On the $A_{\hat{I}}\pm$-spectra of graphs

Abstract

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 $\hat{I}\pm \in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_{\hat{I}}\pm (G)$ as $$A_{\hat{I}}\pm (G)=\hat{I}\pm D(G)+(1-\hat{I}\pm )A(G).$$ In this paper, we give some results on the eigenvalues of $A_{\hat{I}}\pm (G)$ with $\hat{I}\pm >1/2$. In particular, we show that for each $e\notin E(G)$, $\hat{I}{»}_i(A_{\hat{I}}\pm (G+e))\ge \hat{I}{»}_i(A_{\hat{I}}\pm (G))$. By utilizing the result, we prove have $\hat{I}{»}_k(A_{\hat{I}}\pm (G))\le \hat{I}\pm n-1$ for $2\le k\le n$. Moreover, we characterize the extremal graphs with equality holding. Finally, we show that $\hat{I}{»}_n(A_{\hat{I}}\pm (G))\ge 2\hat{I}\pm -1$ if $G$ contains no isolated vertices.

12 pages