  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.
