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

Abstract

AbstractThe sum-connectivity index of a graph G is defined as the sum of weights [Formula: see text] 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 [Formula: see text] 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, [Graphic: see text]q(G)SCI(G)≤n(n+2mn−1−2)n(n−1)+2m−2n+2.

Keywords

• Sum-connectivity index
• spectral radius
• signless Laplacian matrix
• 05C50