On maximum spectral radius of $\{H(3,3),~H(4,3)\}$-free graphs

Let $G$ be a simple connected graph of size $m$. Let $A$ be the adjacency matrix of $G$ and let $\rho(G)$ be the spectral radius of $G$. A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. Let $H(\ell,3)$ be the graph formed by taking a cycle of length $\ell$ and a triangle on a common vertex. Recently, Li, Lu and Peng [Y. Li, L. Lu, Y. Peng, Spectral extremal graphs for the bowtie, Discrete Math. 346(12) (2023) 113680.] showed that the unique $m$-edge $H(3,3)$-free spectral extremal graph is the join of $K_2$ with an independent set of $\frac{m-1}{2}$ vertices if $m\ge 8$ and the condition $m\ge 8$ is tight. In particular, if $G$ does not contain $H(3,3)$ as induced subgraph, they proved that $\rho(G) \leq \frac{1+\sqrt{4m-3}}{2}$ and equality holds when $G$ is isomorphic to $S_{\frac{m+3}{2},2}$. Note that Li et al. denoted $H(3,3)$ by $F_2$. In this paper, we find the maximum spectral radius and identify the graph with the largest spectral radius among all \{$H(3,3), H(4,3)$\}-free graphs of size odd $m$, where $m\geq 259$. Coincidentally, we show that $\rho(G) \leq \frac{1+\sqrt{4m-3}}{2}$ when $G$ forbids both $H(3,3)$ and $H(4,3)$. In our case, the equality holds when $G$ is isomorphic to the same graph.