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The harmonic index of a conected graph G is dened as H(G) = ∑ uv∈E(G)    2d(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.

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