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

58CHAPTER 3.HOW TO COUNT THE NUMBER OF SPANNING TREES INGRAPHSProof: First we notice that F n,1 is the set of all rooted trees (since they only consist ofone component). Note that |F n,1 | = nT n , since in every tree there are n choices for theroot. We now regard F n,k ∈ F n,k as a directed graph with all of the edges directed awayfrom the roots. Now let F k be a fixed forest in F n,k and denote by N(F k ) the number ofrooted trees containing F k , and by N ∗ (F k ) the number of refining sequences ending in F k .We count N ∗ (F k ) in two ways, first by starting at a tree and secondly by starting atF k . Suppose F 1 ∈ F n,1 contains F k . Since we may delete the k - 1 edges of F 1 \F k in anypossible order to get a refining sequence from F 1 to F k , we findN ∗ (F k ) = N(F k )(k − 1)!.Let us now start at the other end. To produce from F k an F k−1 we have to add a directededge, from any vertex v, to any of the k - 1 roots of the trees that do not contain v. Thuswe have n(k − 1) choices. Similarly, for F k−1 we may produce a directed edge from anyvertex u to any of the k - 2 roots of the trees not containing u. For this we have n(k − 2)choices. Continuing this way, we arrive atand thus we have thatN ∗ (F k ) = n k−1 (k − 1)!,N(F k ) = n k−1 for any F k ∈ F n,k .For k = n, F n consists of just n isolated vertices. Hence N(F n counts the number of allrooted trees, thus |F n,1 | = n n−1 , and thus T n = (1/n)(|F n,1 |) = n n−2 .□The method of Prüfer’s code has also been used to investigate the spanning treeformulae for bipartite graphs [42], k-partite graphs [96], extended graphs [61] andcomplete multipartite graphs [41, 72].3.4 Counting the number of spanning trees in graphsalgebraicallyAlgebraic graph theory is a branch of mathematics that studies graphs by using algebraicproperties of associated matrices. More in particular, spectral graph theory studies therelation between graph properties and the spectrum of the adjacency matrix or Laplacematrix. There are some known algebraic methods for counting the number of spanningtrees.3.4.1 The Matrix-Tree TheoremAnother way to compute the number of spanning trees in a graph is using linear algebra.By a theorem of Gustav Robert Kirchoff in 1847, the number of spanning trees can becomputed by finding the cofactor of a special matrix called the Laplacian matrix L. Oneof the earliest uses of the matrix L proper was the Matrix-Tree Theorem, proposed by