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 be the spectral radius of its adjacency matrix. Recently we showed that if G has no 4-cycle, then 2 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 2 m; with equality if G is a star. For 4 m 8 this assertion fails. Keywords: 4-cycles; graph spectral radius; graphs with no 4-cycles; friendship graph. AMS classi…cation: 05C50, 05C35. This note is part of an ongoing project aiming to build extremal graph theory on spectral grounds, see, e.g., [3] and [6, 14]. Suppose G is a graph with n vertices and m edges and let (G) be the spectral radius of its adjacency matrix. How large can (G) be if G has no cycles of length 4? This question was partially answered in [10], Theorem 3: Let G be a graph of order n with (G) = . If G has no 4-cycles; then 2 n 1: (1) Equality holds if and only if every two vertices of G have exactly one common neighbor. The condition for equality in (1) is a popular topic: as shown in [4] and [5], the only graph satisfying this condition is the friendship graph a set of bn=2c triangles sharing a single common vertex. Thus equality is possible only for n odd, and (1) may be improved for even n: Conjecture 1 Let G be a graph of even order n with (G) = . If G has no 4-cycles, then 3 2 (n 1) + 1 0: (2) Equality holds if and only if G is a star of order n with n=2 1 disjoint additional edges.