Maximum degree and spectral radius of graphs in terms of size

Denote by \(\rho (G)\) $\rho (G)$ and \(\kappa (G)\) $\kappa (G)$ the spectral radius and the signless Laplacian spectral radius of a graph G, respectively. Let \(k\ge 0\) $k\ge 0$ be a fixed integer and G be a graph of size m which is large enough. We show that if \(\rho (G)\ge \sqrt{m-k}\) $\rho (G)\ge \sqrt{m-k}$ , then \(C_4\subseteq G\) $C_4\subseteq G$ or \(K_{1,m-k}\subseteq G\) $K_{1,m-k}\subseteq G$ . Moreover, we prove that if \(\kappa (G)\ge m-k+1\) $\kappa (G)\ge m-k+1$ , then \(K_{1,m-k}\subseteq G\) $K_{1,m-k}\subseteq G$ . Both these results extend some known results.