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

List of Figures1.1 An example of graph G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2 Some examples of graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3 Example of graph with adjacent vertices . . . . . . . . . . . . . . . . . . . 231.4 An example of vertex degrees . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 An example of vertex degrees . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 An example of complete graph K 5 . . . . . . . . . . . . . . . . . . . . . . . 251.7 From left to right, the graphs K 4 , K 2,2 , P 4 , C 4 . . . . . . . . . . . . . . . . . 261.8 A graph G and subgraphs of G. . . . . . . . . . . . . . . . . . . . . . . . . 261.9 Some r-regular graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.10 Complete Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.11 Null Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.12 Cycle Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.13 Two graphs G and H are not the same, but they are isomorphic. . . . . . . 291.14 Two graphs G and H are not isomorphic. . . . . . . . . . . . . . . . . . . . 291.15 The graph K 4 drawn as a plane graph without edge crossing. . . . . . . . . 301.16 One graph gives two planar maps . . . . . . . . . . . . . . . . . . . . . . . 311.17 (a) A representation of a graph; its set of vertices is {1, 2, 3, 4}, and(multi)set of edges is {{1, 2}, {2, 3}, {2, 4}, {2, 4}, {3, 3}, {3, 4}}. (b) Twoembeddings of this graph in the sphere, which are not homeomorphic sincethe second has a triangular face, unlike the first. . . . . . . . . . . . . . . . 311.18 The degree of the faces of this planar map are written inside the faces . . . 321.19 An example of map C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.20 Path graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.21 Simple trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.22 Path and Star trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.23 A graph G with its 3 spanning trees . . . . . . . . . . . . . . . . . . . . . . 381.24 An example of a graph G whose diameter is 2 . . . . . . . . . . . . . . . . 392.1 A graph(left) and all spanning trees (right) of this graph. . . . . . . . . . . 422.2 A graph G gives rise to five spanning trees . . . . . . . . . . . . . . . . . . 432.3 A graph G and three of its spanning trees . . . . . . . . . . . . . . . . . . 432.4 A spanning tree of graph G . . . . . . . . . . . . . . . . . . . . . . . . . . 442.5 A spanning tree of graph G . . . . . . . . . . . . . . . . . . . . . . . . . . 449