Let $A(G)$ be the adjacency matrix of a graph G and $\rho (G)$ be its spectral radius. Given a graph H and a family $\mathcal{F}$ of graphs, let $e{x}_{sp}(n,H;\mathcal{F})=\max \{\rho (G)||V(G)|=n,H\subseteq G,G\text{does not contain any graph of}\mathcal{F}\}$. Let $S_{2k-1}(K_{s,t})$ be the graph obtained by replacing an edge of $K_{s,t}$ with a copy of $P_{2k+1}$, where $k\ge 2$. In this paper, we show that $e{x}_{sp}(n,C_{2k+3};\{C_3,C_5,\dots ,C_{2k+1}\})=\rho (S_{2k-1}(K_{\lceil \frac{n-2k+1}{2}\rceil ,\lfloor \frac{n-2k+1}{2}\rfloor }))$ and the unique extremal graph is $S_{2k-1}(K_{\lceil \frac{n-2k+1}{2}\rceil ,\lfloor \frac{n-2k+1}{2}\rfloor })$, which solves a question proposed in [Eigenvalues and triangles in graphs, Comb. Probab. Comput. 30 (2021) 258–270].

让 $\mathrm{一种}(G)$是图G的邻接矩阵和$\rho (G)$是它的光谱半径。给定一个图H和一个家庭$\mathcal{F}$ 的图形，让 $\mathrm{电子}{X}_{秒磷}(n,H;\mathcal{F})=\mathrm{最大限度}\{\rho (G)||伏(G)|=n,H\subseteq G,G\text{不包含任何图形}\mathcal{F}\}$. 让${秒}_{2克-1}({钾}_{秒,吨})$ 是通过替换边获得的图 ${钾}_{秒,吨}$ 带有副本 ${磷}_{2克+1}$， 在哪里 $克\ge 2$. 在本文中，我们表明$\mathrm{电子}{X}_{秒磷}(n,C_{2克+3};\{C_3,C_5,\dots ,C_{2克+1}\})=\rho ({秒}_{2克-1}({钾}_{\lceil \frac{n-2克+1}{2}\rceil ,\lfloor \frac{n-2克+1}{2}\rfloor }))$ 唯一的极值图是 ${秒}_{2克-1}({钾}_{\lceil \frac{n-2克+1}{2}\rceil ,\lfloor \frac{n-2克+1}{2}\rfloor })$，它解决了[图中的特征值和三角形，梳子。可能。计算。30 (2021) 258–270]。