Some results involving the $A_{\hat{I}}\pm$-eigenvalues for graphs and line graphs

Abstract

Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_{\hat{I}}\pm$-matrix of $G$, $A_{\hat{I}}\pm (G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_{\hat{I}}\pm (G):=\hat{I}\pm A(G)+(1-\hat{I}\pm )D(G),$ where $\hat{I}\pm \in [0,1]$. In this paper, we present some bounds for the eigenvalues of $A_{\hat{I}}\pm (G)$ and for the largest and smallest eigenvalues of $A_{\hat{I}}\pm (l(G))$. Extremal graphs attaining some of these bounds are characterized.

18 pages, 5 figures, 3 tables