On the $A_{\alpha }$-spectra of graphs

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_{\alpha }(G)$ as $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)\mathrm{.}$$ In this paper, we give some results on the eigenvalues of $A_{\alpha }(G)$ with $\alpha >1\mathrm{/}2$. In particular, we show that for each $e\mathrm{\notin }E(G)$, ${\lambda }_i(A_{\alpha }(G+e))\ge {\lambda }_i(A_{\alpha }(G))$. By utilizing the result, we prove have ${\lambda }_k(A_{\alpha }(G))\le \alpha n-1$ for $2\leq k\leq n$. Moreover, we characterize the extremal graphs with equality holding. Finally, we show that ${\lambda }_n(A_{\alpha }(G))\ge 2\alpha -1$ if $G$ contains no isolated vertices.