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 μ be the spectral radius of its adjacency matrix.Recently we showed that if G has no 4-cycle, then μ2-μ⩽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 μ2⩽m, with equality if G is a star. For 4⩽m⩽8 this assertion fails.