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 (q2q2) and the index (λ1λ1) 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-1K1,n-1 for the lower bound, and if and only if G   is the complete bipartite graph Kn-2,2Kn-2,2 for the upper bound. In this paper we prove the lower bound and characterize the extremal graphs.