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 ) denote the largest eigenvalue of the signless Laplacian matrix of G . We show that q ( G ) ≤ Δ − μ 4 + ( Δ − μ 4 ) 2 + ( 1 + λ ) Δ + μ ( n − 1 ) − Δ 2 , with equality if and only if G is a strongly regular graph with parameters ( n , Δ , λ , μ ) . [ABSTRACT FROM AUTHOR]