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⁡(itA),t∈RH(t):=exp⁡(itA),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∈Rτ∈R such that the uv  -th entry of H(τ)H(τ) has unit modulus. The graph G   is said to be periodic at τ∈Rτ∈R if there exist γ∈Cγ∈C with |γ|=1|γ|=1 such that H(τ)=γIH(τ)=γI, 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.