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arXiv · 2210.04160

Regular graphs with a complete bipartite graph as a star complement

Xiaona Fang·Lihua You·Rangwei Wu·Yufei Huang
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Abstract

Let $G$ be a graph of order $n$ and $μ$ be an adjacency eigenvalue of $G$ with multiplicity $k\geq 1$. A star complement $H$ for $μ$ in $G$ is an induced subgraph of $G$ of order $n-k$ with no eigenvalue $μ$, and the vertex subset $X=V(G-H)$ is called a star set for $μ$ in $G$. The study of star complements and star sets provides a strong link between graph structure and linear algebra. In this paper, we study the regular graphs with $K_{t,s}\ (s\geq t\geq 2)$ as a star complement for an eigenvalue $μ$, especially, characterize the case of $t=3$ completely, obtain some properties when $t=s$, and propose some problems for further study.

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Xiaona Fang, Lihua You, Rangwei Wu, Yufei Huang. 2022-10-09. Regular graphs with a complete bipartite graph as a star complement. https://arxiv.org/abs/2210.04160

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