On the Laplacian spectrum of $k$-symmetric graphs

TL;DR

This paper investigates the Laplacian spectrum of $k$-symmetric graphs, focusing on eigenvalue 1 multiplicities, connectivity, and integer eigenvalues, expanding understanding of spectral properties in symmetric graph classes.

Contribution

It characterizes 2-connected $k$-symmetric graphs with Laplacian eigenvalue 1 and identifies classes with all integer Laplacian eigenvalues.

Findings

Analysis of eigenvalue 1 in 2-connected $k$-symmetric graphs.

Identification of $k$-symmetric graphs with all integer Laplacian eigenvalues.

Extension of Faria's eigenvalue multiplicity results to specific symmetric graph classes.

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.