enumeration of the number of spanning trees in some ... - Toubkal

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16 LIST OF TABLESand in random walks in graphs [114]. The research of the number of spanning trees in agraph has a long history. The cornerstone of the research, the Matrix Tree Theorem, datedback to 1847, which is attributed to Kirchhoff. The Matrix Tree Theorem is a famousand classic result on the study of τ(G). It is known that Kirchhoff Matrix Tree Theorem[81, 85], can be applied to any map C to determine τ(C) by taking the determinant ofLaplacian matrix L of C, i.e., all cofactors of L are equal, and their common value is τ(C),but this requires evaluation of a determinant of a corresponding characteristic matrix.However, for a few special families of graphs there exist simple formulae which makeit much easier to calculate and determine the number of corresponding spanning treesespecially when these numbers are very large. One of the first such results is due toCayley [25] who showed that complete graph on n vertices, K n , has n n−2 spanning trees[65], that is he showed τ(K n ) = n n−2 for n ≥ 2. Another combinatorial method ofcounting the number of spanning trees in a graph G is the Feussner’s recursive formula[49], for counting τ(G) in a graph G, is quite intuitive. For an undirected simple graph G,let e be any edge of G. All spanning trees in G can be separated into two parts: one partcontains all spanning trees without e as a tree edge; another contains all spanning treeswith e as a tree edge. The first part has the same number of spanning trees as graph G 1 ,where G 1 is the graph G with e deleted; the second part has the same number of spanningtrees as graph G 2 , where G 2 is the graph created from G by shrinking the two vertices of einto one vertex. Both G 1 and G 2 have fewer edges than G, so the number of spanning treesin G can be counted recursively in this way. It is then clear that τ(G) = τ(G 1 ) + τ(G 2 ).In this thesis, we have generalized this formula by replacing the edge e by a simple pathp that contains k edges, see [89]. Let G be a graph with multiple edges and self-loops.When we count τ(G), we first neglect all self-loops in G because they have no contributionto any spanning tree. If G itself is a tree then τ(G) = 1, and if G is disconnected, thenτ(G) = 0.Our research theme in this thesis focuses on the counting of the number of spanningtrees in connected planar maps, a subject in combinatorial graph theory; and to find newmethods to calculate the number of spanning trees in any map (network). Most researchabout the number of spanning trees is devoted to determining exact formulae for thenumber of spanning trees in many kinds of special graphs, see [6, 14, 62, 63, 65, 66, 127].In this thesis, we start by stating the general methods for counting the number ofspanning trees in graphs; we then provide our new results.Spanning trees are relevant to various aspects of graphs (networks). Generally, thenumber of spanning trees in a network can be obtained by computing a related determinantof the Laplacian matrix L of the network. However, for a large map (network),evaluating the relevant determinant is computationally intractable. In this thesis, we givenew methods to facilitate the calculation of the number of spanning trees in planar mapsand prove novel simplified results. Then, we apply these methods on certain planar mapsto derive several explicit simple formulae for calculating the number of spanning treesin some special families of planar maps which are called the n-Fan chains, the n- Gridchains, the n-tent chains, the n-Hexagonal chains, the n-Eight chains, the n-Home chains,
LIST OF TABLES 17the n-Barrel chains, the n-Light chains, the n-Kite chains, the n-Envelope chains and then-Diphenylene, ... etc.Outline of the thesisThis thesis is organized as follows. In the first chapter, some general graph theoryis described. In the second chapter, we introduce the background and research historyof our problem and definitions and general properties of basic objects studied (spanningtrees, complexity, matrices associated to a graph, ... etc). Some basic results are alsointroduced. The third chapter introduces the background and research of the calculationof the number of spanning trees in graphs and some methods for counting spanning treesare introduced. In this chapter, we firstly state the general methods for counting thenumber of spanning trees in graphs, and then our new results are discussed. In the fourthchapter, we provide new non-trivial methods for calculating the number of spanningtrees in planar maps in general then in particular in some special connected planarmaps and prove new simplified results, we then apply these methods on some specialsplanar maps to give explicit simple formulae to calculate the number of spanning treesin certain families of planar maps in chapter five. In the sixth chapter, we consider thestar flower planar map, and derive a simple explicit formulae for counting the numberof spanning tree in it. in this chapter we use one of the methods which mentioned inthe fourth chapter to find the number of spanning tree in the star flower planar map.Finally in the last chapter, we are going to focus on the maximal planar maps. Thischapter will be divided into two sections; we devote the first section for calculating theWeiner index (the sum of distances between all pairs of vertices) in the case of planarmaps in general then in particular in the maximal planar maps and in the other sectionshall study how to count the number of spanning trees in this type of this maps, as wellas enumeration of spanning trees in the maximal planar maps. For that, we develop amethod, by using the Matrix Tree Theorem as the base and manipulating the matrices,to prove that the number of spanning trees in a maximal planar map which satisfies arecurrence relation. This recurrence relation can be determined exactly, i.e., we reach ata formula to calculate the number of spanning trees in the maximal planar maps. In theend, some possible future works are proposed.New resultsThe new results obtained in the framework of this thesis are concentrated in Chapters 4,5, 6 and 7.The results of Chapter 4 have been partially published in [89]. This paper is thecornerstone of the research subject in this thesis, as it includes new methods to calculatethe number of spanning trees in connected planar maps.The results of Chapter 5 are published in the articles [85, 87, 88, 89]. In these articles,we have applied our new methods that we have reached in the article [89] on certain
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Page 6 and 7: 4my wife who has been very supporti
Page 8 and 9: 6 CONTENTSII Theoretical Part 533 H
Page 10 and 11: 8 CONTENTS
Page 14 and 15: 12 LIST OF FIGURES7.11 The maps F n
Page 16: 14 LIST OF TABLES
Page 21: Part IPreliminaries19
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Page 44 and 45: 42 CHAPTER 2. SPANNING TREESFigure
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Page 48 and 49: 46 CHAPTER 2. SPANNING TREESExample
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Page 57 and 58: Chapter 3How to count the number of
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3.4. Counting the number of spannin
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Chapter 4New methods to compute the
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4.3. Main Results 69one vertex (any
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4.3. Main Results 71Lemma 4.3.8 Let
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4.3. Main Results 73This recursion
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4.3. Main Results 75Proof: It is th
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4.3. Main Results 77Figure 4.12: An
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4.3. Main Results 79Theorem 4.3.31
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4.3. Main Results 81Property 4.3.33
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Part IIIUse of Derived Theoretical
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CHAPTER 5.86THE NUMBER OF SPANNING
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CHAPTER 6.108COUNTING THE NUMBER OF
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116 CHAPTER 7. MAXIMAL PLANAR MAPSF
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120 CHAPTER 7. MAXIMAL PLANAR MAPSR
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134 CHAPTER 7. MAXIMAL PLANAR MAPSI
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136 BIBLIOGRAPHY[13] N. Biggs, E. L
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138 BIBLIOGRAPHY[46] P. Erdös, and
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140 BIBLIOGRAPHY[77] D. Lotfi, M. E
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142 BIBLIOGRAPHY[107] R. Shrock and
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Auteur :Titre :Directeurs de thèse
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