Some Spectral Properties of the $B_{\alpha }$ Matrix of a Graph

Abstract

Recently, Samanta et al. introduced a unifying perspective to common graph-associated matrices through the matrix $B_{\alpha }(G)=\alpha A(G)+(1-\alpha )L(G)$ where $\alpha \in [0,1]$. We resolve an open problem regarding the positive semi-definiteness of $B_{\alpha }$ and similar matrices. We also explore eigenvector properties of $B_{\alpha }$ through edge and vertex removal. As a consequence, we provide a basis for results on the common graph-associated matrices.