Generic spectrum of the weighted Laplacian operator on Cayley graphs

TL;DR

This paper studies the spectrum of weighted Laplacians on Cayley graphs, providing conditions for generic irreducibility of eigenspaces and introducing an operator with similar spectral properties.

Contribution

It offers new criteria for generic irreducibility of eigenspaces of weighted Laplacians on Cayley graphs and compares these with an auxiliary operator.

Findings

Criteria for generic irreducibility of eigenspaces

Analysis of spectrum on left-invariant Cayley graphs

Introduction of a comparable operator with similar spectral properties

Abstract

In this paper, we investigate the spectrum of a class of weighted Laplacians on Cayley graphs and determine under what conditions the corresponding eigenspaces are generically irreducible. Specifically, we analyze the spectrum on left-invariant Cayley graphs endowed with an invariant metric, and we give some criteria for generically irreducible eigenspaces. Additionally, we introduce an operator that is comparable to the Laplacian and show that the same criterion holds.