Some Spectral Properties of the $B_α$ Matrix of a Graph

Garrett Kepler

Some Spectral Properties of the $B_α$ Matrix of a Graph

Combinatorics 2026-07-28 v1

Authors: Garrett Kepler

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.

Keywords

graph spectra B-alpha matrix positive semi-definiteness eigenvectors Laplacian

Cite

@article{arxiv.2607.25221,
  title  = {Some Spectral Properties of the $B_α$ Matrix of a Graph},
  author = {Garrett Kepler},
  journal= {arXiv preprint arXiv:2607.25221},
  year   = {2026}
}