Speaker
Istvan Estelyi
(University of Primorska)
Description
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
Klotz and Sander [1] in the abelian case, and by Abdollahi and Jazaeri [2], and independently by
Ahmady, Bell and Mohar [3] in the non-abelian case. In this talk we will generalize this class of groups by introducing the class $\mathcal{G}_k$ of finite groups $G$ for which all graphs $\mbox{Cay}(G,S)$ 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 $\mbox{Dic}(E_{3^n} \times \mathbb{Z}_6),$ where $n \ge 1$.
\bigskip
\noindent
[1] W. Klotz, T. Sander, Integral Cayley graphs over abelian groups, {\it Electronic J. Combin.} {\bf 17} (2010), \#R81.
\smallskip
\noindent
[2] A. Abdollahi, M. Jazaeri, Groups all of whose undirected Cayley graphs are integral, {\it Europ. J. Combin.} {\bf 38} (2014), 102--109.
\smallskip
\noindent
[3] A. Ahmady, J. P. Bell, B. Mohar, Integral Cayley graphs and groups, preprint arXiv: \\ 1209.5126v1 [math.CO] 2013.
\smallskip
\noindent
[4] I. Est\'elyi, I. Kov\'acs, On groups all of whose undirected Cayley graphs of boun\-ded valency are integral, preprint arXiv:1403.7602 [math.GR] 2014.
Primary author
Istvan Estelyi
(University of Primorska)
