Integral Cayley graphs over Abelian groups (Q976740)

Summary: Let \(\Gamma\) be a finite, additive group, \(S \subseteq \Gamma\), \(0\not\in S\), \(-S=\{-s:s\in S\}=S\). The undirected Cayley graph Cay\((\Gamma,S)\) has vertex set \(\Gamma\) and edge set \(\{\{a,b\}:a,b\in \Gamma\), \(a-b \in S\}\). A graph is called integral, if all of its eigenvalues are integers. For an abelian group \(\Gamma\) we show that Cay\((\Gamma,S)\) is integral, if \(S\) belongs to the Boolean algebra \(B(\Gamma)\) generated by the subgroups of \(\Gamma\). The converse is proven for cyclic groups. A finite group \(\Gamma\) is called Cayley integral, if every undirected Cayley graph over \(\Gamma\) is integral. We determine all abelian Cayley integral groups.