A generalization on spectral extrema of $K_{s,t}$-minor free graphs

The spectral extrema problems on forbidding minors have aroused wide attention. Very recently, Zhai and Lin [J. Combin. Theory Ser. B 157 (2022) 184–215] determined the extremal graph with maximum adjacency spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order. The matrix $A_{\alpha }(G)$ is a generalization of the adjacency matrix $A(G)$, which is defined by Nikiforov \cite{Nikiforov2} as $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G),$$ where $0\le \alpha \le 1$. Given a graph $F$, the $A_{\alpha }$-spectral extrema problem is to determine the maximum spectral radius of $A_{\alpha }(G)$ or characterize the extremal graph among all graphs with no subgraph isomorphic to $F$. For $\alpha =0$, the matrix $A_{\alpha }(G)$ is exactly the adjacency matrix $A(G)$. Motivated by the nice work of Zhai and Lin, in this paper we determine the extremal graph with maximum $A_{\alpha }$-spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order, where $0<\alpha <1$ and $2\le s\le t$. As by-products, we completely solve the Conjecture posed by Chen and Zhang in [Linear Multilinear Algebra 69 (10) (2021) 1922–1934].