The maximum spectral radius of graphs of given size with forbidden subgraph

Let $G$ be a graph of size $m$ and $\rho (G)$ be the spectral radius of its adjacency matrix. A graph is said to be $F$-free if it does not contain a subgraph isomorphic to $F$. In this paper, we prove that if $G$ is a $K_{2,r+1}$-free non-star graph with $m\ge (4r+2{)}^2+1$, then $\rho (G)\le \rho (S_m^1)$, with equality if and only if $G\cong S_m^1$. Recently, Li, Sun and Wei showed that for any ${\theta }_{1,2,3}$-free graph of size $m\ge 8$, $\rho (G)\le \frac{1+\sqrt{4m-3}}{2}$, with equality if and only if $G\cong S_{\frac{m+3}{2},2}$. However, this bound is not attainable when $m$ is even. We proved that if $G$ is ${\theta }_{1,2,3}$-free and $G≆S_{\frac{m+3}{2},2}$ with $m\ge 22$, then $\rho (G)\le \rho (F_{m,1})$ if $m$ is even, with equality if and only if $G\cong F_{m,1}$, and $\rho (G)\le \rho (F_{m,2})$ if $m$ is odd, with equality if and only if $G\cong F_{m,2}$.