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Integral Cayley graphs over a group of order $6n$

Jing Wang, Xiaogang Liu, Ligong Wang

Abstract

In this paper, we study the integral Cayley graphs over a non-abelian group $U_{6n}=\langle a,b\mid a^{2n}=b^3=1, a^{-1}ba=b^{-1}\rangle$ of order $6n$. We give a necessary and sufficient condition for the integrality of Cayley graphs over $U_{6n}$. We also study relationships between the integrality of Cayley graphs over $U_{6n}$ and the Boolean algebra of cyclic groups. As applications, we construct some infinite families of connected integral Cayley graphs over $U_{6n}$.

Integral Cayley graphs over a group of order $6n$

Abstract

In this paper, we study the integral Cayley graphs over a non-abelian group of order . We give a necessary and sufficient condition for the integrality of Cayley graphs over . We also study relationships between the integrality of Cayley graphs over and the Boolean algebra of cyclic groups. As applications, we construct some infinite families of connected integral Cayley graphs over .
Paper Structure (5 sections, 19 theorems, 29 equations, 1 table)

This paper contains 5 sections, 19 theorems, 29 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The necessary and sufficient conditions for the integrality of $\mathrm{Cay}(U_{6n},S)$
  4. Relationships between the integrality of $\mathrm{Cay}(U_{6n}, S)$ and the Boolean algebra of cyclic groups
  5. Conclusion

Key Result

Lemma 2.1

(AbdollahiV2009) Let $\omega=\exp(\frac{\pi \imath}{n})$, where $\imath^2=-1$. Then

Theorems & Definitions (29)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 19 more