Algebraic degree of Cayley colour graphs

TL;DR

This work determines the splitting field and algebraic degree for Cayley colour graphs $\Gamma_f=\mathrm{Cay}(G,f)$ when $f$ is a class function, showing the splitting field is the cyclotomic field $\mathbb{Q}(\zeta_n)$ fixed by the subgroup $H_f=\{h\in\mathbb{Z}_n^{*}: f^h=f\}$, and that $\mathrm{Deg}(\Gamma_f)=\varphi(n)/|H_f|$. It extends these results to normal Cayley multigraphs and introduces the distance algebraic degree for normal Cayley graphs, expressed via $H^* = \{h: S^h=S\}$ and $H' = \{h: S_i^h=S_i\ \forall i\}$ respectively. The paper also analyzes how algebraic integrality over a field $K$ depends on Galois invariants and shows that, for normal Cayley graphs, algebraic integrality and distance algebraicity coincide, with implications for equality of splitting fields and degrees. Overall, it provides explicit, group-theoretic criteria to compute splitting fields and algebraic degrees for a broad class of Cayley-type graphs, facilitating classification of algebraically integral and distance-integral instances.

Abstract

The splitting field of a graph $Γ$ with respect to a square matrix $M$ associated with $Γ$, 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 $Γ$ 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.