Laplacian coefficients, Kirchhoff index and the number of spanning trees of graphs

Abstract

Let [Formula: see text], [Formula: see text], be a simple connected graph of order [Formula: see text] and size [Formula: see text]. Denote with [Formula: see text] eigenvalues of the Laplacian matrix [Formula: see text] of [Formula: see text]. The Kirchhoff index and the number of spanning trees of [Formula: see text] expressed in terms of Laplacian eigenvalues are given by [Formula: see text] and [Formula: see text], respectively. The characteristic polynomial of [Formula: see text] is given by [Formula: see text]. The first five Laplacian coefficients have been computed in the literature. In this study, we compute the sixth Laplacian coefficient of [Formula: see text]. Then, we use it to improve the previously obtained results on [Formula: see text] and [Formula: see text]. In addition, we present new Nordhaus–Gaddum-type inequalities for [Formula: see text] and [Formula: see text].