A sharp upper bound on the spectral radius of $\theta (1,3,3)$-free graphs with given size

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\rho (G)$ be the spectral radius of a graph $G$. Let $\theta (1,p,q)$ denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths $1,p,q$, where $p\le q$. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_k$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denote the graph obtained from $S_{n,k}$ by deleting an edge incident to a vertex of degree $k$, respectively. In this paper, we show that if $\rho (G)\ge \rho (S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\ge 92$, then $G$ contains a $\theta (1,3,3)$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$.