  Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency
matrix and the degree matrix of $G$ respectively. Define the convex linear
combinations
  $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha
D(G)+(1-\alpha)A(G)$$ for any real number $0\leq\alpha\leq1$. The
\emph{$\alpha$-index} of $G$ is the largest eigenvalue of $A_\alpha(G)$. In
this paper, we determine the maximum $\alpha$-index and characterize all
extremal graphs for $K_r$ minor-free graphs, $K_{s,t}$ minor-free graphs, and
star-forest-free graphs for any $0<\alpha<1$ by unified eigenvector approach,
respectively.
