For a simple connected graph G of size m, let A be the adjacency matrix and let \(\rho (G)\) be the spectral radius. 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. Let \(S_{n,2}\) be the graph obtained by joining each vertex of \(K_2\) to \(n-2\) isolated vertices and let \(S_{n,2}^-\) be the graph obtained from \(S_{n,2}\) by deleting an edge incident to a vertex of degree 2. The authors (2024) have shown that if G is a graph of odd size that does not contain the subgraphs H(3, 3) and H(4, 3), then \(\rho (G)\le \frac{1+\sqrt{4m-3}}{2}\) , with equality if and only if \(G\cong S_{\frac{m+3}{2},2}\) . In this paper, we show that if G is a \(\{H(3,3), H(4,3)\}\) -free graph of even size, then \(\rho (G)\le \rho ^\prime (m)\) , where \(\rho ^\prime (m)\) is the largest root of \(x^4-mx^2-(m-2)x+\frac{m}{2}-1=0\) , and equality holds if and only if \(G\cong S_{\frac{m+4}{2},2}^-\) . Furthermore, the condition on the number of edges to be greater than or equal to 10 is essential.
