On Groups all of whose Undirected Cayley Graphs of Bounded Valency are Integral

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 nonabelian case. In this paper we generalize this class of groups by introducing the class Gk of finite groups G for which all graphs Cay(G,S) are integral if |S| 6 k. It will be proved that Gk consists of the Cayley integral groups if k > 6; and the classes G4 and G5 are equal, and consist of: (1) the Cayley integral groups, (2) the generalized dicyclic groups Dic(E3n × Z6), where n > 1.