The spectral radius of signless Laplacian matrix and sum-connectivity index of graphs

Abstract

The sum-connectivity index of a graph G is defined as the sum of weights over all edges uv of G, where du and dv are the degrees of the vertices u and v in G, respectively. The sum-connectivity index is one of the most important indices in chemical and mathematical fields. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The sum-connectivity index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the sum-connectivity index of a graph G and the spectral radius of the matrix Q(G). We prove that for some connected graphs with n vertices and m edges,