENTS

List **of** Figures1.1 An example **of** graph G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2 Some examples **of** graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3 Example **of** graph with adjacent vertices . . . . . . . . . . . . . . . . . . . 231.4 An example **of** vertex degrees . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 An example **of** vertex degrees . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 An example **of** complete graph K 5 . . . . . . . . . . . . . . . . . . . . . . . 251.7 From left to right, **the** graphs K 4 , K 2,2 , P 4 , C 4 . . . . . . . . . . . . . . . . . 261.8 A graph G and subgraphs **of** G. . . . . . . . . . . . . . . . . . . . . . . . . 261.9 Some r-regular graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.10 Complete Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.11 Null Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.12 Cycle Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.13 Two graphs G and H are not **the** same, but **the**y are isomorphic. . . . . . . 291.14 Two graphs G and H are not isomorphic. . . . . . . . . . . . . . . . . . . . 291.15 The graph K 4 drawn as a plane graph without edge cross**in**g. . . . . . . . . 301.16 One graph gives two planar maps . . . . . . . . . . . . . . . . . . . . . . . 311.17 (a) A representation **of** a graph; its set **of** vertices is {1, 2, 3, 4}, and(multi)set **of** edges is {{1, 2}, {2, 3}, {2, 4}, {2, 4}, {3, 3}, {3, 4}}. (b) Twoembedd**in**gs **of** this graph **in** **the** sphere, which are not homeomorphic s**in**ce**the** second has a triangular face, unlike **the** first. . . . . . . . . . . . . . . . 311.18 The degree **of** **the** faces **of** this planar map are written **in**side **the** faces . . . 321.19 An example **of** map C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.20 Path graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.21 Simple **trees** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.22 Path and Star **trees** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.23 A graph G with its 3 **spann ing**

12 LIST OF FIGURES7.11 The maps F n and G n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.12 The crystal planar map C n . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.13 The maps E n , E n − e and E n .e . . . . . . . . . . . . . . . . . . . . . . . . . 132

List **of** Tables5.1 Some values **of** τ(F n ), τ(G n ), τ(T n ), τ(H n ) and τ(E n ) . . . . . . . . . . . . 935.2 Some values **of** τ(H n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Some values **of** τ(B n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4 Some values **of** τ(L n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.5 Some values **of** τ(C n ) and τ(Q n ) . . . . . . . . . . . . . . . . . . . . . . . . 995.6 Some values **of** τ(K n ) and τ(Q n ) . . . . . . . . . . . . . . . . . . . . . . . . 1025.7 Some values **of** τ(E n ) and τ(Q n ) . . . . . . . . . . . . . . . . . . . . . . . . 1045.8 Some values **of** τ(D n ) and τ(Q n ) . . . . . . . . . . . . . . . . . . . . . . . . 1066.1 Some values **of** τ(S n,1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.2 Some values **of** τ(S n,2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.1 Some values **of** τ(E n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Some values **of** τ(F n ) and τ(G n ) . . . . . . . . . . . . . . . . . . . . . . . . 1317.3 Some values **of** τ(C n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13213

14 LIST OF TABLES

16 LIST OF TABLESand **in** random walks **in** graphs [114]. The research **of** **the** **number** **of** **spann ing**

LIST OF TABLES 17**the** n-Barrel cha**in**s, **the** n-Light cha**in**s, **the** n-Kite cha**in**s, **the** n-Envelope cha**in**s and **the**n-Diphenylene, ... etc.Outl**in**e **of** **the** **the**sisThis **the**sis is organized as follows. In **the** first chapter, **some** general graph **the**oryis described. In **the** second chapter, we **in**troduce **the** background and research history**of** our problem and def**in**itions and general properties **of** basic objects studied (**spann ing**

18 LIST OF TABLESspecial panar maps. We were able to obta**in** explicit simple formulae for calculat**in**g **the****number** **of** **spann ing**

Part IPrelim**in**aries19

Chapter 1Def**in**itions and PropertiesThis chapter **in**cludes a brief **in**troduction to graph **the**ory. Some **of** **the** basics concern**in**gthis **the**ory are presented **in** [19] and [119]. In case **of** rare term**in**ologies, **the** reader isreferred to consult [69] and [117].1.1 GraphsIn this section, we are go**in**g to present **some** useful def**in**itions related to our work asfollows. A graph will usually be denoted by a capital letter G.Def**in**ition 1.1.1 An undirected graph G is a triplet (V G ,E G ,δ) where V G is **the** set **of**vertices **of** **the** graph G, E G is **the** set **of** edges **of** **the** graph G (E G ⊆ V 2 , mean**in**g thatan edge e ∈ E G is a 2-element subset {v i ; v j } **of** V G ) and δ is **the** application:δ : E G → P(V G )e i ↦→ δ(e i ) = {v j , v k }with v j and v k are end vertices **of** **the** edge e i (not necessarily dist**in**ct). When u and vare **the** endpo**in**ts **of** an edge, **the**y are neighbors. We notice that **the** set {v j , v k } as amultiset (if v j = v k , **the** same vertex appears twice **in** δ(e i )). A loop is an edge e i ∈ E Gwith v j = v k (an edge whose endpo**in**ts are equal), if δ(e i ) = δ(e j ) with i ≠ j **the**n **the**edges e i and e j are called multiple edges (edges hav**in**g **the** same pair **of** endpo**in**ts).Example 1.1.2 In **the** graph shown **in** Figure 1.1, We have: E G = {e 1 , e 2 , e 3 , e 4 }, V G ={v 1 , v 2 , v 3 }, δ(e 1 ) = {v 1 , v 1 } (multiset) δ(e 2 ) = {v 1 , v 2 } and δ(e 3 ) = δ(e 4 ) = {v 2 , v 3 }, **the**edges e 3 and e 4 are multiple edges, **the** edge e 1 is a loop, **the**n **the** graph G admits a loopand two multiple edges.In a graph G, a path is a sequence **of** vertices and edges p = v 0 , e 1 , v 1 , e 2 , ..., v n−1 , e n , v nsuch that δ(e i ) = {v i−1 , v i }. We say that this path attached both ends v 0 and v n . Acycle is a path such that v 0 = v n . A graph G is f**in**ite if its vertex set and edge set are f**in**ite.21

22 CHAPTER 1. DEFINITIONS AND PROPERTIESFigure 1.1: An example **of** graph GWe adopt **the** convention that every graph mentioned **in** this **the**sis is f**in**ite, unlessexplicitly constructed o**the**rwise.Remark 1.1.3 For brevity we denote **the** edge {u, v} ∈ E G simply by uv. In particularwhen uv ∈ E G we say that **the** vertices u and v are adjacent. Then **the** degree **of** a vertexv, written deg(v), is **the** **number** **of** vertices **of** G which are adjacent to v. When **the**re isno ambiguity, we **some**times write V (G) and E(G) **in**stead **of** V G and E G , respectively.In a graph, as already expla**in**ed, two or more edges jo**in****in**g **the** same pair **of** verticesare multiple edges. An edge jo**in****in**g a vertex to itself is a loop. A graph with no multipleedges or loops is a simple graph. For example, graph (a) below (see Figure 1.2) hasmultiple edges and graph (b) has a loop; **the**refore, it is not a simple graph. Graph (c)has no multiple edges or loops, and is **the**refore a simple graph.Figure 1.2: Some examples **of** graphDef**in**ition 1.1.4The vertices v and u **of** a graph are adjacent vertices if **the**y are jo**in**ed by an edge e. Thevertices v and u are **in**cident with **the** edge e, and **the** edge e is **in**cident with **the** verticesv and u.Example 1.1.5 In **the** graph below (see Figure 1.3), **the** vertices u and x are adjacent,vertex w is **in**cident with edges 2, 3, 4 and 5, and edge 6 is **in**cident with **the** vertex x.

1.1. Graphs 23Figure 1.3: Example **of** graph with adjacent verticesDef**in**ition 1.1.6 A graph is f**in**ite if its vertex set and edge set are f**in**ite. We adopt **the**convention that every graph mentioned **in** this **the**sis is f**in**ite, unless explicitly constructedo**the**rwise.Def**in**ition 1.1.7 We say that a graph G is connected if any two **of** its vertices may beconnected by a path (if each pair **of** vertices **in** G belongs to a path); o**the**rwise, G isdisconnected [69, 84].Remark 1.1.8 If G has a u, v-path (Tow vertices u and v connected by a path **in** G),**the**n u is connected to v **in** G.The connection relation on V (G) consists **of** **the** ordered pairs (u, v) such that u isconnected to v. "Connected" is an adjective we apply only to graphs and to pairs **of**vertices (we never say "v is disconnected" when v is a vertex).For a graph G, and a set S **of** vertices and edges **in** G, we shall denote G − S as **the**graph obta**in**ed by remov**in**g **the** elements **of** S from G. Remov**in**g a vertex from a graph,is def**in**ed as remov**in**g **the** vertex, and all edges **in**cident to it.Def**in**ition 1.1.9 A graph G is said to be k-connected, if **the** removal **of** any (k − 1)vertices will not disconnect G, but **the**re exist a set S **of** k vertices such that G − S isdisconnected.The graphs that we consider are **in** most cases connected but may conta**in** multipleedges.1.1.1 Vertex DegreesIn many applications **of** graph **the**ory we need a term for **the** **number** **of** edges meet**in**g at avertex. For example, we may wish to specify **the** **number** **of** roads meet**in**g at a particular**in**tersection, **the** **number** **of** wires meet**in**g at a given term**in**al **of** an electrical network, or**the** **number** **of** chemical bonds jo**in****in**g a given atom to its neighbors. Such situations areillustrated below (see Figure 1.4):

26 CHAPTER 1. DEFINITIONS AND PROPERTIES3. A path P n is a graph with vertices {v 1 , ..., v n } and edges {v 1 v 2 , v 2 v 3 , ..., v n−1 v n }. Itcan also be called a path from v 1 to v n .4. For n ≥ 3, C n is **the** graph P n with one additional edge: v n v 1 . Consequently it iscalled a closed path or a cycle.Figure 1.7: From left to right, **the** graphs K 4 , K 2,2 , P 4 , C 4 .In ma**the**matics, we **of**ten study complicated objects by look**in**g at simpler objects **of** **the**same type conta**in**ed **in** **the**m - subsets **of** sets, subgroups **of** groups, and so on. In graph**the**ory we make **the** follow**in**g def**in**ition.Def**in**ition 1.1.16 A subgraph **of** a graph G is a graph, such that, all **of** whose verticesare vertices **of** G and all **of** whose edges are edges **of** G, i.e., a graph H is a subgraph **of**ano**the**r graph G if V H ⊆ V G and E H ⊆ E G and **the** assignment **of** endpo**in**ts to edges **in**H is **the** same as **in** G. We denote this relation by H ⊆ G and say that G conta**in**s H.For example, P 3 ⊆ C 3 and K 2,2 ⊆ K 2,4 .Remark 1.1.17 Note that G is a subgraph **of** itself.Example 1.1.18 The follow**in**g graphs (see Figure 1.8) are subgraphs **of** **the** graph G on**the** left, with vertices {u, v, w, x} and edges {1, 2, 3, 4, 5}.Figure 1.8: A graph G and subgraphs **of** G.A graph **in** which all **the** vertex degrees are **the** same is given a special name.Def**in**ition 1.1.19 A graph G is regular if its vertices all have **the** same degree.

1.1. Graphs 27Def**in**ition 1.1.20 A graph G is said to be r-regular, or regular **of** degree r, or simplyregular, if every vertex **in** G has degree r, that is, every vertex has r edges **in**cident to it.For example K 3 is 2-regular and K n is n − 1-regular. A regular 3-regular graph is calledcubic. A cubic graph has an even **number** **of** vertices.Example 1.1.21 In **the** follow**in**g graphs, we illustrate **some** r-regular graphs, for variousvalues **of** r:Figure 1.9: Some r-regular graphs.A useful consequence **of** **the** proposition (Degree-Sum Formula) 1.1.12 is **the** follow**in**gresult.Theorem 1.1.22 Let G be an r-regular graph with n vertices. Then G has nr/2 edges.Pro**of**: Let G be a graph with n vertices, each **of** degree r; **the**n **the** sum **of** **the** degrees**of** all **the** vertices is nr. By **the** Proposition (Degree-Sum Formula) 1.1.12, **the** **number** **of**edges is one-half **of** this sum, which is nr/2.□Examples **of** Regular GraphsWe now consider **some** important classes **of** regular graphs.1. Complete Graphs, a complete graph is a graph **in** which each vertex is jo**in**ed to each**of** **the** o**the**rs by exactly one edge. The complete graph with n vertices is denotedby K n . The graph K n is regular **of** degree n − 1, and **the**refore has n(n − 1)/2 edges,by Theorem 1.1.22.

28 CHAPTER 1. DEFINITIONS AND PROPERTIESFigure 1.10: Complete Graphs.2. Null Graphs, A null graph is a graph with no edges. The null graph with n verticesis denoted by N n . The graph N n is regular **of** degree 0.Figure 1.11: Null Graphs.3. Cycle Graphs, A cycle graph is a graph consist**in**g **of** a s**in**gle cycle **of** vertices andedges. The cycle graph with n vertices is denoted by C n . The graph C n is regular**of** degree 2, and has n edges. For n ≥ 3, C n can be drawn **in** **the** form **of** a regularpolygon.Figure 1.12: Cycle Graphs.Def**in**ition 1.1.23 If V = {v 1 , ..., v n } **the**n **the** degree sequence **of** G is deg(v 1 ), ..., deg(v n )arranged **in** decreas**in**g order. For example **the** degree sequence **of** P 5 is 2, 2, 2, 1, 1.Def**in**ition 1.1.24 An isomorphism from a simple graph G to a simple graph H is abijection f : V (G) → V (H) such that uv ∈ E(G) if and only if f(u)f(v) ∈ E(H). Wesay "G is isomorphic to H", written G ∼ = H, if **the**re is an isomorphism from G to H.Def**in**ition 1.1.25 Two graphs are isomorphic to each o**the**r, written G ∼ = H, if **the**re isa bijection f : V G → V H such that a, b ∈ V G are adjacent if and only if f(a), f(b) ∈ V Hare adjacent. For example K 3∼ = C3 and both are called triangles. Also K 2∼ = K1,1∼ = P2 .

1.1. Graphs 29Two graphs G and H are isomorphic if H can be obta**in**ed by relabell**in**g **the** vertices**of** G that is, if **the**re is a one-one correspondence between **the** vertices **of** G and those **of**H, such that **the** **number** **of** edges jo**in****in**g each pair **of** vertices **in** G is equal to **the** **number****of** edges jo**in****in**g **the** correspond**in**g pair **of** vertices **in** H. Such a one-one correspondenceis an isomorphism.Example 1.1.26 For example, **the** graphs G and H represented by **the** diagramsFigure 1.13: Two graphs G and H are not **the** same, but **the**y are isomorphic.are not **the** same, but **the**y are isomorphic, s**in**ce we can relabel **the** vertices **in** **the** graphG to get **the** graph H, us**in**g **the** follow**in**g one-one correspondence:G ↔ H, u ↔ 4, v ↔ 3, w ↔ 2 and x ↔ 1Note that edges **in** G correspond to edges **in** H, for example: **the** two edges jo**in****in**gu and v **in** G correspond to **the** two edges jo**in****in**g 4 and 3 **in** H; **the** edge uw **in** Gcorresponds to **the** edge 42 **in** H; **the** loop ww **in** G corresponds to **the** loop 22 **in** H.Remark 1.1.27 Note that if G ∼ = H **the**n |V G | = |V H |, |E G | = |E H |, and **the**ir degreesequences must be identical. However none **of** **the**se is a sufficient condition for isomorphism.Example 1.1.28 For example, **the** graphs G and H represented by **the** diagrams below(see Figure 1.14); which have |V G | = |V H | and |E G | = |E H |, but **the**y are not isomorphic.Figure 1.14: Two graphs G and H are not isomorphic.

30 CHAPTER 1. DEFINITIONS AND PROPERTIES1.1.2 Planar GraphsDef**in**ition 1.1.29 A graph is planar if it can be drawn **in** **the** plane such that no edgesare cross**in**g each o**the**r (its edges do not cross). This particular draw**in**g **of** **the** graphis called a plane graph. For example K 4 is planar graph but K 3,3 is not planar graph.Although **the** complete graph with four vertices K 4 is usually pictured with cross**in**g edgesas **in** Figure. 1.15(a), it can also be drawn with noncross**in**g edges as **in** Figure. 1.15(b);hence K 4 is planar.Figure 1.15: The graph K 4 drawn as a plane graph without edge cross**in**g.Note that if G is disconnected, **the**n G is planar if and only if each component is planar,hence we may assume well that G is connected throughout this **the**sis.Proposition 1.1.30 If H ⊆ G and H is not planar **the**n nei**the**r is G. In particular K m,nis not planar if m, n ≥ 3.Def**in**ition 1.1.31 A planar graph partitions **the** plane **in**to subsets called regions (faces).For example **the** plane graph **of** K 4 has four regions, one **of** which is exterior to **the** graph.1.2 MapsThe aim **of** this section is to provide a short and accessible presentation **of** planar maps.For a more detailed **in**troduction, see **the** **in**troductory chapter **in** [69] and **the** **the**sis **of** ElMarraki [43].1.2.1 Prelim**in**aries and notations on mapsWe beg**in** with **some** vocabulary on maps. A map is a proper embedd**in**g **of** a connectedgraph **in**to **the** two-dimensional sphere, considered as cont**in**uous deformations. A map isrooted if one **of** its edges is dist**in**guished as **the** root-edge and attributed an orientation.Unless o**the**rwise specified, all maps under consideration **in** this **the**sis are rooted. Theface at **the** right **of** **the** root-edge is called **the** root-face and **the** o**the**r faces are said tobe **in**ternal. Similarly, **the** vertices **in**cident to **the** root-face are said to be external and**the** o**the**rs are said to be **in**ternal. Graphically, **the** root-face is usually represented as **the****in**f**in**ite face when **the** map is projected on **the** plane.

1.2. Maps 31Def**in**ition 1.2.1 (Map) A map C is a graph G drawn on a surface X or embedded **in**toit (that is, a compact 2-dimensional orientable variety) **in** such a way that:• **the** vertices **of** graph are represented as dist**in**ct po**in**ts **of** **the** surface.• **the** edges are represented as curves on **the** surface that **in**tersect only at **the** vertices.• if we cut **the** surface along **the** graph thus drawn, what rema**in**s (that is, **the** set X\G)is a disjo**in**t union **of** connected components, called faces, each homeomorphic to anopen disk (for more **in**formation on **the** faces **of** a map see [43] and [69]).A planar map is a map drawn on **the** plane. Through this **the**sis, all maps are planar andconnected.Figure 1.16: One graph gives two planar mapsA planar map (hereafter shortly called a map) is an isotopy class **of** planar embedd**in**gs**of** a connected planar graph. Notice that **the** graphs embedded are unlabelled. To stateit simply, a planar map is a connected unlabelled graph drawn **in** **the** plane without edgecross**in**gsand up to cont**in**uous deformation. Planar maps are **of**ten called plane graphs **in****the** literature [43, 69]. As illustrated **in** Figure 1.17 (a)-(b), a planar graph can have nonisotopicplanar embedd**in**gs, so that it gives rise to several maps. Due to **the** topologicalembedd**in**g, a map has more structure than a graph. In particular, a map has faces, eachface correspond**in**g to a connected component **of** **the** plane splits by **the** embedd**in**g.Figure 1.17: (a) A representation **of** a graph; its set **of** vertices is {1, 2, 3, 4}, and (multi)set**of** edges is {{1, 2}, {2, 3}, {2, 4}, {2, 4}, {3, 3}, {3, 4}}. (b) Two embedd**in**gs **of** this graph**in** **the** sphere, which are not homeomorphic s**in**ce **the** second has a triangular face, unlike**the** first.

32 CHAPTER 1. DEFINITIONS AND PROPERTIESThe unique unbounded face is called **the** outer (or **in**f**in**ite) face. Vertices, edges, andfaces are called **the** 0-cells, 1- cells, and 2-cells **of** **the** map, respectively. The **number**s |V |,|E|, and |F | **of** vertices, edges, and faces (**in**clud**in**g **the** outer (external) face) **of** a mapare related by **the** well known Euler’s relation:|V | − |E| + |F | = 2.1.2.2 Faces and Euler’s formulaLet C be a planar map. If we omit **the** l**in**e segments **of** C from **the** plane surface on whichC is drawn, **the** rema**in**der splits **in**to a **number** **of** connected open regions; **the** closure **of**such a region is called a face.An edge is **in**cident to a face if it belongs to **the** boundary **of** this face. If both "banks"**of** **the** edge belong to **the** same face, **the**n such an edge is called an isthmus; we say thatan isthmus is **in**cident to **the** correspond**in**g face twice.Def**in**ition 1.2.2 The degree **of** a face f denoted by deg(f) is **the** **number** **of** edges**in**cident to this face (isthmus be**in**g counted twice).The notion **of** face degree is illustrated **in** Figure.1.18). If we go around **the** boundary **of**a face slightly **in**side **the** face, **the**n **the** **number** **of** times we pass along an edge is exactly**the** degree **of** **the** face.Figure 1.18: The degree **of** **the** faces **of** this planar map are written **in**side **the** facesThe follow**in**g proposition is similar to **the** previous proposition 1.1.12.Proposition 1.2.3 The sum **of** **the** degrees **of** all faces **of** a map C is equal to twice **the****number** **of** its edges, i.e.,∑f∈F Cdeg(f) = 2|E C |,where **the** sum is taken over **the** set F C **of** **the** faces **of** **the** map.

1.2. Maps 33Theorem 1.2.4 (Euler characteristic) Let us associate to a map C **the** **number**χ(C) = |V C | + |F C | − |E C |which is called its Euler characteristic. Then χ(C) does not depend on **the** map C itselfbut only on its genus g and is equal to 2 − 2g.1.2.3 Euler’s formulaEuler’s formula for planar maps is:|V C | + |F C | − |E C | = 2 − 2g,where F C is **the** set **of** faces **of** **the** planar map C and g is **the** genus **of** a map C, **in** **the**planar case g = 0. A map **of** genus zero is called planar map. This formula is valid forplanar maps (graphs embedded **in** **the** plane or **in** **the** surface without edge-cross**in**gs).For **the** genus zero case this **the**orem was already observed by Descartes, and was provedby Euler **in** 1752 [125].The follow**in**g **the**orem gives ano**the**r famous result due to Euler.Theorem 1.2.5 (Euler’s formula) Let G be a connected planar map with n vertices, medges and f faces (regions). Then n − m + f = 2.Pro**of**: Let’s apply **the** **in**duction on m . For m = 0 we have n = 1 and f = 1, so that**the** statement holds. Now let m ≠ 0. If G conta**in**s a cycle, we discard one **of** **the** edgesconta**in**ed **in** this cycle and get a graph G ′ with n ′ = n, m ′ = m − 1 and f ′ = f − 1. By**in**duction hypo**the**sis, n ′ −m ′ +f ′ = 2 and hence n−m+f = 2. If G is acyclic, **the**n G is atree so that m = n−1, by Theorem 1.2.8; as f = 1, we still obta**in** n−m+f = 2. □Example 1.2.6 In **the** map C shown **in** Figure. 1.19, we have: **the** **number** **of** **the** verticesis 11, **the** **number** **of** **the** faces is 13, **the** **number** **of** **the** edges is 22, **the**n 11 + 13 − 22 = 2.Figure 1.19: An example **of** map C

34 CHAPTER 1. DEFINITIONS AND PROPERTIESCorollary 1.2.71. If G is planar with |V | ≥ 3 **the**n |E| ≤ 3|V | − 6. In particular K 5 is not planar andnei**the**r is K n for all n ≥ 6.2. If G is planar **the**n **the**re is a vertex **of** degree 5 or less.3. If G is planar and conta**in**s no triangles **the**n |E| ≤ 2|V | − 4.Remark 1.2.8 In this **the**sis, we are only **in**terested **in** planar graphs. For fur**the**r detailand more explanation on graphs we refer **the** reader to [1, 10, 13, 19, 20, 26, 39, 56, 117,118, 119, 121].1.3 Trees and ForestsIn this section, we focus our attention on one particularly important and useful type **of**graph - a tree. Although **trees** are relatively simple structures, **the**y form **the** basis **of**many **of** **the** practical techniques used to model and to design large-scale systems.The concept **of** a tree is one **of** **the** most important and commonly used ideas **in** graph**the**ory, especially **in** **the** applications **of** **the** subject. It arose **in** connection with **the** work**of** Gustav Kirchh**of**f on electrical networks **in** **the** 1840, and later with Arthur Cayley’swork on **the** **enumeration** **of** molecules **in** **the** 1870. More recently, **trees** have proved to be**of** value **in** such areas as computer science, decision mak**in**g, l**in**guistics, and **the** design **of**gas pipel**in**e systems.The word "tree" suggests branch**in**g out from a root and never complet**in**g a cycle. Trees asgraphs have many applications, especially **in** data storage, search**in**g, and communication.Remark 1.3.1 A path graph P n is a tree consist**in**g **of** a s**in**gle path through all itsvertices. The graph P n has n − 1 edges, and is obta**in**ed from **the** cycle graph C n byremov**in**g any **of** its edges (see Figure 1.20).Figure 1.20: Path graphs.Def**in**ition 1.3.2 (Tree) A tree is a connected simple graph that conta**in**s no cycle, i.e.,without cycle. For example P 4 .Def**in**ition 1.3.3 (A plan tree) A plan tree is a tree designed **in** **the** plane or is a mapwith only one face, **the** outer face (see Figure 1.21).

1.3. Trees and Forests 35Figure 1.21: Simple **trees**Remark 1.3.4 Tree graphs form an important class **of** planar graphs.In general a graph which conta**in**s no cycles is called acyclic. A tree is a connected acyclicgraph. A tree is a path if and only if its maximum degree is 2. A star is a tree consist**in**g**of** one vertex adjacent to all **the** o**the**rs K 1,n−1 , see Figure 1.22.Figure 1.22: Path and Star **trees**An acyclic graph, one not conta**in****in**g any cycles, is called a forest. A connected forestis called a tree (Thus, a forest is a graph whose components are **trees**). The vertices **of**degree 1 **in** a tree are its leaves. Every nontrivial tree has at least two leaves-take, forexample, **the** ends **of** a longest path. This little fact **of**ten comes **in** handy, especially **in****in**duction pro**of**s about **trees**: if we remove a leaf from a tree, what rema**in**s is still a tree.Example 1.3.5 A tree is a connected forest, and every component **of** a forest is a tree.1.3.1 Properties **of** TreesTrees have many equivalent characterizations, any **of** which could be taken as **the**def**in**ition. Such characterizations are useful because we need only verify that a graphsatisfies anyone **of** **the**m to prove that it is a tree, after which we can use all **the** o**the**rproperties.We first prove that delet**in**g a leaf from a tree yields a smaller tree.Proposition 1.3.6 A tree conta**in**s a vertex **of** degree 1, which is called a leaf.Lemma 1.3.7 Every tree with at least two vertices has at least two leaves. Delet**in**g aleaf from an n-vertex tree produces a tree with n − 1 vertices.

36 CHAPTER 1. DEFINITIONS AND PROPERTIESPro**of**: A connected graph with at least two vertices has an edge. In an acyclic graph,an endpo**in**t **of** a maximal nontrivial path has no neighbor o**the**r than its neighbor on **the**path. Hence **the** endpo**in**ts **of** such a path are leaves. Let v be a leaf **of** a tree G, andlet G ′ = G − v. A vertex **of** degree 1 belongs to no path connect**in**g two o**the**r vertices.Therefore, for u, w ∈ V (G ′ ), every u, w-path **in** G is also **in** G ′ . Hence G ′ is connected.S**in**ce delet**in**g a vertex cannot create a cycle, G ′ also is acyclic. Thus G ′ is a tree withn−1 vertices.□Remark 1.3.8 The previous Lemma implies that every tree with more than one vertexarises from a smaller tree by add**in**g a vertex **of** degree 1 (all our graphs are f**in**ite).This rescues **some** pro**of**s from **the** **in**duction trap: grow**in**g an n + 1-vertex tree from anarbitrary n-vertex tree by add**in**g a new neighbor at an arbitrary old vertex generates all**trees** with n + 1 vertices. The word "arbitrary" means that **the** discussion considers allways **of** mak**in**g **the** choice.The pro**of** **of** equivalence **of** characterizations **of** **trees** uses **in**duction, prior results,a count**in**g argument, extremality, and contradiction.Theorem 1.3.9 For an n-vertex graph G (with n ≥ 1), **the** follow**in**g are equivalent (andcharacterize **the** **trees** with n vertices).(A) G is connected and has no cycles (G is acyclic).(B) G is connected and has n − 1 edges (|V | = |E| + 1).(C) G has n − 1 edges and no cycles.(D) For u, v ∈ V (G), G has exactly one u, v-path.Pro**of**: We first demonstrate **the** equivalence **of** A, B, and C by prov**in**g that any two**of** connected, acyclic, n − 1 edges toge**the**r imply **the** third. A ⇒ B, C. We use **in**ductionon n. For n = 1, an acyclic 1-vertex graph has no edge. For n > 1, we suppose that **the**implication holds for graphs with fewer than n vertices. Given an acyclic connected graphG, **the** previous Lemma provides a leaf v and states that G ′ = G − v also is acyclic andconnected (see figure above). Apply**in**g **the** **in**duction hypo**the**sis to G ′ yields E G ′ = n − 2.S**in**ce only one edge is **in**cident to v, we have E G = n − 1.B ⇒ A, C. Delete edges from cycles **of** G one by one until **the** result**in**g graph G ′ is acyclic.S**in**ce no edge **of** a cycle is a cut-edge (**the** previous Theorem), G ′ is connected. Now **the**

1.3. Trees and Forests 37preced**in**g paragraph implies that E G ′ = n − 1. S**in**ce we are given E G = n − 1, no edgeswere deleted. Thus G ′ = G, and G is acyclic.C ⇒ A, B. Let G l , ..., G k be **the** components **of** G. S**in**ce every vertex appears **in** onecomponent, ∑ n(G i ) = n. S**in**ce G has no cycles, each component satisfies property A.iThus E Gi = n(G i ) − 1. Summ**in**g over i yields E G = ∑ [n(G i ) − 1] = n − k. We are giveniE G = n − 1, so k = 1, and G is connected.A ⇒ D. S**in**ce G is connected, each pair **of** vertices is connected by a path. If **some** pairis connected by more than one, we choose a shortest (total length) pair P , Q **of** dist**in**ctpaths with **the** same endpo**in**ts. By this extremal choice, no **in**ternal vertex **of** P or Q canbelong to **the** o**the**r path (see figure below).This implies that P ∪ Q is a cycle, which contradicts **the** hypo**the**sis A.D ⇒ A. If **the**re is a u, v-path for every u, v ∈ V G , **the**n G is connected. If G has a cycleC, **the**n G has two u, v-paths for u, v ∈ V C ; hence G is acyclic (this also forbids loops). □Def**in**ition 1.3.10 A cut-edge or cut-vertex **of** a graph is an edge or vertex whose deletion**in**creases **the** **number** **of** components. We write G − e or G − M for **the** subgraph **of** Gobta**in**ed by delet**in**g an edge e or set **of** edges M. We write G − v or G − S for **the**subgraph obta**in**ed by delet**in**g a vertex v or set **of** vertices S.Next, we characterize cut-edges **in** terms **of** cycles.Theorem 1.3.11 An edge is a cut-edge if and only if it belongs to no cycle.Corollary 1.3.12a) Every edge **of** a tree is a cut-edge.b) Add**in**g one edge to a tree forms exactly one cycle.c) Every connected graph conta**in**s a **spann ing** tree.Pro

38 CHAPTER 1. DEFINITIONS AND PROPERTIESTheorem 1.3.13 The follow**in**g assertions are equivalent for a graph T :(i) T is a tree;(ii) any two vertices **of** T are l**in**ked by a unique path **in** T ;(iii) T is m**in**imally connected, i.e. T is connected but T − e is disconnected for everyedge e ∈ T ;(iv) T is maximally acyclic, i.e. T conta**in**s no cycle but T + xy does, for any twonon-adjacent vertices x, y ∈ T .1.3.2 Spann**in**g TreesAn important concept that we need later is that **of** a **spann ing** tree

1.4. Distance **in** **trees** and graphs 39Def**in**ition 1.4.1 (Walk) A walk is a sequence **of** edges v 1 v 2 , v 2 v 3 , ..., v n−1 v n which arenot necessarily dist**in**ct (unless **the** walk is a path). In this case we say that **the** walk isfrom v 1 to v n **of** length n − 1.Def**in**ition 1.4.2 (Distance) The distance between two dist**in**ct vertices v i and v j **of**a graph G, denoted by d(v i , v j ) is equal to **the** length **of** **the** shortest path (**number** **of**edges **in**) that connects v i and v j (**the** least length between v i and v j ). Conventionally,d(v i , v i ) = 0. If G has a u, v-path, **the**n **the** distance from u to v, written d G (u, v) or simplyd(u, v), is **the** least length **of** a u, v-path. If G has no such path, **the**n d(u, v) = ∞ [87].The distance between two vertices v and u, written d(v, u), is **the** length **of** **the**shortest walk from v to u, if it exists, o**the**rwise let d(v, u) = ∞. Note that **the** shortestwalk is necessarily a path. Fur**the**rmore, **in** a weighted graph, d(v, u) is understood to be**the** m**in**imum total weights **of** all possible walks from v to u.Def**in**ition 1.4.3 (Weight) The weight, denoted by p(v i , v j ) is **the** **number** **of** edges thatconnects v i with v j .We use **the** word "diameter" due to its use **in** geometry, where it is **the** greatest distancebetween two elements **of** a set.Def**in**ition 1.4.4 (Diameter) The diameter **of** a graph G is def**in**ed as **the** maximum **of****the** set **of** all shortest walks jo**in****in**g any two vertices, i.e.,Diam(G) = max{d(v, u)|v, u ∈ V }.For example diam(G) = 1 if and only if G is complete but **the** diameter **in** a plane tree is**the** **number** **of** edges **of** **the** longest path. Note also that diam(G) = ∞ if and only if G isdisconnected.Example 1.4.5 The diameter **of** **the** graph below is 2 (see Figure. 1.24). This is becausefor any vertex not directly connected to ano**the**r, **the**re is a path **of** length two connect**in**g**the** two. By look**in**g at **the** graph, it can be seen that this is true.Figure 1.24: An example **of** a graph G whose diameter is 2

40 CHAPTER 1. DEFINITIONS AND PROPERTIES

Chapter 2Spann**in**g **trees**2.1 IntroductionSpann**in**g **trees** have always been **of** great **in**terest **in** various areas **of** computer science.The same is true for **the** idea **of** shortest paths **in** a graph. The **number** **of** **spann ing**

42 CHAPTER 2. SPANNING TREESFigure 2.1: A graph(left) and all **spann ing**

2.2. Basic concepts and research background 43Thus, with low reliability **of** each **of** **the** l**in**e, **the** network’s reliability is determ**in**ed,basically, by **the** **number** **of** **spann ing**

44 CHAPTER 2. SPANNING TREESThe **number** **of** **spann ing**

2.3. Spann**in**g Trees and Enumeration 45Remark 2.2.5 If two **spann ing**

46 CHAPTER 2. SPANNING TREESExample 2.3.4 Below is **the** kite. To count **the** **spann ing**

2.4. Matrices associated to a graph 47Figure 2.7: A graph G with its adjacency matrix.Remark 2.4.1 An adjacency matrix is determ**in**ed by a vertex order**in**g. Every adjacencymatrix is symmetric (a ij = a ji for all i, j). An adjacency matrix **of** a simple graph G hasentries 0 or 1, with 0s on **the** diagonal. The degree **of** v is **the** sum **of** **the** entries **in** **the**row for v **in** ei**the**r A or M.Example 2.4.2 Consider **the** labelled graph G **in** Figure 2.8.Figure 2.8: A graph with 6 vertices.which has **the** adjacency matrix A2.4.2 Degree Matrix⎡⎤0 1 0 0 1 01 0 1 0 1 1A =0 1 0 1 0 1⎢0 0 1 0 0 1⎥⎣1 1 0 0 0 1⎦0 1 1 1 1 0The degree **of** a vertex v ∈ V **of** G denoted by deg G (v); is **the** **number** **of** edges **of** G whichare **in**cident with v. The degree matrix **of** G is **the** diagonal V -by-V matrix D = D(G)such that D vv = deg G (v) for all v ∈ V , and D vu = 0 if v ≠ u; more precisely, **the** degreematrix D **of** a graph, is a diagonal matrix with **the** vertex degrees **in** **the** diagonal.

48 CHAPTER 2. SPANNING TREESFor example, a graph G that shown **in** Figure 2.7 has degree matrix as follows:⎛⎞2 0 0 0D = ⎜ 0 3 0 0⎟⎝ 0 0 4 0 ⎠0 0 0 1Example 2.4.3 The degree matrix D **of** **the** graph G shown **in** Figure 2.8 is as follows:2.4.3 Incidence Matrix⎡⎤2 0 0 0 0 00 4 0 0 0 0D =0 0 3 0 0 0⎢0 0 0 2 0 0⎥⎣0 0 0 0 3 0⎦0 0 0 0 0 4Let G be a f**in**ite undirected graph without loops. The (vertex-edge) **in**cidence matrix **of**G is **the** 0-1 matrix M, with rows **in**dexed by **the** vertices and columns **in**dexed by **the**edges, where M ve = 1 when vertex v is an endpo**in**t **of** edge e. For example **the** **in**cidencematrix **of** P 4 can be given by:Figure 2.9: P 4 and its **in**cidence matrix.The **in**cidence matrix M is **the** n-by-m matrix **in** which entry m ij is 1 if v i is an endpo**in**t**of** e j and o**the**rwise is 0. If vertex v is an endpo**in**t **of** edge e, **the**n v and e are **in**cident.Example 2.4.4 For **the** loopless graph G below (see Figure 2.10), we exhibit **the** **in**cidencematrix that result from **the** vertex order**in**g w, x, y, z and **the** edge order**in**g a, b,c, d, e. The degree **of** y is 4, by view**in**g **the** graph or by summ**in**g **the** row for y **in** ei**the**rmatrix.

2.4. Matrices associated to a graph 49Figure 2.10: The **in**cidence matrix **of** a graph GDef**in**ition 2.4.5 The **in**cidence matrix M for a graph with n vertices and m edges is an × m matrix that **in**dicates which edges are **in**cident on which vertices. We assume thatboth **the** edges and vertices are given an order**in**g. Us**in**g **the** order**in**g **of** **the** vertices, weimpose a direction on each edge such that **the** edge po**in**ts from **the** lower-ordered vertexto **the** higher ordered vertex. The entries **of** **the** **in**cidence matrix are def**in**ed as follows:⎧⎨ 1 if edge e j po**in**ts out from v ia i,j := −1 if edge e j po**in**ts to v i⎩0 o**the**rwise.Example 2.4.6 The underly**in**g graph **of** **the** digraph below, is **the** graph **of** Example2.4.4; note **the** similarity and difference **in** **the**ir matrices.Figure 2.11: The **in**cidence matrix M **of** a directed graph G2.4.4 Laplacian MatrixIn **the** ma**the**matical field **of** graph **the**ory **the** Laplacian matrix or **the** matrix Laplace,**some**times called admittance matrix or Kirchh**of**f matrix, is a matrix representation**of** a graph. Toge**the**r with Kirchh**of**f’s **the**orem it can be used by Kirchh**of**f [65, 81]to calculate **the** **number** **of** **spann ing**

50 CHAPTER 2. SPANNING TREESIn **the** follow**in**g work, we consider only **the** graphs that do not have loops.Def**in**ition 2.4.7 The Laplacian matrix L **of** an undirected graph G is def**in**ed as **the**difference between its degree matrix D and its adjacency matrix A.L = D − A.More formally, an undirected graph G = (V G , E G ) **of** n vertices, weighted by **the** weightfunction at any edge (v i , v j ) associated weight p(v i , v j ).The Laplacian matrix **of** G verifies :⎧⎨ deg(v i ) = ∑ nk=1 p(v i, v k ) if i = jL i,j := −p(v i , v j )if i ≠ j and (v i , v j ) ∈ E G⎩0 o**the**rwise.That is, **the** diagonal elements have values equal to **the** degree **of** **the** correspond**in**g vertices,and **the** **of**f-diagonal elements are −p(v i , v j ) (**the** **number** **of** edges that connects v iwith v j ) if an edge connects **the** two vertices, and 0 o**the**rwise.Example 2.4.8 The Laplacian matrix **of** a labeled graph G shown **in** Figure 2.7 is givenby:⎛⎞ ⎛⎞ ⎛⎞2 0 0 0 0 1 1 0 2 −1 −1 0L = ⎜ 0 3 0 0⎟⎝ 0 0 4 0 ⎠ − ⎜ 1 0 2 0⎟⎝ 1 2 0 1 ⎠ = ⎜ −1 3 −2 0⎟⎝ −1 −2 4 −1 ⎠0 0 0 1 0 0 1 0 0 0 −1 1Example 2.4.9 The graph G pictured **in** Figure 2.8 gives:⎡⎤2 −1 0 0 −1 0−1 4 −1 0 −1 −1L =0 −1 3 −1 0 −1⎢ 0 0 −1 2 0 −1⎥⎣−1 −1 0 0 3 −1⎦0 −1 −1 −1 −1 4Remark 2.4.10 Let G be a f**in**ite undirected graph without loops. The Laplace matrix **of**G is **the** matrix L **in**dexed by **the** vertex set **of** G, with zero row sums, where L vu = −A vufor v ≠ u. If D is **the** diagonal matrix, **in**dexed by **the** vertex set **of** G such that D vv is**the** degree (valency) **of** v, **the**n L = D − A. The matrix Q = D + A is called **the** signlessLaplace matrix **of** G. An important property **of** **the** Laplace matrix L and **the** signlessLaplace matrix Q is that **the**y are positive semidef**in**ite. Indeed, one has Q = MM T andL = NN T , if M is **the** **in**cidence matrix **of** G and N **the** directed **in**cidence matrix **of** **the**directed graph obta**in**ed by orient**in**g **the** edges **of** G **in** an arbitrary way.

2.4. Matrices associated to a graph 51Depth-First Search AlgorithmGoal: To determ**in**e whe**the**r or not a given graph G is connected, **in** which case it producesa **spann ing** tree

52 CHAPTER 2. SPANNING TREES

Part IITheoretical Part53

Chapter 3How to count **the** **number** **of** **spann ing**

56CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHSFigure 3.1: A map C gives rise to eight **spann ing**

3.3. Count**in**g **the** **number** **of** **spann ing**

58CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHSPro**of**: First we notice that F n,1 is **the** set **of** all rooted **trees** (s**in**ce **the**y only consist **of**one component). Note that |F n,1 | = nT n , s**in**ce **in** every tree **the**re are n choices for **the**root. We now regard F n,k ∈ F n,k as a directed graph with all **of** **the** edges directed awayfrom **the** roots. Now let F k be a fixed forest **in** F n,k and denote by N(F k ) **the** **number** **of**rooted **trees** conta**in****in**g F k , and by N ∗ (F k ) **the** **number** **of** ref**in****in**g sequences end**in**g **in** F k .We count N ∗ (F k ) **in** two ways, first by start**in**g at a tree and secondly by start**in**g atF k . Suppose F 1 ∈ F n,1 conta**in**s F k . S**in**ce we may delete **the** k - 1 edges **of** F 1 \F k **in** anypossible order to get a ref**in****in**g sequence from F 1 to F k , we f**in**dN ∗ (F k ) = N(F k )(k − 1)!.Let us now start at **the** o**the**r end. To produce from F k an F k−1 we have to add a directededge, from any vertex v, to any **of** **the** k - 1 roots **of** **the** **trees** that do not conta**in** v. Thuswe have n(k − 1) choices. Similarly, for F k−1 we may produce a directed edge from anyvertex u to any **of** **the** k - 2 roots **of** **the** **trees** not conta**in****in**g u. For this we have n(k − 2)choices. Cont**in**u**in**g this way, we arrive atand thus we have thatN ∗ (F k ) = n k−1 (k − 1)!,N(F k ) = n k−1 for any F k ∈ F n,k .For k = n, F n consists **of** just n isolated vertices. Hence N(F n counts **the** **number** **of** allrooted **trees**, thus |F n,1 | = n n−1 , and thus T n = (1/n)(|F n,1 |) = n n−2 .□The method **of** Prüfer’s code has also been used to **in**vestigate **the** **spann ing** treeformulae for bipartite graphs [42], k-partite graphs [96], extended graphs [61] andcomplete multipartite graphs [41, 72].3.4 Count

3.4. Count**in**g **the** **number** **of** **spann ing**

60CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHSFigure 3.5: A graph G and its graph orientedFigure 3.5 (b) shows **the** graph **of** Figure 3.5 (a) with **the** edges oriented arbitrarily andlabelled with **the** letters from a to h. Relative to this orientation **the** signed **in**cidencematrix **of** **the** graph is:⎡⎤−1 1 −1 0 0 0 0 01 0 0 −1 −1 0 0 0M =⎢ 0 0 1 0 1 0 1 0⎥⎣ 0 0 0 0 0 −1 −1 1 ⎦0 −1 0 1 0 1 0 −1(The rows are **in**dexed by 1 ... 5 and **the** columns by a ... h.) For any matrix Mwe denote **the** conjugate transpose **of** M by M T .Lemma 3.4.3 Let G = (V, E) be a graph, orient G arbitrarily, and let M be **the** correspond**in**gsigned **in**cidence matrix. ThenL(G) = MM T .The pro**of** **of** Matrix Tree Theorem can be shown to follow from **the** basic comb**in**atorialidea, **the** multil**in**earity **of** **the** determ**in**ant and **in**duction on **the** **number** **of** edges **in** **the**graph.Pro**of**: First, we show that if G ′ is an orientation **of** G and M is **the** **in**cidence matrix**of** G ′ **the**n L = MM T . Label **the** now directed edges by e 1 , ..., e m . By def**in**ition **of** **the****in**cidence matrix, s**in**ce every entry **in** **the** n by n matrix MM T is **the** dot product **of** rows**of** M, diagonal entries **in** **the** product count vertex degrees and **of**f-diagonal entries count−1 for every edge **of** G between two vertices.

3.4. Count**in**g **the** **number** **of** **spann ing**

62CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHS⎛D =⎜⎝2 0 0 0 0 0 0 00 2 0 0 0 0 0 00 0 3 0 0 0 0 00 0 0 3 0 0 0 00 0 0 0 3 0 0 00 0 0 0 0 3 0 00 0 0 0 0 0 2 00 0 0 0 0 0 0 2⎞⎛, A =⎟ ⎜⎠ ⎝Then **the** matrix L = D − A is **the** 8 × 8 matrix⎛L =⎜⎝2 −1 0 −1 0 0 0 0−1 2 −1 0 0 0 0 00 −1 3 −1 0 −1 0 0−1 0 −1 3 −1 0 0 00 0 0 −1 3 −1 0 −10 0 −1 0 −1 3 −1 00 0 0 0 0 −1 2 −10 0 0 0 −1 0 −1 20 1 0 1 0 0 0 01 0 1 0 0 0 0 00 1 0 1 0 1 0 01 0 1 0 1 0 0 00 0 0 1 0 1 0 10 0 1 0 1 0 1 00 0 0 0 0 1 0 10 0 0 0 1 0 1 0Let now s = 3. By delet**in**g **the** s-th row and s-th column **of** L we have **the** 7 × 7 matrix⎛L ∗ =⎜⎝2 −1 −1 0 0 0 0−1 2 0 0 0 0 0−1 0 3 −1 0 0 00 0 −1 3 −1 0 −10 0 0 −1 3 −1 00 0 0 0 −1 2 −10 0 0 −1 0 −1 2The determ**in**ant **of** this matrix is 56. Thus **the** Matrix Tree Theorem states that **the****number** **of** **spann ing**

3.4. Count**in**g **the** **number** **of** **spann ing**

64CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHScolumn. By **the** matrix-tree **the**orem, **the** product n−1 ∏**of** **spann ing**

3.4. Count**in**g **the** **number** **of** **spann ing**

66CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHSRemark 3.4.10 As we have seen, calculat**in**g **the** **number** **of** **spann ing**

Chapter 4New methods to compute **the** **number****of** **spann ing**

68CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSRemark 4.2.1Figure 4.1: A map C and its complexity which has five **spann ing**

4.3. Ma**in** Results 69one vertex (any path connect**in**g a vertex **of** C 1 to a vertex **of** C 2 must pass through thisvertex); see Figure 4.3.Figure 4.3: A map C = C 1 • C 2Property 4.3.2 Let C be a map **of** type C = C 1 • C 2- C 1 and C 2 have a common vertex v 1 and a common face (**the** external face).- V C =V C1 +V C2 -1, E C =E C1 +E C2 and F C =F C1 +F C2 -1- A path from a vertex **of** C 1 to a vertex **of** C 2 must pass through v 1 .- If we remove **the** vertex v 1 **of** **the** map C, **the** result**in**g map is not connected.Theorem 4.3.3 If we have a map C such that C = C 1 • C 2 , **the**nτ(C) = τ(C 1 • C 2 ) = τ(C 1 ) × τ(C 2 ).Pro**of**: Each path that connects a vertex **of** C 1 to a vertex **of** C 2 must pass through v 1 .The laplacian matrix associated with a previous map C = C 1 • C 2 is as follows:After delet**in**g **the** row and **the** column **of** **the** vertex v 1 , we obta**in** **the** follow**in**g matrix:⎛⎞M n1 ,n 10⎝⎠0 M n2 ,n 2

70CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSIn calculat**in**g **the** determ**in**ant, we obta**in**:τ(C) = τ(C 1 ) × τ(C 2 ).□Theorem 4.3.4 (Generalization **of** Theorem 4.3.3) Let C be one **of** **the** follow**in**g maps(see Figure 4.4), **the**nn∏τ(C) = τ(C i ).i=1Figure 4.4: Star map and cha**in** map4.3.2 Count**in**g **the** **number** **of** **spann ing**

4.3. Ma**in** Results 71Lemma 4.3.8 Let C be a map and let e = v i v j be an edge **of** C (v i ≠ v j ) **the**n: **the****number** **of** **spann ing**

72CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSPro**of**: Let T be **the** set **of** **spann ing**

4.3. Ma**in** Results 73This recursion shows that **the** function τ is recursively enumerable, but **the** result**in**galgorithm **in** general requires on **the** order **of** 2 |E(C)| arithmetic operations, and so it isnot suitable for large computations. It is a fortune that we have several computationallysimplified and totally generalized formulae for τ(C) which we are go**in**g to derive **the**m **in**this chapter and **in** upcom**in**g chapters.Example 4.3.13 Here is an example **of** a map C with Recursive calculation **of** τ(C)Figure 4.8: Recursive calculation **of** τ(C)We consider f**in**ite undirected simple maps and multimaps with no loops; **the** term multimapis used when multiple edges are allowed **in** a map. For a map C, we denote by V (C)and E(C) **the** vertex set and edge set **of** C, respectively. The multiplicity **of** a vertex-pair(v, u) **of** a map C, denoted by l C (vu), is **the** **number** **of** edges jo**in****in**g **the** vertices v and u**in** C.Remark 4.3.14 If C is a connected loopless map with no cycle **of** length at least 3, **the**nτ(C) is **the** product **of** **the** edge multiplicities. A disconnected map has no **spann ing**

74CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSDef**in**ition 4.3.15 Let C be a map that conta**in**s a simple path p formed by **the** verticesv 1 , v 2 , ..., v k , v k+1 . We denote by C − p **the** map obta**in**ed by delet**in**g **the** simple pathp and **the** map rema**in**s connected. We denote by C.p or C.v 1 v k+1 **the** map obta**in**ed bydelet**in**g **the** simple path p and past**in**g two vertices v 1 and v k+1 .Example 4.3.16 Here is an example **of** maps C, C − p and C.p; see Figure. 4.9.Figure 4.9: An example **of** maps C, C − p and C.pLemma 4.3.17 (Generalization **of** Lemma 4.3.7) If C is a connected planar map, **the**nfor any simple path p from C, **the**re is a **spann ing** tree conta

4.3. Ma**in** Results 75Pro**of**: It is **the** same pro**of** as that **of** Theorem 4.3.11 with |T 1 | = τ(C.p) and |T 2 | =kτ(C − p) s**in**ce **the**re are k ways to cut **the** path (v 1 , v 2 , ..., v k , v k+1 ), **the**n τ(C) =τ(C.p) + kτ(C − p).Theorem 4.3.21 Let C be a map, **the**n|E C |1 ∑τ(C) =τ(C.e i ), where e i ∈ E C .|V C | − 1i=1Pro**of**: Let C a map, T a **spann ing** tree

76CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPS4.3.3 Count**in**g **the** **number** **of** **spann ing**

4.3. Ma**in** Results 77Figure 4.12: An example **of** maps C= C 1 : C 2 , C 1 , C 2 , C 1 .v 1 v 2 and C 2 .v 1 v 2Theorem 4.3.26 [87, 89] Let C= C 1 : C 2 be a map, v 1 and v 2 two vertices **of** **the** map Cwhich is formed by two maps C 1 and C 2 (see Figure 4.10), **the**nτ(C) = τ(C 1 ) × τ(C 2 .v 1 v 2 ) + τ(C 1 .v 1 v 2 ) × τ(C 2 ).Pro**of**: Let T be **the** set **of** **spann ing**

78CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSFigure 4.13: Case **of** Particular maps4.3.4 Count**in**g **the** **number** **of** **spann ing**

4.3. Ma**in** Results 79Theorem 4.3.31 [87, 89] Let C be a map **of** type C= C 1 | C 2 (v 1 and v 2 two vertices **of****the** map C connected by an edge e) (see Figure 4.14), **the**nτ(C) = τ(C 1 ) × τ(C 2 ) − τ(C 1 − e) × τ(C 2 − e).Pro**of**: We transform **the** map C as follows (see Figure 4.16):τ(C 1 | C 2 )= τ((C 1 − e) : C 2 )From Theorem 4.3.26, we haveFigure 4.16: The map C after **the** tranformationτ(C) = τ((C 1 − e).v 1 v 2 ) × τ(C 2 ) + τ(C 1 − e) × τ(C 2 .v 1 v 2 ).From Theorem 4.3.11, we have τ(C 2 .e) = τ(C 2 ) − τ(C 2 − e) and s**in**ce τ((C 1 − e).v 1 v 2 ) =τ(C 1 .e), C 2 .v 1 v 2 = C 2 .e **the**nτ(C) = τ(C 1 .e) × τ(C 2 ) + τ(C 1 − e) × (τ(C 2 ) − τ(C 2 − e))We take τ(C 2 ) as a factor, **the**n= τ(C 1 .e) × τ(C 2 ) + τ(C 1 − e) × τ(C 2 ) − τ(C 1 − e) × τ(C 2 − e).τ(C) = [τ(C 1 .e) + τ(C 1 − e)]τ(C 2 ) − τ(C 1 − e)τ(C 2 − e)We use Theorem 4.3.11, we obta**in** : τ(C) = τ(C 1 )τ(C 2 )−τ(C 1 −e)τ(C 2 −e).□Example 4.3.32 Here is an example **of** a map C **of** type C= C 1 | C 2 with calculation **of**τ(C) (see Figure 4.17):Figure 4.17: An example **of** a map C **of** type C = C 1 | C 2we apply Theorem 4.3.31 **in** C 1 to calculate τ(C 1 ), we **the**n obta**in** τ(C 1 )= 8, (see Figure4.18):

80CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSFigure 4.18: A map C **of** type C= C 1 | C 2 with calculation **of** τ(C)The same goes for C 2 , we obta**in** τ(C 2 )= 8. Now, we calculate τ(C), we aga**in** applyTheorem 4.3.31, we **the**n obta**in** τ(C) = 55, (see Figure 4.19):Figure 4.19: A map C **of** type C= C 1 | C 2 with calculation **of** τ(C)4.3.5 Count**in**g **the** **number** **of** **spann ing**

4.3. Ma**in** Results 81Property 4.3.33 Let C be a map **of** type C = C 1 ‡ C 2 , where ‡ a simple path thatconta**in**s k + 1 vertices and k edges, **the**n- C 1 and C 2 have k + 1 common vertices v 1 , v 2 ,..., v k , v k+1 , k common edges (simple pathp) and a common face (**the** external face).- V C =V C1 +V C2 -(k + 1), E C =E C1 +E C2 -k and F C =F C1 +F C2 -1.Theorem 4.3.34 (Generalization **of** Theorem 4.3.31) Let C be a map, v 1 and v k+1 twovertices **of** **the** map C connected by a simple path p = v 1 , v 2 , ..., v k , v k+1 that conta**in**s kedges; see Figure 4.20, **the**nτ(C) = τ(C 1 ) × τ(C 2 ) − k 2 τ(C 1 − p) × τ(C 2 − p).Pro**of**:From Theorem 4.3.20, and we use Theorems 4.3.3 and 4.3.26, we have:τ(C) = τ(C.p) + kτ(C − p)= τ(C 1 .p)τ(C 2 .p) + k[τ(C 1 .p)τ(C 2 − p) + τ(C 1 − p)τ(C 2 .p)]= [τ(C 1 .p) + kτ(C 1 − p)]τ(C 2 .p) + kτ(C 1 .p)τ(C 2 − p)= τ(C 1 )[τ(C 2 ) − kτ(C 2 − p)] + kτ(C 1 .p)τ(C 2 − p)= τ(C 1 )τ(C 2 ) − kτ(C 1 )τ(C 2 − p) + kτ(C 1 .p)τ(C 2 − p)= τ(C 1 )τ(C 2 ) − k[τ(C 1 ) − τ(C 1 .p)]τ(C 2 − p)= τ(C 1 )τ(C 2 ) − k 2 τ(C 1 − p)τ(C 2 − p), hence **the** result.Example 4.3.35 Here is an example **of** a map C **of** type C= C 1 ‡ C 2 with calculation **of**τ(C) (see Figure 4.21):□Figure 4.21: An example **of** a map C **of** type C = C 1 ‡ C 2we apply Theorem 4.3.34 **in** C 1 to calculate τ(C 1 ), we **the**n obta**in** τ(C 1 ) = 16, (see Figure4.22):

82CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPSFigure 4.22: A map C **of** type C= C 1 ‡ C 2 with calculation **of** τ(C)and, we apply Theorem 4.3.31 **in** C 2 to calculate τ(C 2 ), (see Figure 4.21); we **the**nobta**in** τ(C 2 )= 15. Now, we calculate τ(C), we aga**in** apply Theorem 4.3.34, we **the**nobta**in** τ(C) = 176, (see Figure 4.23):Figure 4.23: A map C **of** type C= C 1 ‡ C 2 with calculation **of** τ(C)In **the** next chapter, we are go**in**g to apply our methods which we have seen **in** this chapteron **some** special planar maps (maps complicated with n vertices) to give **the** **number** **of****spann ing**

Part IIIUse **of** Derived Theoretical Results83

Chapter 5The Number **of** Spann**in**g Trees **of**Certa**in** Families **of** Planar MapsIn this chapter, we are go**in**g to derive several easily computable formulae for certa**in**families **of** planar maps, and similar formulae for related generat**in**g series.5.1 IntroductionThe **number** **of** **spann ing**

CHAPTER 5.86THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSFigure 5.1: Example **of** A map C which has two faces (Cycle)5.3 The case **of** two cyclesTheorem 5.3.1 Let C be **the** follow**in**g map; see Figure 5.2 (It conta**in**s two cycles C 1with h 1 edges and C 2 with h 2 edges, and k is **the** length **of** **the** simple path p), **the**nτ(C) = τ(C 1 ) × τ(C 2 ) − k 2 .Figure 5.2: Case **of** two cyclesPro**of**:From Theorem 4.3.20, we haveτ(C) = τ(C.p) + kτ(C − p) = (h 1 − k)(h 2 − k) + k[(h 1 − k) + (h 2 − k)]= h 1 h 2 − k 2 = τ(C 1 ) × τ(C 2 ) − k 2 .□Particular case:In **the** previous Theorem 5.3.1, if k = 1, **the**n τ(C) = τ(C 1 ) × τ(C 2 ) − 1.Example 5.3.2 Consider **the** follow**in**g simple map (see Figure. 5.3); us**in**g **the** previousTheorem 5.3.1, it is clear that **the** **number** **of** **spann ing**

5.4. The case **of** n cycles (n ≥ 3) 87Figure 5.3: A simple planar map has 23 **spann ing**

CHAPTER 5.88THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSTheorem 5.4.1 For **the** sequence maps C n **in** Figure 5.4, we have:⎧⎨ τ(C n ) = h n τ(C n−1 ) − τ(C n−2 )kn−1, 2 with n ≥ 3(1) τ(C 1 ) = h 1 ,⎩τ(C 2 ) = h 1 h 2 − k12where k n is **the** **number** **of** common edges between **the** i − th and **the** (i + 1) − th cycles**of** C n and h i is **the** **number** **of** edges **of** i − th cycle (h i ≥ 3, for i = 1, ..., n).Pro**of**: τ(C 1 ) = h 1 , τ(C 2 ) = h 1 h 2 − k1, 2 **in** **the** sequence **of** maps C n , we cut **the** last cycle**of** length h n (see Figure 5.4) and we use Theorem 4.3.34, **the**n we obta**in**:τ(C n ) = τ(C n−1 )h n − τ(C n−2 )k 2 n−1 = h n τ(C n−1 ) − τ(C n−2 )k 2 n−1,hence we obta**in** **the** system (1).□5.5 Particular casesIn this section, we apply **the** previous Theorem 5.4.1 to obta**in** **some** particular cases **of**k i and h i .5.5.1 The case **of** k i = k and h i = h (h ≥ 2k + 1)Theorem 5.5.1 In **the** previous Theorem 5.4.1, if we take k i = k for i = 1, ..., n − 1 andh i = h (h ≥ 2k + 1) for i = 1, ..., n, n ≥ 1, we obta**in** **the** sequence **of** maps C n **in** Figure5.5, **the**nFigure 5.5: Case **of** n cycles whose lengths are **the** same with h i = h and k i = kτ(C n ) =(1√h2 − 4k 2( h + √ h 2 − 4k 22) n+1 − ( h − √ h 2 − 4k 2)), n+1 n ≥ 1.2

5.5. Particular cases 89Pro**of**: τ(C 1 ) = h, τ(C 2 ) = h 2 − k 2 , **in** **the** sequence **of** maps C n , we cut **the** last cycle **of**length h (see Fig. 5.5) and we use Theorem 4.3.34, **the**n we obta**in**: τ(C n ) = τ(C n−1 )h −τ(C n−2 )k 2 = hτ(C n−1 ) − τ(C n−2 )k 2 , hence we obta**in** **the** system:⎧⎨⎩τ(C n ) = hτ(C n−1 ) − τ(C n−2 )k 2 , with n ≥ 3τ(C 1 ) = h,τ(C 2 ) = h 2 − k 2The characteristic equation is r 2 − hr + k 2 = 0, ∆ = h 2 − 4k 2 , if h ≥ 2k ⇒ ∆ ≥ 0,**the**refore **the** solutions **of** this equation are: r 1 = h−√ h 2 −4k 2and r2 2 = h+√ h 2 −4k 2, hence:2τ(C n ) = α( h−√ h 2 −4k 2) n + β( h+√ h 2 −4k 2) n , α, β ∈ R, n ≥ 1 Us**in**g **the** **in**itial conditions2 2τ(C 1 ) = h and τ(C 2 ) = h 2 − k 2 , we obta**in**:α = −h2 +h √ √ h 2 −4k 2 +2k 2h 2 −4k 2 (h− √ and β = h2 +h √ √ h 2 −4k 2 −2k 2h 2 −4k 2 ) h 2 −4k 2 (h+ √ , hence **the** result.h 2 −4k 2 )□5.5.2 The case **of** k i = 1 and h i = hTheorem 5.5.2 In **the** previous Theorem 5.5.1, if we take k i = 1 for i = 1, ..., n − 1 andh i = h (h ≥ 3) for i = 1, ..., n, n ≥ 1, we obta**in** **the** sequence **of** maps C n **in** Figure 5.6,Figure 5.6: Case **of** n cycles whose lengths are **the** same with h i = h and k i = 1**the**nτ(C n ) =(1√h2 − 4( h + √ h 2 − 42) n+1 − ( h − √ h 2 − 4)), n+1 n ≥ 1.2Pro**of**: By replac**in**g k i = 1 and h i = h for i = 1, ..., n **in** **the** system (1) **of** Theorem5.4.1, we get :⎧⎨ τ(C n ) = hτ(C n−1 ) − τ(C n−2 )τ(C 1 ) = h⎩τ(C 2 ) = h 2 − 1

CHAPTER 5.90THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSThe characteristic equation is r 2 − hr + 1 = 0, ∆ = h 2 − 4, if h ≥ 2 ⇒ ∆ ≥ 0 **the**refore**the** solutions are: r 1 = h−√ h 2 −4and r2 2 = h+√ h 2 −4, hence:2τ(C n ) = α( h − √ h 2 − 42) n + β( h + √ h 2 − 4) n , α, β ∈ R, and n ≥ 12Us**in**g **the** **in**itial condition τ(C 1 ) = h, τ(C 2 ) = h 2 − 1, we obta**in**:α = (h+√ h 2 −4)(−h 2 +2+h √ h 2 −4)4 √ h 2 −4, β = (h−√ h 2 −4)(h 2 −2+h √ h 2 −4)4 √ , hence **the** result.h 2 −4□Remark 5.5.3 In Theorem 5.5.1, if we replace k = 1, we obta**in** **the** sequence **of** mapsC n shown **in** Figure 5.6, **the**nτ(C n ) =(1√h2 − 4( h + √ h 2 − 425.6 O**the**r values **of** k i = 1 and h i = h) n+1 − ( h − √ h 2 − 4)), n+1 n ≥ 1.2In this section, we fix **the** value **of** k i by 1 and change **the** value **of** h i ; subsequently, wederive a simple formula for **the** **number** **of** **spann ing**

5.6. O**the**r values **of** k i = 1 and h i = h 91Corollary 5.6.1 (The n-Fan cha**in**s) The complexity **of** **the** n-Fan cha**in**s F n is given by**the** follow**in**g formula:τ(F n ) = 1 √5(( 3 + √ 52) n+1 − ( 3 − √ 5)), n+1 n ≥ 1.2• The n-Grid cha**in**s (k = 1 and h = 4)If we take h = 4 **in** **the** sequence **of** maps C n **in** Figure 5.6, we obta**in** **the** sequencen-Grid cha**in**s G n , where n is **the** **number** **of** squares (|V Gn | = 2n + 2) [125, 106];(see Figure 5.8).Figure 5.8: The n-Grid cha**in**s G nCorollary 5.6.2 (The n-Grid cha**in**s) The complexity **of** **the** n-Grid cha**in**s G n (|V Gn | =2n + 2), is given by **the** follow**in**g formula:τ(G n ) = 1 ((22 √ + √ 3) n+1 − (2 − √ )3) n+1 , n ≥ 1.3• The n-Tent cha**in**s (k = 1 and h = 5)If we take h = 5 **in** **the** sequence **of** maps C n **in** Figure 5.6, we obta**in** **the** sequence n-Tentcha**in**s T n (see Figure. 5.9).Figure 5.9: The n-Tent cha**in**s T n

CHAPTER 5.92THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSCorollary 5.6.3 (The n-Tent cha**in**s) The complexity **of** **the** n-Tent cha**in**s T n is givenby **the** follow**in**g formula:τ(T n ) = √ 1 (21( 5 + √ 212• The n-Hexagonal cha**in**s (k = 1 and h = 6)) n+1 − ( 5 − √ 21)), n+1 n ≥ 1.2If we take h = 6 **in** **the** sequence **of** maps C nn-Hexagonal cha**in**s H n (see Figure 5.10).**in** Figure 5.6, we obta**in** **the** sequenceFigure 5.10: The n-Hexagonal cha**in**s H nCorollary 5.6.4 (The n-Hexagonal cha**in**s) The complexity **of** **the** n-Hexagonal cha**in**sH n is given by **the** follow**in**g formula:τ(H n ) = 1 ((34 √ + 2 √ 2) n+1 − (3 − 2 √ )2) n+1 , n ≥ 1.2• The n-Eight cha**in**s (k = 1 and h = 8)If we take h = 8 **in** **the** sequence **of** maps C n **in** Figure 5.6, we obta**in** **the** sequence n-Eightcha**in**s E n (see Figure 5.11).Figure 5.11: The n-Eight cha**in**s E nCorollary 5.6.5 (The n-Eight cha**in**s) The complexity **of** **the** n-Eight cha**in**s E n is givenby **the** follow**in**g formula:τ(E n ) = 1 ((42 √ + √ 15) n+1 − (4 − √ )15) n+1 , n ≥ 1.15

5.7. O**the**r uses 93Numerical results The Table 5.1 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

CHAPTER 5.94THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSFigure 5.12: The n-Home cha**in**s H nτ(H n ) = 11τ(H n−1 ) − 3 × 3τ(H n−2 ), hence we obta**in** **the** system:⎧⎨ τ(H n ) = 11τ(H n−1 ) − 9τ(H n−2 )τ(H 1 ) = 11⎩τ(H 2 ) = 112The characteristic quadratic equation is r 2 − 11r + 9 = 0, so **the** solutions **of** this equationare: r 1 = 11−√ 85and r2 2 = 11+√ 85, hence:2τ(H n ) = α( 11 − √ 852) n + β( 11 + √ 85) n , α, β ∈ R, n ≥ 1.2Us**in**g **the** **in**itial condition τ(H 1 ) = 11, τ(H 2 ) = 112, we obta**in**: α = 1 − 11√ 85,2 170β = 1 + 11√ 85, hence **the** result. □2 170Numerical results The Table 5.2 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

5.7. O**the**r uses 955.7.2 Formula for **the** Number **of** Spann**in**g Trees **in** The n-Barrelcha**in**sIn this section, we derive a simple formula for **the** **number** **of** **spann ing**

CHAPTER 5.96THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSNumerical result The Table 5.3 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

5.8. The **number** **of** **spann ing**

CHAPTER 5.98THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPS5.8.1 Formulae for **the** **number** **of** **spann ing**

5.8. The **number** **of** **spann ing**

CHAPTER 5.100THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSTheorem 5.8.2 (The n-Kite cha**in**s) The complexity **of** **the** n-Kite cha**in**s K n and Q n(see Figure. 5.16) is given by **the** follow**in**g formulae:τ(K n ) = 3 × ((43 6n−14 √ + 30 √ 2)(3 + 2 √ 2) n−1 − (43 − 30 √ 2)(3 − 2 √ )2) n−1 , n ≥ 12andτ(Q n ) =21 × 6n−14 √ 2((3 + 2 √ 2) n − (3 − 2 √ 2) n ), n ≥ 1.Figure 5.16: The n-Kite cha**in**s K nPro**of**: We put K n = τ(K n ) and Q n = τ(Q n ). K 1 = 45, Q 1 = 21, **in** **the** sequence **of**maps K n ; we cut **the** last Kite (see Figure 5.18), and use Theorem 4.3.31 (**the** same goesfor **the** sequence **of** maps Q n ), **the** obta**in**ed is: τ(Q n ) = 21τ(K n−1 ) − 9τ(Q n−1 ), τ(K n ) =45τ(K n−1 ) − 21τ(Q n−1 ) **the**refore, we have **the** follow**in**g system:{Kn = 45K n−1 − 21Q n−1Q n = 21K n−1 − 9Q n−1 with K 1 = 45 and Q 1 = 21(Kn)= MQ n(Kn)= MQ n(Kn−1Q n−1), where M =( 45 -2121 -9(Kn−1Q n−1)= ... = M n−1 (K1Q 1),)

5.8. The **number** **of** **spann ing**

CHAPTER 5.102THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSNumerical results The Table 5.6 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

5.8. The **number** **of** **spann ing**

CHAPTER 5.104THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSNumerical results The Table 5.7 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

5.8. The **number** **of** **spann ing**

CHAPTER 5.106THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSn τ(D n ) τ(Q n )1 23 62 505 1323 11087 28984 243409 636245 5343911 13968306 117322633 306666367 2575754015 6732691628 56549265697 147812549289 1241508091319 32451433925410 27256628743321 7124534208660Table 5.8: Some values **of** τ(D n ) and τ(Q n )

Chapter 6Count**in**g **the** **number** **of** **spann ing**

CHAPTER 6.108COUNTING THE NUMBER OF SPANNING TREES IN THE STARFLOWER PLANAR MAP6.2 The star flower planar mapLet C n be a cycle with n vertices. The Star flower planar map is a simple graph G formedfrom a cycle C n by add**in**g a vertex adjacent to every edge **of** C n and we connect thisvertex with two end vertices **of** each edge **of** C n , i.e., we replace each edge **of** C n by atriangulation. If **the**re are k edges between every two vertices **of** each edge **of** **the** cycleC n , **the**n we obta**in** **the** star flower planar map **in** **the** general case. In this chapter, wedenote **the** star flower planar map by S n,k where n is **the** **number** **of** triangles **of** **the** starflower planar map, k is **the** **number** **of** edges between each two vertices **of** each edge **of** **the**cycle C n ; and derive **the** explicit formula for τ(S n,k ) **the** **number** **of** **spann ing**

6.3. Ma**in** Results 109Figure 6.2: A cycle C n and **the** star flower planar mapIf **the**re are k edges (simple path **in** **the** general case) between every two vertices **of** eachedge **of** **the** cycle C n , **the**n we obta**in** **the** star flower planar map shown **in** Figure. 6.3.In this work, we denote **the** star flower planar map by S n,k where n is **the** **number** **of**triangles **of** **the** star flower planar map, k is **the** **number** **of** edges (**the** length **of** simplepath) between each two vertices **of** each edge **of** **the** cycle C n ; **the** star flower planar maphas n(k + 1) vertices (n vertices **of** degree 4 and n + (k − 1)n **of** degree 2), n(k + 2) edgesand n + 2 faces (n faces **of** degree k + 2, one face **of** degree nk and **the** o**the**r face **of** degree2n)(see Figure. 6.3).Figure 6.3: The star flower planar map S n,kExample 6.3.1 Here is an example **of** a star flower planar map S 2,k with n = 2 (2triangles) and k edges, and a star flower planar map S 3,k with n = 3 (3 triangles) and kedges (see Figure. 6.4).

CHAPTER 6.110COUNTING THE NUMBER OF SPANNING TREES IN THE STARFLOWER PLANAR MAPFigure 6.4: The star flower planar maps S 2,k and S 3,k6.4 An explicit formula for **the** **number** **of** **spann ing**

6.4. An explicit formula for **the** **number** **of** **spann ing**

CHAPTER 6.112COUNTING THE NUMBER OF SPANNING TREES IN THE STARFLOWER PLANAR MAPFigure 6.7: The star flower planar maps S 2,1 and S 3,1Corollary 6.4.3 The **number** **of** **spann ing**

6.4. An explicit formula for **the** **number** **of** **spann ing**

CHAPTER 6.114COUNTING THE NUMBER OF SPANNING TREES IN THE STARFLOWER PLANAR MAPPro**of**: See **the** pro**of** **of** Corollary 6.4.3. □Example 6.4.6 In **the** star flower planar map shown **in** Figure. 6.1; we have n = 8 andk = 2, we apply Corollary 6.4.5 to f**in**d **the** **number** **of** **spann ing**

Chapter 7Maximal Planar MapsIn this chapter, we shall focus on **the** maximal planar maps. This chapter will be divided**in**to two sections; we devote **the** first section for calculat**in**g **the** We**in**er **in**dex **in** **the** case**of** planar maps **in** general **the**n **in** particular **in** **the** maximal planar maps and **in** **the** o**the**rsection will study how to count **the** **number** **of** **spann ing**

116 CHAPTER 7. MAXIMAL PLANAR MAPSFor example, recall **the** utilities graph, **in** which three houses A, B and C are jo**in**edto **the** three utilities gas (g), water (w) and electricity (e). This graph is specifiedcompletely by **the** follow**in**g sets:vertices: {A, B, C, g, w, e},edges: {Ag, Aw, Ae, Bg, Bw, Be, Cg, Cw, Ce},and can be drawn **in** many ways, such as **the** follow**in**g:Figure 7.1: The utilities graph can be drawn **in** many ways.Each **of** **the**se diagrams has six vertices and n**in**e edges, and conveys **the** same **in**formation.Each house is jo**in**ed to each utility, but no two houses are jo**in**ed, and no twoutilities are jo**in**ed. It follows that **the**se two dissimilar diagrams represent **the** same graph.On **the** o**the**r hand, two diagrams may look similar, but represent different graphs. Forexample, **the** diagrams below look similar, but **the**y are not **the** same graph: for example,AB is an edge **of** **the** second graph, but not **the** first.Figure 7.2: Two diagrams look similar, but **the**y are not **the** same graph.We express this similarity by say**in**g that **the** graphs represented by **the**se two diagramsare isomorphic. This means that **the** two graphs have esse1ltially **the** same structure: wecan relabel **the** vertices **in** **the** first graph to get **the** second graph - **in** this case, we simply**in**terchange **the** labels w and B.This leads to **the** follow**in**g def**in**ition.Def**in**ition 7.1.1 Two graphs G and H are isomorphic if H can be obta**in**ed by relabell**in**g**the** vertices **of** G that is, if **the**re is a one-one correspondence between **the** vertices **of** Gand those **of** H, such that **the** **number** **of** edges jo**in****in**g each pair **of** vertices **in** G is equalto **the** **number** **of** edges jo**in****in**g **the** correspond**in**g pair **of** vertices **in** H. Such a one-onecorrespondence is an isomorphism.

7.1. Introdution 117Example 7.1.2 For example, **the** graphs G and H represented by **the** diagramsFigure 7.3: Two graphs G and H are not **the** same, but **the**y are isomorphic.are not **the** same, but **the**y are isomorphic, s**in**ce we can relabel **the** vertices **in** **the** graphG to get **the** graph H, us**in**g **the** follow**in**g one-one correspondence:G ↔ H, u ↔ 4, v ↔ 3, w ↔ 2 and x ↔ 1Note that edges **in** G correspond to edges **in** H, for example: **the** two edges jo**in****in**g u andv **in** G correspond to **the** two edges jo**in****in**g 4 and 3 **in** H; **the** edge uw **in** G correspondsto **the** edge 42 **in** H; **the** loop ww **in** G corresponds to **the** loop 22 **in** H.Remark 7.1.3 To check whe**the**r two graphs are **the** same, we must check whe**the**r all **the**vertex labels correspond. However, to check whe**the**r two graphs are isomorphic, we must**in**vestigate whe**the**r we can relabel **the** vertices **of** one graph to give those **of** **the** o**the**r.In order to do this, we first check that **the** graphs have **the** same **number**s **of** vertices andedges, and **the**n look for special features **in** **the** two graphs, such as a loop, multiple edges,or **the** **number** **of** edges meet**in**g at a vertex. For example, **the** follow**in**g two graphs bothhave five vertices and six edges, but are not isomorphic, as **the** first has two vertices wherejust two edges meet, whereas **the** second has only one.Figure 7.4: Two graphs both have five vertices and six edges, but are not isomorphic.Example 7.1.4 The graphs G and H, represented **in** **the** follow**in**g Figure 7.5 are not**the** same, but **the**y are isomorphic.The graphs G and H represent **the** maximal planar map with 5 vertices. Hereafter, wedenote **the** maximal planar map with n vertices by E n .

118 CHAPTER 7. MAXIMAL PLANAR MAPSFigure 7.5: The graphs G and H are not **the** same, however, **the**y are isomorphic.7.2 Calculat**in**g **the** Wiener **in**dex **in** **the** maximal planarmapsIn this section, we shall focus on **the** Wiener **in**dex **in** case **of** planar maps and give aformula which calculates **the** Wiener **in**dex **in** **the** maximal planar map, **the**n we give an**in**equality, which m**in**imizes and maximizes any planar map by **the** maximal planar mapwith n vertices and **the** path **of** n vertices; like **the** **in**equality **in** **the** **trees**.7.2.1 IntrodutionIn a communication network, large diameter may be acceptable if most pairs can communicatevia short paths. This leads us to study **the** average distance **in**stead **of** **the**maximum. S**in**ce **the** average is **the** sum divided by ( n2)(**the** **number** **of** vertex pairs), itis equivalent to study D(G) = ∑ {v i ,v j }⊆V (G) d(v i, v j ). The sum D(G) has been called **the**Wiener **in**dex **of** G (also written W (G)). It is used by Wiener to study **the** boil**in**g po**in**t**of** paraff**in**. Molecules can be modeled by graphs with vertices for atoms and edges foratomic bonds. Many chemical properties **of** molecules are related to **the** Wiener **in**dex**of** **the** correspond**in**g graphs. We study **the** extreme values **of** W (G). In **the** next, wefocus on **the** Wiener **in**dex **in** planar maps, **the** particular case **of** **trees** has been echoedby several people [39, 69, 125]. We study **the** extreme values **of** W (G). Refer to [2] and[117] for **the** significance **of** **the** Wiener **in**dex.7.2.2 Calculation **of** **the** Wiener **in**dex **in** **the** planar mapsHere**in**, we give **some** basic def**in**itions and properties about **the** Wiener **in**dex. An importantconcept that we need **in** this section is that **of** **the** Wiener **in**dex **in** **the** case **of** planarmaps.Def**in**ition 7.2.1 The distance between two dist**in**ct vertices v i and v j **of** a map C, denotedby d(v i , v j ) is equal to **the** length **of** (**number** **of** edges **in**) **the** shortest path thatconnects v i and v j . Conventionally, d(v i , v i ) = 0.

7.2. Calculat**in**g **the** Wiener **in**dex **in** **the** maximal planar maps 119Def**in**ition 7.2.2 We def**in**e a complete vertex **in** a planar map C by **the** vertex v 0 suchthat d(u, v 0 ) = 1 for each u ∈ V (C). In a complete graph, all **the** vertices are complete.The Wiener **in**dex **of** a connected graph is **the** sum **of** distances between all pairs **of**∑vertices [39, 87, 69, 125], **the** Wiener **in**dex **of** a connected graph G is def**in**ed as W (G) ={v i ,v j }⊆V (G) d(v i, v j ). The Wiener **in**dex **of** a vertex u **in** G denoted by W (u, G), is **the** sum**of** distances **of** vertex u to each vertex **of** vertices **of** G, i.e., W (u, G) = ∑ v∈V (G)d(u, v).Def**in**ition 7.2.3 In **the** same way as graphs, we def**in**e **the** Wiener **in**dex for planar mapsas follows:∑W (C) = d(v i , v j ) and W (u, C) = ∑d(u, v){v i ,v j }⊆V (G)Remark 7.2.4 We notice that:W (C) = 1 2= 1 2∑∑u∈V (C) v∈V (C)∑u∈V (C)W (u, C)d(u, v)v∈V (C)Example 7.2.5 Let C 5 be a planar map with |V (C 5 )| = 5 (see Figure 7.6), we have:deg(v 1 ) = 2, deg(v 2 ) = 4, deg(v 4 ) = 4, W (v 1 , C 5 ) = 6, W (v 2 , C 5 ) = 4, W (v 3 , C 5 ) = 6,W (v 4 , C 5 ) = 4, W (v 5 , C 5 ) = 6, W (C 5 ) = 13, v 2 and v 4 are complete vertices.Figure 7.6: Example **of** a map C 5Let C be a planar map and e be an edge **in** C (e ∈ E(C)), we denote by C − e **the** mapobta**in**ed after delet**in**g **the** edge e from **the** map C and **the** result**in**g map is connected.Lemma 7.2.6 Let C be a planar map and let e 1 , e 2 be two edges **in** C that connect **the**vertices v 1 and v 2 (multiple edges **in** C), **the**nW (C − e i ) = W (C), i = 1, 2where C − e i is **the** map obta**in**ed by delet**in**g **the** edge e i from **the** map C.

120 CHAPTER 7. MAXIMAL PLANAR MAPSRemark 7.2.7 We notice that delet**in**g one edge **of** multiple edges does not affect **the**Wiener **in**dex; through this section, we consider only **the** simple planar maps, i.e., mapswithout loops and multiple edges.Lemma 7.2.8 Let C n be a simple planar map with n vertices and v be a complete vertex**in** **the** map C, **the**ndeg(v) = n − 1 and W (v, C n ) = n − 1.Lemma 7.2.9 Let C n be a simple planar map with n vertices (n ≥ 2) and let v be avertex not complete **of** C n , **the**nW (v, C n ) ≥ n.Remark 7.2.101. Let C n be a simple planar map and v be a vertex **of** C n , **the**n W (v, C n ) ≥ n − 1.2. Let C n be a simple planar map with n vertices, e be an edge **of** C n and let C n − e be**the** map obta**in**ed by delet**in**g **the** edge e such that **the** map C n −e rema**in**s connected,**the**n W (C n − e) ≥ W (C n ).Theorem 7.2.11 Let C n be a simple planar map with n vertices, **the**nW (C n ) ≥n(n − 1).2Pro**of**: Let C n be a planar map with n vertices and let v be a vertex **of** C n . By Remark7.2.10, we have: W (v, C n ) ≥ n − 1;W (C n ) = 1 2≥ 1 2≥∑v∈V (C n )∑v∈V (C n )n(n − 1).2W (v, C n )(n − 1) = 1 2 (n − 1) ∑v∈V (C n )Def**in**ition 7.2.12 (A maximal planar map [39]) Let E n be a family **of** planar maps thatconta**in**s:• n vertices, two complete vertices **of** degree n − 1, two vertices **of** degree 3 and n − 4vertices **of** degree 4,• 2(n − 2) faces **of** degree 3 (all faces hav**in**g degree 3),• 3(n − 2) edges,1□

7.2. Calculat**in**g **the** Wiener **in**dex **in** **the** maximal planar maps 121**the**n **the** family **of** this maps is called **the** maximal planar maps if to which no new edgecan be added without violat**in**g **the** planarity **of** this maps.The maps E 3 and E 4 are presented **in** **the** example 7.2.13. For E 5 , E 6 , and E n (see Figure.7.7)Figure 7.7: The maps E 5 , E 6 and E nExample 7.2.13 In **the** planar maps E 3 and E 4 , all **the** vertices are completes. In o**the**rwords, we say that **the** map is complete (see Figure 7.8).Figure 7.8: The maps E 3 and E 4Remark 7.2.14 For all u, v ∈ V (E n ), we have d(u, v) ≤ 2.Proposition 7.2.15 Let E n be **the** maximal planar map and v be a vertex **of** E n , **the**nwe have:1. W (v, E n ) = 2n − deg(v) − 2, 2. W (E n ) = (n − 2) 2 + 2.

122 CHAPTER 7. MAXIMAL PLANAR MAPSPro**of**:1. W (v, E n ) = ∑u∈V (E n)= ∑u∈V (En)d(u,v)=1d(u, v) (we use Remark 7.2.14)d(u, v) += deg(v) + 2 ∑u∈V (En)d(u,v)=2∑u∈V (En)d(u,v)=21d(u, v)= deg(v) + 2(n − deg(v) − 1)= 2n − deg(v) − 22. W (E n ) = 1 2= 1 2∑W (v, E n )v∈V (E n )∑v∈V (E n )= (n − 1)(2n − deg(v) − 2) (from 1)∑1 − 1 2v∈V (E n)∑deg(v)v∈V (E n)= n(n − 1) − 1 2 × 2|E(E n)|= (n − 2) 2 + 2□Lemma 7.2.16 Let C n be a simple planar map with n vertices (n ≥ 2) and let v be avertex **of** C n , **the**nW (v, C n ) ≥ 2n − deg(v) − 2.Pro**of**:W (v, C n ) = ∑u∈V (C n )= ∑u∈V (Cn)d(u,v)=1d(u, v)d(u, v) +∑u∈V (Cn)d(u,v)≥2d(u, v)≥ deg(v) + 2(n − deg(v) − 1)≥ 2n − deg(v) − 2□Let T n be a tree with n vertices, **the**n **the** Wiener **in**dex W (T n ) = ∑ u,vd(u, v) is m**in**imizedby star tree with n vertices and maximized by path with n vertices, both uniquely, i.e.,

7.3. Formulae for **the** Number **of** Spann**in**g Trees **in** a Maximal Planar Map 123(n − 1) 2 ≤ W (T n ) ≤ n(n2 −1)6[117]. The goal **of** this work is to give an **in**equality similar **in****the** case **of** planar maps. Let C n be a planar map with n vertices, **the**n **the** Wiener **in**dex**of** W (C n ) is m**in**imized by **the** maximal planar map W (E n ) with n vertices and maximizedby **the** path W (P n ) with n vertices. Now we can state **the** follow**in**g **the**orem.Theorem 7.2.17 Let C n be a simple planar map with n vertices, **the**nW (E n ) ≤ W (C n ) ≤ W (P n )Pro**of**: By Remark 7.2.10, In each deleted edge **of** C, we expand **the** Wiener **in**dex. Theconnected planar map obta**in**ed after delet**in**g all possible edges is a **spann ing** tree

124 CHAPTER 7. MAXIMAL PLANAR MAPS7.3.1 Ma**in** ResultsIt is known that Kirchh**of**f Matrix Tree Theorem [81, 85], can be applied to any map Cto determ**in**e τ(C) by tak**in**g a determ**in**ant **of** Laplacian matrix **of** C, but this requiresevaluat**in**g a determ**in**ant **of** a correspond**in**g characteristic matrix. However, for a fewspecial families **of** maps, **the**re exist simple formulae which make it much easier to calculateand determ**in**e **the** **number** **of** correspond**in**g **spann ing**

7.3. Formulae for **the** Number **of** Spann**in**g Trees **in** a Maximal Planar Map 125n − 1 −1 −1 −1 −1 −1 . . . −1 −1−1 n − 1 −1 −1 −1 −1 . . . −1 −1−1 −1 3 −1 0 0 . . . 0 0−1 −1 −1 4 −1 0 . . . 0 0τ(E n ) =−1 −1 0 −1 4 −1 . . . 0 0−1 −1 0 0 −1 4 . . . 0 0. . . . . . . .. . .−1 −1 0 0 0 0 . . . 4 −1∣ −1 −1 0 0 0 0 . . . −1 4 ∣we denote r i by **the** i-th row and c i by **the** i-th column **of** **the** determ**in**ant. In previousdeterm**in**ant, we replace c 1 by c 1 + c 2 + ... + c n−1 , i.e., we add to **the** first column **the** sum**of** o**the**r (transformation is symbolized as follows: c 1 ← ∑ n−1i=1 c i, ; this does not change**the** determ**in**ant, **the**n we obta**in**:1 −1 −1 −1 −1 −1 . . . −1 −11 n − 1 −1 −1 −1 −1 . . . −1 −10 −1 3 −1 0 0 . . . 0 00 −1 −1 4 −1 0 . . . 0 0τ(E n ) =0 −1 0 −1 4 −1 . . . 0 00 −1 0 0 −1 4 . . . 0 0. . . . . ... . . .0 −1 0 0 0 0 . . . 4 −1∣1 −1 0 0 0 0 . . . −1 4 ∣Next, we replace c j by c 1 + c j for j = 2, ..., n − 1, i.e., c j ← c 1 + c j , we obta**in**:1 0 0 0 0 0 . . . 0 01 n 0 0 0 0 . . . 0 00 −1 3 −1 0 0 . . . 0 00 −1 −1 4 −1 0 . . . 0 0τ(E n ) =0 −1 0 −1 4 −1 . . . 0 00 −1 0 0 −1 4 . . . 0 0. . . . . . . .. . .0 −1 0 0 0 0 . . . 4 −1∣1 0 1 1 1 1 . . . 0 5 ∣Expand**in**g L n−1 along **the** first row we obta**in** **the** determ**in**ant **of** order (n − 2) × (n − 2)

126 CHAPTER 7. MAXIMAL PLANAR MAPSas follows:n 0 0 0 0 . . . 0 0−1 3 −1 0 0 . . . 0 0−1 −1 4 −1 0 . . . 0 0−1 0 −1 4 −1 . . . 0 0τ(E n ) =−1 0 0 −1 4 . . . 0 0. . . . . . .. . .−1 0 0 0 0 . . . 4 −1∣ 0 1 1 1 1 . . . 0 5 ∣and expand**in**g **the** determ**in**ant obta**in**ed along **the** first row we obta**in** **the** determ**in**ant**of** order (n − 3) × (n − 3) as follows:3 −1 0 0 . . . 0 0−1 4 −1 0 . . . 0 00 −1 4 −1 . . . 0 0τ(E n ) = n0 0 −1 4 . . . 0 0. . . . . .. . .0 0 0 0 . . . 4 −1∣ 1 1 1 1 . . . 0 5 ∣hence **the** result.□Lemma 7.3.2 Let E n be a maximal planar map with n vertices, **the**nτ(E n )n= 4τ(E n−1)n − 1− τ(E n−2)n − 2 , n ≥ 5.Pro**of**: By Lemma 7.3.1, we have **the** determ**in**ant **of** order (n − 3) × (n − 3):τ(E n )n3 −1 0 0 . . . 0 0−1 4 −1 0 . . . 0 00 −1 4 −1 . . . 0 0=0 0 −1 4 . . . 0 0. . . . . .. . .0 0 0 0 . . . 4 −1∣ 1 1 1 1 . . . 0 5 ∣In **the** previous determ**in**ant, we denote by c i by **the** i-th column and r i by **the** i-th row **of****the** determ**in**ant, c i ← c i −c i+1 for i = 1, ..., n−4, we obta**in** **the** determ**in**ant as follows:

7.3. Formulae for **the** Number **of** Spann**in**g Trees **in** a Maximal Planar Map 127τ(E n )n4 −1 0 0 . . . 0 0−5 5 −1 0 . . . 0 01 −5 5 −1 . . . 0 0=0 1 −5 5 . . . 0 0. . . . . . . . .0 0 0 0 . . . 5 −1∣ 0 0 0 0 . . . −5 5 ∣In **the** end, r i ← ∑ ij=1 r j for i = 2, ..., n − 3, we obta**in** **the** determ**in**ant as follows:τ(E n )n4 −1 0 0 . . . 0 0−1 4 −1 0 . . . 0 00 −1 4 −1 . . . 0 0=0 0 −1 4 . . . 0 0. . . . . .. . .0 0 0 0 . . . 4 −1∣ 0 0 0 0 . . . −1 4 ∣**the**n τ(En)n= 4∆ n−1 − ∆ n−2 , where ∆ i = det(D i ) (∆ i = 4∆ i−1 − ∆ i−2 because ∆ iis tri-diagonal matrix), and D i is def**in**ed as follow**in**g:⎡⎤4 −1−1 4 −1−1 4 −1D i =−1 4 −1. .. . .. . ..⎢⎥⎣−1 4 −1⎦−1 4i×iWith **the** same technique, we obta**in**:result.τ(E n−1 )n−1= ∆ n−1 and τ(E n−2)n−2= ∆ n−2 , hence **the**□7.3.2 An explicit formula for **the** **number** **of** **spann ing**

128 CHAPTER 7. MAXIMAL PLANAR MAPSTheorem 7.3.3 (A maximal planar map) The complexity **of** **the** maximal planar mapE n (see Figure. 7.7) is given by **the** follow**in**g formula:τ(E n ) =n ((22 √ + √ 3) n−2 − (2 − √ )3) n−2 , n ≥ 3.3Pro**of**: τ(E 3 ) = 3, τ(E 4 ) = 16, τ(E 5 ) = 75, τ(E n)nhence we obta**in** **the** system:⎧⎨⎩τ(E n )n= 4τ(E n−1)n−1τ(E 3 ) = 3τ(E 4 ) = 16= 4τ(E n−1)n−1− τ(E n−2)n−2 , with− τ(E n−2)n−2(by Lemma 7.3.2),Then we get a sequence such that u n = 4u n−1 − u n−2 , n ≥ 3, **the**refore **the** characteristicquadratic equation is r 2 − 4r + 1 = 0, so **the** solutions **of** this equation are: r 1 = 2 − √ 3and r 2 = 2 + √ 3, hence:τ(E n ) = α(2 − √ 3) n + β(2 + √ 3) n , α, β ∈ R, n ≥ 1.Us**in**g **the** **in**itial conditions τ(E 3)3= 1 and τ(E 4)4= 4, we obta**in**:α = −2− 7 6√3, β = −2+76√3, hence **the** result. □Example 7.3.4 The follow**in**g figure (see Figure. 7.10), illustrates a maximal planar mapwith 4 vertices E 4 and its all **spann ing**

7.3. Formulae for **the** Number **of** Spann**in**g Trees **in** a Maximal Planar Map 129Numerical results The Table 7.1 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

130 CHAPTER 7. MAXIMAL PLANAR MAPSPro**of**: We put f n = τ(F n ) and g n = τ(G n ). f 2 = 2, g 2 = 1, **in** **the** map F n , we cut **the**first cycle (see Figure. 7.11), and we use Theorem 4.3.31 (**the** same goes for **the** mapG n ), **the**n we obta**in**: τ(G n ) = 3τ(F n−1 ) − τ(G n−1 ), τ(F n ) = 2τ(G n ) − τ(F n−1 ) **the**refore,we have **the** follow**in**g system:{fn = 2g n − f n−1g n = 3f n−1 − g n−1 with f 2 = 2 and g 2 = 1,we replace by **the** value **of** g n **in** **the** first equation, we get:{fn = 5f n−1 − 2g n−1g n = 3f n−1 − g n−1 with f 2 = 2 and g 2 = 1(fn)= Mg n)= Mg n(fn(fn−1), where M =g n−1)g n−1(fn−1( 5 -23 -1= ... = M n−2 (f2g 2),**the**n, we compute M n−2 :det (M − λI 2 ) = λ 2 −4λ + 1 = 0, λ 1 = 2 − √ 3 and λ 2 = 2 + √ 3, λ 1 ≠ λ 2 **the**n **the**re isP **in**vertible such that M = P DP −1 where( )λ1 0D =,0 λ 2P is **the** transformation matrix formed by eigenvectors( 1 1P =M n−2 = P D n−2 P −1 whereD n−2 =3+ √ 32(P −1 = √ −133− √ 33− √ 322-1−3− √ 312),)( √ )(2 − 3)n−200 (2 + √ 3) n−2 ,from which we obta**in** M n−2 , **the**n(fng n)= M n−2 (f2g 2),,),hence **the** result.□

7.3. Formulae for **the** Number **of** Spann**in**g Trees **in** a Maximal Planar Map 131Numerical results The Table 7.2 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

132 CHAPTER 7. MAXIMAL PLANAR MAPSFigure 7.13: The maps E n , E n − e and E n .eNumerical results The Table 7.3 illustrates **some** **of** **the** values **of** **the** **number** **of** **spann ing**

ConclusionThis **the**sis presents our contribution **in** **the** research area **of** calculat**in**g **the** **number** **of****spann ing**

134 CHAPTER 7. MAXIMAL PLANAR MAPSIn future, we plan to work on **the** planar maps **of** type C= C 1 ‡ C 2 with ‡ is notat all a simple path, and shall try to generalize **the** **the**orem deletion - contraction bywork**in**g on deletion more than one edge, i.e., **the** **the**orem deletion - contraction workedon one edge but we shall work on n edges and associate it with our results **in** this **the**sis.Keywords graphs, maps, **trees**, **spann ing**

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Auteur :Titre :Directeurs de thèse :Abdulhafidh MODABISHÉnumération de nombre d’arbres couvrants dans certa**in**escartes planaires spéciales.Pr. Mohamed EL MARRAKIRésuméLe nombre d’arbres couvrants dans une carte planaire - graphe plongé dans une surfacesans croisement d’arêtes - (réseau) est un important bien étudié la quantité et **in**variantdu graphe (réseau); de plus c’est aussi une mesure importante de la fiabilité d’un réseauqui joue un rôle central dans la théorie classique de Kirchh**of**f des réseaux électriques.Dans un graphe (réseau) qui contient plusieurs cycles, il faut supprimer les redondancesdans ce réseau, i.e., on obtient un arbre couvrant. Un arbre couvrant dans une carte Cest un arbre qui a le même ensemble de sommets en tant que C (arbre qui passe par tousles sommets de la carte C).Notre thème de recherche dans cette thèse se concentre sur le calcul du nombred’arbres couvrants dans les cartes planaires connexes, un sujet dans la théorie des graphescomb**in**atoire; a**in**si que, pour trouver de nouvelles méthodes pour calculer le nombred’arbres couvrants dans une carte planaire (réseau).Arbres couvrants sont pert**in**ents pour les différents aspects de graphes (réseaux). Engénéral, le nombre d’arbres couvrants dans un réseau peut être obtenu par le calcul ledéterm**in**ant de la matrice laplacien liée ou le calcul du spectre de Laplace du réseau.Cependant, pour une grande carte (réseau), l’évaluation du déterm**in**ant pert**in**ent est uncalcul difficile. Dans ce travail, nous fournissons de nouvelles méthodes pour faciliter lecalcul du nombre d’arbres couvrants dans les cartes planaires et de prouver de nouveauxrésultats simplifiée et généralisée. Enf**in**, nous appliquons ces méthodes sur certa**in**escartes planaires à f**in** de dériver plusieurs formules explicites pour calculer le nombred’arbres couvrants dans certa**in**es familles particulières des cartes planaires.Mots-clés : graphes, cartes, arbres, arbres couvrants, complexité, laplacien matrice,théorème de Kirchh**of**f, les chaînes de n-Fan, les chaînes de n-Grille, fleur d’étoile carteplanaire, maximale carte planaire, **in**dice de Wiener.

ENUMERATION OF THE NUMBER OF SPANNING TREESIN SOME SPECIAL PLANAR MAPSAbstractThe **number** **of** **spann ing**