The \(A_\alpha \)-spectral radius of graphs with chorded cycles

Sufficient conditions for the existence of various kinds of subgraphs in a graph have been extensively studied. Recently, Gould (2022) studied the existence of a chorded cycle in a graph form spectral perspectives and proposed a question that “What spectral conditions imply the existence of a chorded cycle in a graph?" Zheng et al. (2023) and Xu et al. (2023) respectively gave the answers to this question by providing the sufficient conditions involving the spectral radius and the signless Laplacian spectral radius for the existence of a chorded cycle in a graph. In this paper, we generalize Xu et al.’s result to \(\lambda _1^{\alpha }(G)\) ${\lambda }_1^{\alpha }(G)$ for \(\alpha \in [\frac{1}{2}, 1)\) $\alpha \in [\frac{1}{2},1)$ , where \(\lambda _1^{\alpha }(G)\) ${\lambda }_1^{\alpha }(G)$ is the spectral radius of \(A_\alpha (G):=\alpha D(G)+(1-\alpha )A(G)\) $A_{\alpha }(G):=\alpha D(G)+(1-\alpha )A(G)$ , D(G) and A(G) being respectively the adjacency matrix and the diagonal degree matrix of G.