On the $A_{\alpha }$-characteristic polynomial of a graph

Xiaogang Liu, Shunyi Liu

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$$ for any r…