Laplacian spectra and spanning trees of threshold graphs

Abstract

In this paper we study the Laplacian spectra, the Laplacian polynomials, and the number of spanning trees of a special class of graphs called threshold graphs. We give formulas for the Laplacian spectrum, the Laplacian polynomial, and the number of spanning trees of a threshold graph, in terms of so-called composition sequences of threshold graphs. It is shown that the degree sequence of a threshold graph and the sequence of eigenvalues of its Laplacian matrix are “almost the same”. On this basis, formulas are given to express the Laplacian polynomial and the number of spanning trees of a threshold graph in terms of its degree sequence. Moreover, threshold graphs are shown to be uniquely defined by their spectrum, and a polynomial time procedure is given for testing whether a given sequence of numbers is the spectrum of a threshold graph.