Algebraic degrees of quasi-abelian semi-Cayley digraphs

TL;DR

This paper investigates the algebraic degrees and splitting fields of quasi-abelian semi-Cayley digraphs, extending previous results from Cayley graphs over abelian groups to a broader class of semi-Cayley digraphs.

Contribution

It provides a general method to determine the splitting field and algebraic degree of quasi-abelian semi-Cayley digraphs using irreducible characters of groups, broadening prior work.

Findings

Determines the splitting field of quasi-abelian semi-Cayley digraphs.

Calculates the algebraic degree in terms of group characters.

Generalizes previous results on Cayley graphs and semi-Cayley digraphs.

Abstract

For a digraph $\Gamma$, if $F$ is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of $\Gamma$, then $F$ is called the splitting field of $\Gamma$. The extension degree of $F$ over the field of rational numbers $\mathbb{Q}$ is said to be the algebraic degree of $\Gamma$. A digraph is a semi-Cayley digraph over a group $G$ if it admits $G$ as a semiregular automorphism group with two orbits of equal size. A semi-Cayley digraph $\mathrm{SC}(G,T_{11},T_{22},T_{12},T_{21})$ is called quasi-abelian if each of $T_{11},T_{22},T_{12}$ and $T_{21}$ is a union of some conjugacy classes of $G$. This paper determines the splitting field and the algebraic degree of a quasi-abelian semi-Cayley digraph over any finite group in terms of irreducible characters of groups. This work generalizes the previous works on algebraic degrees of Cayley graphs over abelian groups and any group having a subgroup of index 2, and semi-Cayley digraphs over abelian groups.