On two conjectures of Randic index and the largest signless Laplacian eigenvalue of graphs

The Randic index R of a graph G is defined as the sum of (d(i)d(j))(-1/2) over all edges v(i)v(j) of G, where d(i) denotes the degree of a vertex v(i) in G. q(1) is the largest eigenvalue of the signless Laplacian matrix Q = D + A of G, where D is the diagonal matrix with degrees of the vertices on the main diagonal and A is the adjacency matrix of G. Hansen and Lucas [18] conjectured (1) q(1) - R <= 3/2n - 2 and equality holds for G congruent to K-n and (2) q(1)/R <= {4n-4/n, 4 <= n <= 12, n/root n-1, n >= 13 4nn-4 4 < n < 12, with equality if and only if G congruent to K-n for 4 <= n <= 12 and G congruent to S-n for n >= 13, respectively. In this paper, we prove the conjecture (1) and obtain a result very close to the conjecture (2). Moreover, we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix. (C) 2013 Elsevier Inc. All rights reserved.