MAXIMA OF THE Q-INDEX: GRAPHS WITH BOUNDED CLIQUE NUMBER

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 omega, then the largest eigenvalue q (G) of the matrix Q = A + D satisfies q (G) <= 2 (1 - 1/omega) n. If G is a complete regular omega-partite graph, then equality holds in the above inequality.