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Abstract

A graph is sad to be $H$-free if it does not contain $H$ as a subgraph. Let $H(k,3)$ be the graph formed by taking a cycle of length $k$ and a triangle on a common vertex. Li, Lu and Peng [Discrete Math. 346 (2023) 113680] proved that if $G$ is an $H(3,3)$-free graph of size $m \geq 8$, then the spectral radius $\rho(G) \leq \frac{1+\sqrt{4 m-3}}{2}$ with equality if and only if $G \cong S_{\frac{m+3}{2}, 2}$, where $S_{\frac{m+3}{2}, 2}=K_2 \vee \frac{m-1}{2}K_1$. Note that the bound is attainable only when $m$ is odd. Recently, Pirzada and Rehman [Comput. Appl. Math. 44 (2025) 295] proved that if $G$ is an $\{H(3,3),H(4,3)\}$-free graph of even size $m \geq 10$, then $\rho(G) \leq \rho^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$, where $\rho^{\prime}(m)$ is the largest root of $x^4-m x^2-(m-2) x+\frac{m}{2}-1=0$, and $S_{\frac{m+4}{2}, 2}^{-}$ is the graph obtained from $S_{\frac{m+4}{2}, 2}$ by deleting an edge incident to a vertex of degree two. In this paper, we improve the result of Pirzada and Rehman by showing that if $G$ is an $H(4,3)$-free graph of even size $m \geq 38$ without isolated vertices, then $\rho(G) \leq \rho^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$.