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

Chapter 3How to count the number of spanningtrees in graphsIn this chapter, we shall focus on the general methods for counting the number of spanningtrees in graphs while discussing the spanning trees, as well as the enumeration of spanningtrees.3.1 IntroductionSpanning trees are relevant to various aspects of graphs (networks). Generally, the numberof different spanning trees in a graph G turns out to be a very useful number leading tointeresting results and applications. Most of the classical theories of interest concerningspanning trees is the number of spanning trees of a graph G. We denote the number ofspanning trees in a graph G by τ(G) and then, we shall investigate different methods forcalculating τ(G). The research of the number of spanning trees in a graph has a longhistory. The cornerstone of the research, the Matrix Tree Theorem, dated back to 1847,is attributed to Kirchhoff. Most research about the number of spanning trees is devotedto determining exact formulae for the number of spanning trees in many kinds of specialgraphs, see [6, 14, 62, 63, 65, 66, 127]. We first state the problem of counting the numberof spanning trees in graphs then display the general methods for counting the number ofspanning trees in graphs.3.2 ProblematicHow to count the number of spanning trees in graphs?Statement of problem: given a graph, how many spanning trees does it have?As an example, a graph G with all its spanning trees is displayed in Figure 3.155