Extension on spectral extrema of gem-free graph with given size

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\mathcal{G}(m,F)$ denote the family of $F$-free graphs with $m$ edges and without isolated vertices. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_k$ to $n-k$ isolated vertices and $S_{n,k}^t$ denote the graph obtained from $S_{n-t,k}$ by attaching $t$ pendant vertices to the maximal degree vertex of $S_{n-t,k}$, respectively. Denote by $H_n$ the fan graph obtain from $n-1$-vertex path plus a vertex adjacent to each vertex of the path. Particularly, the graph $H_5$ is also known as the gem. Zhang and Wang [Discrete Math. 347(2024)114171] and Yu, Li and Peng [arXiv: 2404. 03423] showed that every gem-free graph $G$ with $m$ edges satisfies $\rho (G)\le \rho (S_{\frac{m+3}{2},2})$. In this paper, we show that if $G\in \mathcal{G}(m,H_5)\setminus S_{\frac{m+3}{2},2}$ be a graph of odd size $m\ge 23$, then $\rho (G)\le \rho (S_{\frac{m+5}{2},2}^2)$, and equality holds if and only if $G\cong S_{\frac{m+5}{2},2}^2$.