Maximizing the signless Laplacian spectral radius of graphs with given diameter or cut vertices

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan, sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. The signless Laplacian matrix of a graph is defined to be the sum of its adjacency matrix and degree matrix. Let G d n be the set of all the connected graphs of order n and diameter d and G n,k the set of all connected graphs with order n and k cut vertices. In this article, we determine the graphs that have the maximal signless Laplacian spectral radius and give the upper bounds of graphs in these two sets. 1. Introduction All graphs considered here are simple. For a graph G, let M be a responding graph matrix defined in a prescribed way. The M-spectrum of G is a multiset consisting of the eigenvalues of its graph matrix M. The M-spectral radius (or M-index) of G is the largest eigenvalue of its graph matrix M. It is well-known that there are several graph matrices two of which named adjacency matrix A(G) and Laplacian matrix L(G) ¼ D(G) À A(G) where D(G) is a diagonal matrix of vertices degrees, are investigated extensively, and the other one named signless Laplacian Q(G) ¼ A(G) þ D(G). Recently, Cvetkovicét al. [6] intended to build a spectral theory for the signless Laplacian matrices (see [6–9,18] for more results). For this purpose, in this article we will focus our attention on Q(G)-spectrum of a graph G. Brualdi and Solheid [4] posed the following problem concerning the spectral radius of graphs: