Maximum Aα-spectral radius of {C(3,3),C(4,3)}-free graphs

For a simple graph G and for any $\alpha \in [0,1]$, Nikiforov defined the generalized adjacency matrix as $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$, where $A\left(G\right)$ and $D\left(G\right)$ are the adjacency and degree diagonal matrices of G, respectively. The largest eigenvalue of $A_{\alpha }\left(G\right)$ is called the generalized adjacency spectral radius (or $A_{\alpha }$-spectral radius) of G. Let $C(l,t)$ denote the graph obtained from $C_l$ and $C_t$ by superimposing an edge of $C_l$ with an edge of $C_t$. If a graph is free of both $C(3,3)$ and $C(4,3)$, we call it a $\left\{C\right(3,3),C(4,3\left)\right\}$-free graph. In this paper, we give a sharp upper bound on the $A_{\alpha }$-spectral radius of $\left\{C\right(3,3),C(4,3\left)\right\}$-free graphs for $\alpha \in [0,\frac{1}{2})$. We show that the extremal graph attaining the bound is the 2-partite Turán graph.