A characterization of strongly regular graphs in terms of the largest signless Laplacian eigenvalues

Let G be a simple graph of order n with maximum degree Δ. Let λ (resp. μ) denote the maximum number of common neighbors of a pair of adjacent vertices (resp. nonadjacent distinct vertices) of G  . Let q(G)q(G) denote the largest eigenvalue of the signless Laplacian matrix of G. We show thatq(G)≤Δ−μ4+(Δ−μ4)2+(1+λ)Δ+μ(n−1)−Δ2, with equality if and only if G   is a strongly regular graph with parameters (n,Δ,λ,μ)(n,Δ,λ,μ).