The maximum spectral radius of C4-free graphs of given order and size

Suppose that G is a graph with n vertices and m edges, and let mu be the spectral radius of its adjacency matrix., n Recently we showed that if G has no 4-cycle, then mu(2) - mu <= n - 1, with equality if and only if G is the friendship graph. Here we prove that if m >= 9 and G has no 4-cycle, then mu(2) <= M, with equality if G is a star. For 4 <= m <= 8 this assertion fails. (C) 2009 Elsevier Inc. All rights reserved.