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 $Dic(E_{3^n} \times \mathbb{Z}_6),$ where $n \ge 1$.
@article{10_37236_4238,
author = {Istv\'an Est\'elyi and Istv\'an Kov\'acs},
title = {On groups all of whose undirected {Cayley} graphs of bounded valency are integral},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {4},
doi = {10.37236/4238},
zbl = {1305.05100},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4238/}
}
TY - JOUR
AU - István Estélyi
AU - István Kovács
TI - On groups all of whose undirected Cayley graphs of bounded valency are integral
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/4238/
DO - 10.37236/4238
ID - 10_37236_4238
ER -
%0 Journal Article
%A István Estélyi
%A István Kovács
%T On groups all of whose undirected Cayley graphs of bounded valency are integral
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/4238/
%R 10.37236/4238
%F 10_37236_4238
István Estélyi; István Kovács. On groups all of whose undirected Cayley graphs of bounded valency are integral. The electronic journal of combinatorics, Tome 21 (2014) no. 4. doi: 10.37236/4238
