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

Chapter 6Counting the number of spanning treesin the star flower planar mapIn this chapter, we shall consider the star flower planar maps (graphs embedded in theplane without edge-crossings). We look at this map classes, and find a formula for countingthe number of spanning trees in the star flower planar map. Next, we show that thenumber of spanning trees in this type of planar maps always satisfy recurrence relationsand describe how to derive this relation, i.e., we find the simple explicit formulae forcalculating the number of spanning trees in the star flower planar map.6.1 IntroductionAs mentioned in Chapter 3, the number of spanning trees in a graph (network) is animportant quantity to estimate the reliability of the graph (network). A famous andclassic result on the study of the number of spanning trees in graph τ(G) is the followingtheorem, known as the Matrix tree Theorem [81, 106]. The Laplacian matrix (also calledKirchhoff matrix) of a graph G is defined as L(G) = D(G) − A(G), where D(G) andA(G) are the degree matrix and the adjacency matrix of G, respectively. In graph theory,Laplacian matrix is a matrix representing a graph. It has been used by Kirchhoff tocalculate the number of spanning trees of a graph.The Matrix Tree Theorem, which is given by Kirchhoff defines the number of spanningtrees in graph G as the determinant of its Laplacian matrix. This theorem simplifies thecomputation of the number of spanning trees but in large graphs, it is not practical. Inthe following we shall describe a general method to count the number of spanning treesin the star flower planar map. Our approach is to use our main results [89] that we havederived in Chapter 4 to calculate the number of spanning trees in the star flower planarmap.107