Spectral extremal results on the $\alpha$-index of graphs without minors and star forests

Abstract

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 $0\le \alpha \le 1$. 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.

16 pages