Algebraic degree of Cayley colour graphs

TL;DR

This paper determines the algebraic degree of Cayley colour graphs by analyzing their splitting fields, generalizing previous results, and exploring the relationship between algebraic integrality and degrees in normal Cayley graphs.

Contribution

It explicitly computes the splitting field and algebraic degree of Cayley colour graphs, extending prior work and establishing new relations between algebraic properties of these graphs.

Findings

Complete determination of the splitting field for Cayley colour graphs

Generalization of Wu et al.'s results on algebraic degrees

Equivalence of algebraic degree and distance algebraic degree in normal Cayley graphs

Abstract

The splitting field of a graph $\Gamma$ with respect to a square matrix $M$ associated with $\Gamma$, is the smallest field extension over the field of rationals $\mathbb{Q}$ that contains all the eigenvalues of $M$. The degree of the extension is called the algebraic degree of $\Gamma$ with respect to $M$. In this paper, we completely determine the splitting field of the adjacency matrix of the Cayley colour graph $\operatorname{Cay}(G,f)$ on a finite group $G$, associated with a class function $f:G\to\mathbb{Q}$ and compute its algebraic degree, which generalize the main results of Wu et al. Moreover, we study the relation between the algebraic integrality of two Cayley colour graphs, and deduce the fact that the algebraic degree and distance algebraic degree of a normal Cayley graph are same, generalizing a result of Zhang et al.