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 ( 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 2 n − 2 and equality holds for G ≅ K n and (2) q 1 R ⩽ { 4 n − 4 n , 4 ⩽ n ⩽ 12 , n n − 1 , n ⩾ 13 with equality if and only if G ≅ K n for 4 ⩽ n ⩽ 12 and G ≅ 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.