ON THE HARMONIC INDEX AND THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

The harmonic index of a conected graph G is defined as H(G) = Sigma(uv is an element of 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 q(G)/H(G) <= { n2/2(n-1), if n >= 6, 16/5, if n = 5, 3, if n = 4, and the bounds are best possible.