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On groups all of whose undirected Cayley graphs of bounded valency are integral

Authors
  • Estélyi, István
  • Kovács, István
Type
Preprint
Publication Date
Mar 29, 2014
Submission Date
Mar 29, 2014
Identifiers
arXiv ID: 1403.7602
Source
arXiv
License
Yellow
External links

Abstract

A finite group $G$ is called Cayley integral if all undirected Cayley graphs over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell and Mohar in the non-abelian case. In this paper we generalize this class of groups by introducing the class $\mathcal{G}_k$ of finite groups $G$ for which all graphs $\mathrm{Cay}(G,S)$ are integral if $|S| \le k$. It will be proved that $\mathcal{G}_k$ consists of the Cayley integral groups if $k \ge 6;$ and the classes $\mathcal{G}_4$ and $\mathcal{G}_5$ are equal, and consist of:\ (1) the Cayley integral groups, (2) the generalized dicyclic groups $\mathrm{Dic}(E_{3^n} \times \mathbb{Z}_6),$ where $n \ge 1$.

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