The Randić index and signless Laplacian spectral radius of graphs.

Abstract Given a connected graph G , the Randić index R (G) is the sum of (d (u) d (v)) − 1 ∕ 2 over all edges { u , v } of G , where d (u) and d (v) are the degrees of vertices u and v respectively. Let q (G) be the largest eigenvalue of the signless Laplacian matrix of G and n = | V (G) |. Hansen and Lucas (2010) made the following conjecture: q (G) R (G) ≤ 4 n − 4 n 4 ≤ n ≤ 12 ; n n − 1 n ≥ 13 , where the equality holds if and only if G = K n for 4 ≤ n ≤ 12 and G = S n for n ≥ 13 , respectively. Here K n is the complete graph with n vertices and S n is the star with n vertices. Deng et al. verified this conjecture for 4 ≤ n ≤ 11. In this paper, we prove the conjecture for n ≥ 12. [ABSTRACT FROM AUTHOR]