Combinatorial explanation of the weighted Laplacian characteristic polynomial of a graph and applications

Abstract

Assume that G is a graph with n vertices v1,v2,…,vn. Given a rooted spanning forest F of G, which has s components and roots vk1,vk2,…,vks, define the weight ω(F) of F to be ∏i=1sxki, where x1,x2,…,xn are n independent commuting variables. In this paper, we give combinatorial explanations of the weighted enumeration of rooted spanning forests of G, the Laplacian characteristic polynomial of G, and the weighted Laplacian det[L+diag(x1,x2,…,xn)], where L is the Laplacian matrix of G and diag(x1,x2,…,xn) is a diagonal matrix. As applications, we count rooted spanning forests in the almost complete bipartite graph, and we also give a new proof of a formula on the number of rooted spanning forests in a complete multipartite graph Ka1,a2,…,as each of which has bi roots in the i-th part for i=1,2,…,s.