On the log-concavity of Laplacian characteristic polynomials of graphs

Abstract

Let G be a graph and L(G) be the Laplacian matrix of G. In this article, we first point out that the sequence of the moduli of Laplacian coefficients ofG is log-concave and hence unimodal. Using this fact, we provide an upper bound for the partial sums of the Laplacian eigenvalues of G, based on coefficients of its Laplacian characteristic polynomial. We then obtain some lower bounds on the algebraic connectivity of G. Finally, we investigate the mode of such sequences. 1. Introduction. Throughout this paper, we consider simple undirected graphs having n vertices and m edges. For a given graph G, let V (G) and E(G) denote the vertex and the edge set of G, respectively. Let A(G) be the adjacency matrix of G. The Laplacian matrix and signless Laplacian matrix of G are defined as L(G) = D(G) − A(G) and Q(G) = D(G)+A(G), respectively, where D(G) is a diagonal matrix whose diagonal entries are vertex degrees of G. The eigenvalues of matrices L(G) and Q(G) are denoted by µ1(G) ≥ µ2(G) ≥ ··· ≥ µn(G) = 0 and �1(G) ≥ �2(G) ≥ ··· ≥ �n(G), respectively. As it is well-known, L(G) and Q(G) are positive semi-definite, and they have the same characteristic polynomial if and only if the graph G is bipartite. The second smallest eigenvalue of L(G), µn−1(G), is called the algebraic connectivity of G, and it is positive if and only if the graph G is connected. For bibliographies on the graph Laplacian, the reader is referred to (11).