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

The Randić index R of a graph G   is defined as the sum of (didj)−12 over all edges vivjvivj of G  , where didi denotes the degree of a vertex vivi in G  . q1q1 is the largest eigenvalue of the signless Laplacian matrix Q=D+AQ=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) q1−R⩽32n−2 and equality holds for G≅KnG≅Kn and (2)q1R⩽{4n−4n,4⩽n⩽12,nn−1,n⩾13 with equality if and only if G≅KnG≅Kn for 4⩽n⩽124⩽n⩽12 and G≅SnG≅Sn for n⩾13n⩾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.