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

1.4. Distance in trees and graphs 39Definition 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 distinct (unless the walk is a path). In this case we say that the walk isfrom v 1 to v n of length n − 1.Definition 1.4.2 (Distance) The distance between two distinct vertices v i and v j ofa graph G, denoted by d(v i , v j ) is equal to the length of the shortest path (number ofedges 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, then 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, then d(u, v) = ∞ [87].The distance between two vertices v and u, written d(v, u), is the length of theshortest walk from v to u, if it exists, otherwise let d(v, u) = ∞. Note that the shortestwalk is necessarily a path. Furthermore, in a weighted graph, d(v, u) is understood to bethe minimum total weights of all possible walks from v to u.Definition 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.Definition 1.4.4 (Diameter) The diameter of a graph G is defined as the maximum ofthe set of all shortest walks joining 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 isthe 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 another, there is a path of length two connectingthe two. By looking at the graph, it can be seen that this is true.Figure 1.24: An example of a graph G whose diameter is 2