ON COEFFICIENTS OF THE CHARACTERISTIC POLYNOMIAL OF THE LAPLACE MATRIX OF A WEIGHTED DIGRAPH AND THE ALL MINORS THEOREM

摘要

Let L be the Laplace, matrix of a weighted digraph. The aim of the paper is to establish a simple way for computing any coefficient of the characteristic polynomial of L as a constant sign sum over the incoming spanning forests. The idea is to express L as the product of generalized (weighted) incidence matrices. It turns out that the minors of them can be studied in terms of the tree-like structure of the digraph. This makes it possible to compute the minors of L.