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

Abstract

Let G be a simple graph of order n with maximum degree Delta. Let lambda (resp. mu) 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) <= Delta - mu/4 + root(Delta - mu/4)(2) + (1 + lambda)Delta + mu(n - 1) -Delta(2), with equality if and only if G is a strongly regular graph with parameters (n, Delta, lambda, mu). (C) 2016 Elsevier Inc. All rights