Maxima of the Q-index: Graphs with bounded clique number

Abstract

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number ω, then the largest eigenvalue q (G) of the matrix Q = A+ D satisfies q(G)≤ 2(1- 1/ω)n. If G is a complete regular ω-partite graph, then equality holds in the above inequality.