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

1.1. Graphs 29Two graphs G and H are isomorphic if H can be obtained by relabelling the verticesof G that is, if there is a one-one correspondence between the vertices of G and those ofH, such that the number of edges joining each pair of vertices in G is equal to the numberof edges joining the corresponding 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 they are isomorphic.are not the same, but they are isomorphic, since we can relabel the vertices in the graphG to get the graph H, using the following 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 joiningu and v in G correspond to the two edges joining 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 then |V G | = |V H |, |E G | = |E H |, and their degreesequences must be identical. However none of these 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 they are not isomorphic.Figure 1.14: Two graphs G and H are not isomorphic.