On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

Abstract

The harmonic index of a conected graph G is defined as H(G) = ∑ uv∈E(G) 2 d(u)+d-(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. 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 Q(G) = D(G) + A(G), 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 harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have ( 2 || ----n----- ||{ 2 (n − 1), if n ≥ 6, -q(G-)- ≤ | 16-, if n = 5, H (G ) || 5 |( 3, if n = 4, and the bounds are best possible.