Perfect State Transfer on gcd-graphs

Abstract

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ is denoted by $H(t)$ and it is defined by $H(t):=\exp{\left(itA\right)},\;t\in\mathbb{R}.$ The graph $G$ has perfect state transfer (PST) from a vertex $u$ to another vertex $v$ if there exist $\tau\left(\neq0\right)\in\mathbb{R}$ such that the $uv$-th entry of $H(\tau)$ has unit modulus. In case when $u=v$, we say that $G$ is periodic at the vertex $u$ at time $\tau$. The graph $G$ is said to be periodic if it is periodic at all vertices at the same time. A gcd-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. We establish a sufficient condition for a gcd-graph to have periodicity and PST at $\frac{\pi}{2}$. Using this we deduce that there exists gcd-graph having PST over an abelian group of order divisible by $4$. Also we find a necessary and sufficient condition for a class of gcd-graphs to be periodic at $\pi$. Using this we characterize a class of gcd-graphs not exhibiting PST at $\frac{\pi}{2^{k}}$ for all positive integers $k$.