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The $A_{\alpha}$-matrix of a graph $G$ is the convex linear combination ofthe 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 theadjacency matrix (resp. signless Laplacian matrix) of $G$. He, Li and Feng[arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacencyspectral radius among all graphs of sufficiently large order withoutintersecting triangles and quadrangles as a minor, respectively. Motivated bythe above results of He, Li and Feng, in this paper we characterize theextremal graphs with maximum $\alpha$-index among all graphs of sufficientlylarge order without intersecting triangles and quadrangles as a minor for any$0<\alpha<1$, respectively. As by-products, we determine the extremal graphswith maximum signless Laplacian radius among all graphs of sufficiently largeorder without intersecting triangles and quadrangles as a minor, respectively.

The $\alpha$-index of graphs without intersecting triangles/quadrangles as a minor