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 $m\geq (4r+2)^2+1$, then $\rho(G)\leq \rho(S_m^1)$, 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 $m\geq 8$, $\rho(G)\leq \frac{1+\sqrt{4m-3}}{2}$, 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 $G\ncong S_{\frac{m+3}{2},2}$ with $m\geq 22$, then $\rho(G)\leq \rho(F_{m,1})$ if $m$ is even, with equality if and only if $G\cong F_{m,1}$, and $\rho(G)\leq \rho(F_{m,2})$ if $m$ is odd, with equality if and only if $G\cong F_{m,2}$.