Integral Cayley graphs defined by greatest common divisors (Q540092)

Summary: An undirected graph is called integral, if all of its eigenvalues are integers. Let \(\Gamma =Z_{m_1}\otimes \dots \otimes Z_{m_r}\) be an abelian group represented as the direct product of cyclic groups \(Z_{m_i}\) of order \(m_i\) such that all greatest common divisors \(\gcd(m_i,m_j)\leq 2\) for \(i\neq j\). We prove that a Cayley graph \(Cay(\Gamma,S)\) over \(\Gamma\) is integral, if and only if \(S\subseteq \Gamma\) belongs to the the Boolean algebra \(B(\Gamma)\) generated by the subgroups of \(\Gamma\). It is also shown that every \(S\in B(\Gamma)\) can be characterized by greatest common divisors.