On the two largest Q-eigenvalues of graphs

Abstract

In this paper, we first give an upper bound for the largest signless Laplacian eigenvalue of a graph and find all the extremal graphs. Secondly, we consider the second-largest signless Laplacian eigenvalue and we characterize the connected graphs whose secondlargest signless Laplacian eigenvalue does not exceed 3. Furthermore, we give the signless Laplacian spectral characterization of the latter graphs. In particular, the well-known friendship graph is proved to be determined by the signless Laplacian spectrum. © 2010 Elsevier B.V. All rights reserved.