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Algebraic degrees of $n$-Cayley digraphs over abelian groups

Hao Li, Xiaogang Liu

Abstract

A digraph is called an $n$-Cayley digraph if its automorphism group has an $n$-orbit semiregular subgroup. We determine the splitting fields of $n$-Cayley digraphs over abelian groups and compute a bound on their algebraic degrees, before applying our results on Cayley digraphs over non-abelian groups.

Algebraic degrees of $n$-Cayley digraphs over abelian groups

Abstract

A digraph is called an -Cayley digraph if its automorphism group has an -orbit semiregular subgroup. We determine the splitting fields of -Cayley digraphs over abelian groups and compute a bound on their algebraic degrees, before applying our results on Cayley digraphs over non-abelian groups.
Paper Structure (7 sections, 12 theorems, 67 equations)

This paper contains 7 sections, 12 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Field and Galois theory
  4. Representation theory of finite groups
  5. Main lemmas
  6. Algebraic degrees of $n$-Cayley digraphs over abelian groups
  7. Applications

Key Result

Lemma 2.1

Let $n\geq1$ be an integer. Let $G$ be a finite abelian group. Let $A=[a_{i,j}]_{n\times n}\in M_{n}(\mathbb{C}[G])$. Then $\forall\chi\in\widehat{G}$,

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 12 more