New results on $B_{\hat{I}}\pm$-eigenvalues of a graph

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and Laplacian matrix $L(G)$. In 2024, Samanta \textit{et} \textit{al.} defined the convex linear combination of $A(G)$ and $L(G)$ as $B_{\hat{I}}\pm (G)=\hat{I}\pm A(G)+(1-\hat{I}\pm )L(G)$, for $\hat{I}\pm \in [0,1]$. This paper presents some results on the eigenvalues of $B_{\hat{I}}\pm (G)$ and their multiplicity when some sets of vertices satisfy certain conditions. Moreover, the positive semidefiniteness problem of $B_{\hat{I}}\pm (G)$ is studied.

19 pages, 10 figures