Proof of conjecture involving the second largest signless laplacian eigenvalue and the index of graphs

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetković et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (^q2) and the index (^λ1) of graph G (see also Aouchiche and Hansen [1]):1-n-1≤^q2-^λ1≤n- 2-2n-4with equality if and only if G is the star K1_,n-1 for the lower bound, and if and only if G is the complete bipartite graph Kn-_2,2 for the upper bound. In this paper we prove the lower bound and characterize the extremal graphs.