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

124 CHAPTER 7. MAXIMAL PLANAR MAPS7.3.1 Main ResultsIt is known that Kirchhoff Matrix Tree Theorem [81, 85], can be applied to any map Cto determine τ(C) by taking a determinant of Laplacian matrix of C, but this requiresevaluating a determinant of a corresponding characteristic matrix. However, for a fewspecial families of maps, there exist simple formulae which make it much easier to calculateand determine the number of corresponding spanning trees especially when thesenumbers are very large, as we have already seen in previous chapters. In this section, weare going to describe a way to calculate the number of spanning trees by an extensionof Kirchhoff’s formula, also known as the matrix tree theorem. In this work, we use thedeterminant of Laplacian matrix of planar maps (Matrix Tree Theorem) to derive theexplicit formula for calculating the number of spanning trees in the maximal planar mapand deduce a formula to calculate the number of spanning trees in the crystal planar map.Before presenting the main results, we need the following lemmas:Lemma 7.3.1 Let E n be a maximal planar map with n vertices, thenn/τ(E n ), n ≥ 3.Proof: We first form the Kirchhoff characteristic matrix L n (n × n) corresponding tothe labeling of E n shown as follows (in Figure 7.9):Figure 7.9: The principal matrix of E nNext, we focus our attention on the principal submatrix L n−1 which obtained by cancelingits last row and column corresponding to vertex v n . So, the number of spanning trees ofa maximal planar map E n equals τ(E n ) = det(L n−1 ).