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

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 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 joining u andv in G correspond to the two edges joining 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 whether two graphs are the same, we must check whether all thevertex labels correspond. However, to check whether two graphs are isomorphic, we mustinvestigate whether we can relabel the vertices of one graph to give those of the other.In order to do this, we first check that the graphs have the same numbers of vertices andedges, and then look for special features in the two graphs, such as a loop, multiple edges,or the number of edges meeting at a vertex. For example, the following 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 following Figure 7.5 are notthe same, but they 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 .