A class of gcd-graphs having Perfect State Transfer

Let \(G\) be a graph with adjacency matrix \(A\). The transition matrix corresponding to \(G\) is defined by \(H(t):=\exp{\left(itA\right)}\), \(t\in\Rl\). The graph \(G\) is said to have perfect state transfer (PST) from a vertex \(u\) to another vertex \(v\), if there exist \(\tau\in\Rl\) such that the \(uv\)-th entry of \(H(\tau)\) has unit modulus. The graph \(G\) is said to be periodic at \(\tau\in\Rl\) if there exist \(\gamma\in\Cl\) with \(|\gamma|=1\) such that \(H(\tau)=\gamma I\), where \(I\) is the identity matrix. A \(\mathit{gcd}\)-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. In this paper, we construct classes of \(\mathit{gcd}\)-graphs having periodicity and perfect state transfer.