The α-index of graphs without intersecting triangles/quadrangles as a minor

The $A_{\alpha }$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A\left(G\right)$ and the diagonal matrix of vertex degrees $D\left(G\right)$, i.e., $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$, where $0\le \alpha \le 1$. The $\alpha$-index of $G$ is the largest eigenvalue of $A_{\alpha }\left(G\right)$. In this paper, we characterize the extremal graphs with the maximum $\alpha$-index among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor for any $0<\alpha <1$, respectively. As by-products, we determine the extremal graphs with the maximum signless Laplacian spectral radius over all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively.