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

Chapter 1Definitions and PropertiesThis chapter includes a brief introduction to graph theory. Some of the basics concerningthis theory are presented in [19] and [119]. In case of rare terminologies, the reader isreferred to consult [69] and [117].1.1 GraphsIn this section, we are going to present some useful definitions related to our work asfollows. A graph will usually be denoted by a capital letter G.Definition 1.1.1 An undirected graph G is a triplet (V G ,E G ,δ) where V G is the set ofvertices of the graph G, E G is the set of edges of the graph G (E G ⊆ V 2 , meaning 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 distinct). When u and vare the endpoints of an edge, they 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 endpoints are equal), if δ(e i ) = δ(e j ) with i ≠ j then theedges e i and e j are called multiple edges (edges having the same pair of endpoints).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 }, theedges e 3 and e 4 are multiple edges, the edge e 1 is a loop, then 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 finite if its vertex set and edge set are finite.21