Supercharacters of finite abelian groups and applications to spectra of $U$-unitary Cayley graphs

TL;DR

This paper introduces supercharacter theory for finite abelian groups to analyze the spectra of super-Cayley graphs, generalizing previous results and applying to $U$-unitary Cayley graphs over rings, revealing their spectral properties.

Contribution

It develops a supercharacter framework for finite abelian groups and applies it to study spectra of super-Cayley graphs, extending prior cyclic group results to more general structures.

Findings

Super-Cayley graphs are spectrally determined by their group structure.

Spectra of $U$-unitary Cayley graphs can be expressed via super-Fourier transforms.

The theory reveals rationality and arithmetical connections in the spectra.

Abstract

We define super-Cayley graphs over a finite abelian group $G$. Using the theory of supercharacters on $G$, we explain how their spectra can be realized as a super-Fourier transform of a superclass characteristic function. Consequently, we show that a super-Cayley graph is determined by its spectrum once an indexing on the underlying group $G$ is fixed. This generalizes a theorem by Sander-Sander, which investigates the case where $G$ is a cyclic group. We then use our theory to define and study the concept of a $U$-unitary Cayley graph over a finite commutative ring $R$, where $U$ is a subgroup of the unit group of $R$. Furthermore, when the underlying ring is a Frobenius ring, we show that there is a natural supercharacter theory associated with $U$. By applying the general theory of super-Cayley graphs developed in the first part, we explore various spectral properties of these $U$-unitary Cayley graphs, including their rationality and connections to various arithmetical sums.