On the $A_α$-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 $α\in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_α(G)$ as $A_α(G)=αD(G)+(1-α)A(G).$ In this paper, we give some results on the eigenvalues of $A_α(G)$ with $α>1/2$. In particular, we show that for each $e\notin E(G)$, $λ_i(A_α(G+e))\geqλ_i(A_α(G))$. By utilizing the result, we prove have $λ_k(A_α(G))\leqαn-1$ for $2\leq k\leq n$. Moreover, we characterize the extremal graphs with equality holding. Finally, we show that $λ_n(A_α({G}))\geq 2α-1$ if $G$ contains no isolated vertices.

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