| Regular Graphs with a Complete Bipartite Graph as a Star Complement |
| Received:August 27, 2022 Revised:February 26, 2023 |
| Key Words:
adjacency eigenvalue star set star complement regular graph
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11971180; 12271337), the Characteristic Innovation Project of General Colleges and Universities in Guangdong Province (Grant No.2022KTSCX225) and the Guangdong Education and Scientific Research Project (Grant No.2021GXJK159). |
| Author Name | Affiliation | | Xiaona FANG | School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China | | Lihua YOU | School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China | | Rangwei WU | School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China | | Yufei HUANG | Department of Mathematics Teaching, Guangzhou Civil Aviation College, Guangdong 510403, P. R. China |
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| Abstract: |
| Let $G$ be a graph of order $n$ and $\mu$ be an adjacency eigenvalue of $G$ with multiplicity $k\geq 1$. A star complement $H$ for $\mu$ in $G$ is an induced subgraph of $G$ with $n-k$ vertices and no eigenvalue $\mu$, and the vertex subset $X=V(G-H)$ is called a star set for $\mu$ in $G$. The star complement technique provides a spectral tool for reconstructing a certain part of a graph from the remaining part. In this paper, we study the regular graphs with $K_{t,s}\ (s\geq t\geq 2)$ as a star complement for an eigenvalue $\mu$, especially, characterize the case of $t=3$ completely, obtain some properties when $t=s$, and propose some problems for further study. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2023.05.001 |
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