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arXiv · 2211.11164

On the Laplacian spectrum of $k$-symmetric graphs

Sunyo Moon·Hyungkee Yoo
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Abstract

For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are integers.

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Sunyo Moon, Hyungkee Yoo. 2022-11-21. On the Laplacian spectrum of $k$-symmetric graphs. https://arxiv.org/abs/2211.11164

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