Proof of a conjecture on the spectral radius of C4-free graphs

Let G be a graph with n vertices and ρ be the spectral radius of its adjacency matrix. Recently, Nikiforov showed that if G has no 4-cycle, then , with equality if and only if G is the friendship graph. However, this bound is not attainable when n is even. He conjectured that if G is a C4-free graph with even number of vertices, then , with equality if and only if G is a star of order n with n/2-1 disjoint additional edges. We prove the conjecture in this paper.