The spectral extrema of graphs of odd size forbidding $H(4,3)$ beyond the book graph

A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. The fish graph, denoted by $H(4,3)$, is a $6-$vertex graph obtained from a cycle of length $4$ and a triangle by sharing a common vertex. Earlier it is shown that $\lambda (G)\le \frac{1+\sqrt{4m-3}}{2}$ holds for all $H(4,3)-$free graphs of odd size $m\ge 44,$ and the equality holds if and only if $G\cong S_{\frac{m+3}{2},2},$ where $S_{\frac{m+3}{2},2}$ is the $m-$edge book graph $K_2\vee \frac{m-1}{2}{K}_1,$ where $K_2\vee \frac{m-1}{2}{K}_1,$ denotes the join of $K_2$ and $\frac{m-1}{2}{K}_1\mathrm{.}$ Let $\mathcal{G}(m,H(4,3))$ denote the family of $H(4,3)$-free graphs with $m$ edges and no isolated vertices. We write $\mathcal{G}(m,H(4,3))\setminus \{K_2\vee \frac{m-1}{2}{K}_1\}$ for the corresponding subfamily obtained by excluding the book graph. In this paper, we establish a sharp upper bound on the spectral radius of graphs over $\mathcal{G}(m,H(4,3))\setminus \{K_2\vee \frac{m-1}{2}{K}_1\}$ for odd $m\ge 58$ and characterize the unique extremal graph attaining this bound.