A method to determine algebraically integral Cayley digraphs on finite Abelian group

TL;DR

This paper develops a method to identify Cayley digraphs on finite Abelian groups with eigenvalues that are algebraic integers in a specific number field, advancing spectral graph theory.

Contribution

It introduces a novel algebraic approach and proves a key theorem to characterize such Cayley digraphs, also computing their total number.

Findings

Established a criterion for eigenvalues to be algebraic integers in a given field.

Derived a formula for counting Cayley digraphs with this property.

Proved a fundamental theorem (Theorem 1) related to the characterization.

Abstract

Researchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite Abelian group G whose eigenvalues are algebraic integers in a given number field K. And we succeed in finding a method to do so by proving Theorem 1. Also, the number of such Cayley digraphs is computed.