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arXiv · 2510.16812

New results on $B_{\alpha}$-eigenvalues of a graph

Germain Pastén·Carla Silva Oliveira·João Domingos G. da Silva Junior·Claudia M. Justel
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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 $B_\alpha(G) = \alpha A(G) + (1-\alpha)L(G)$, for $\alpha \in [0,1]$. This paper presents some results on the eigenvalues of $B_{\alpha}(G)$ and their multiplicity when some sets of vertices satisfy certain conditions. Moreover, the positive semidefiniteness problem of $B_{\alpha}(G)$ is studied.

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Germain Pastén, Carla Silva Oliveira, João Domingos G. da Silva Junior, Claudia M. Justel. 2025-10-19. New results on $B_{\alpha}$-eigenvalues of a graph. https://arxiv.org/abs/2510.16812

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