On groups all of whose undirected Cayley graphs of bounded valency are integral (Q490258)

Summary: 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 \textit{W. Klotz} and \textit{T. Sander} [ibid. 17, No. 1, Research Paper R81, 13 p. (2010; Zbl 1189.05074)] in the abelian case, and by \textit{A. Abdollahi} and \textit{M. Jazaeri} [Eur. J. Comb. 38, 102--109 (2014; Zbl 1282.05063)] and independently by \textit{A. Ahmady} et al. [SIAM J. Discrete Math. 28, No. 2, 685--701 (2014; Zbl 1298.05155)] 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 \(|S| \leq k\). It will be proved that \(\mathcal{G}_k\) consists of the Cayley integral groups if \(k \geq 6;\) 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 \(n \geq 1\).