The $\alpha$-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 $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)$, where $0\leq\alpha \leq1$. The $\alpha$-index of $G$ is the largest eigenvalue of $A_\alpha(G)$. Particularly, the matrix $A_0(G)$ (resp. $2A_{\frac{1}{2}}(G)$) is exactly the adjacency matrix (resp. signless Laplacian matrix) of $G$. He, Li and Feng [arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency spectral radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively. Motivated by the above results of He, Li and Feng, in this paper we characterize the extremal graphs with 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 maximum signless Laplacian radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively.