On the spectral radius of the adjacency matrix and signless Laplacian matrix of a graph

Abstract

Let G be a simple graph of order n. The matrix is called the signless Laplacian matrix of G, where and denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let and be the largest eigenvalue of and , respectively. In this paper, we first present sharp upper and lower bounds for and involving the maximum degree, the minimum degree, order, size and sum-connectivity F-index. Moreover, we investigate the relation between and .