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Let G be a graph with adjacency matrix A . The transition matrix corresponding to G is defined by H ( t ) : = exp ⁡ ( i t A ) , t ∈ R . The graph G is said to have perfect state transfer (PST) from a vertex u to another vertex v , if there exist τ ∈ R such that the uv -th entry of H ( τ ) has unit modulus. The graph G is said to be periodic at τ ∈ R if there exist γ ∈ C with | γ | = 1 such that H ( τ ) = γ I , where I is the identity matrix. A gcd -graph is a Cayley graph over a finite abelian group defined by greatest common divisors. In this paper, we construct classes of gcd -graphs having periodicity and perfect state transfer. [ABSTRACT FROM AUTHOR]