Maximum degree and spectral radius of graphs in terms of size

Denote by rho(G) and kappa(G) the spectral radius and the signless Laplacian spectral radius of a graph G, respectively. Let k >= 0 be a fixed integer and G be a graph of size m which is large enough. We show that if rho(G) >= root m - k, then C-4 subset of G or K-1,K-m-k subset of G. Moreover, we prove that if kappa(G) >= m - k + 1, then K-1,K-m-k subset of G. Both these results extend some known results.