Circulant Digraphs Integral over Number Fields

TL;DR

This paper characterizes when circulant digraphs are integral over a given number field, providing necessary and sufficient conditions, and confirms a conjecture related to this property.

Contribution

It offers a complete characterization of circulant digraphs integral over number fields and proves a previously conjectured statement.

Findings

Established necessary and sufficient conditions for integrality over number fields

Confirmed the conjecture from [XM] as true

Enhanced understanding of eigenvalues in algebraic graph theory

Abstract

A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant digraphs which are integral over a number field K. And we solve the Conjecture3.3 in [XM] and find it is affirmative.