On the log-concavity of Laplacian characteristic polynomials of graphs

Abstract

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 of G 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.

Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues

“…We provide a necessary and sufficient condition under which a bicyclic graph with a perfect matching Mathematics 2019, 7, 1233; doi:10.3390/math7121233 www.mdpi.com/journal/mathematics has 2 as its Laplacian eigenvalue, for more see [1]. For more about Laplacians of some parameters of graphs we refer to [2][3][4][5]. In the last couple of years there has been a renewed interest toward the Laplacian spectral properties of bicyclic graphs (see [6,7]), and it is very likely that many techniques employed in this paper could be also helpful to solve the correspondent problems in the context of signed graphs.…”

The Laplacian eigenvalue 2 of bicyclic graphs

“…We provide a necessary and sufficient condition under which a bicyclic graph with a perfect matching has a Laplacian eigenvalue 2. For more about Laplacian of some parameters of graphs we refer [1,7,8,12].…”