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Research on the relationship of the (signless Laplacian) spectral radius of agraph with its structure properties is an important research project inspectral graph theory. Denote by $\rho(G)$ and $q(G)$ the spectral radius andthe signless Laplacian spectral radius of a graph $G$, respectively. Let $k\ge0$ be a fixed integer and $G$ be a graph of size $m$ which is large enough. Weshow that if $\rho(G)\ge\sqrt{m-k}$, then $C_4\subseteq G$ or$K_{1,m-k}\subseteq G$. Furthermore, we prove that if $q(G)\ge m-k$, then$K_{1,m-k}\subseteq G$. Both these two results extend some known results.

Maximum degree and spectral radius of graphs in terms of size