On conjectures involving second largest signless Laplacian eigenvalue of graphs

Abstract

Let G = ( V , E ) be a simple graph. Denote by D ( G ) the diagonal matrix of its vertex degrees and by A ( G ) its adjacency matrix. Then the Laplacian matrix of G is L ( G ) = D ( G ) - A ( G ) and the signless Laplacian matrix of G is Q ( G ) = D ( G ) + A ( G ) . In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G . In [5], Cvetković et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.