Perfect State Transfer on Cayley graphs over finite abelian groups

Hiranmoy Pal, Bikash Bhattacharjya

Published 2016-01-28Version 1

We study perfect state transfer (with respect to adjacency matrix) on Cayley graph over finite abelian groups. We find that if a Cayley graph over a finite abelian group exhibits perfect state transfer then it must have integral spectrum. A gcd-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. It is well known that all gcd-graphs have integral spectrum. We find a sufficient condition for a gcd-graph to have periodicity and perfect state transfer at $\frac{\pi}{2}$. Also we find a necessary and sufficient condition for a particular class of gcd-graphs to be periodic at $\pi$. Using that we find that there are some gcd-graphs not exhibiting perfect state transfer at $\frac{\pi}{2^{k}}$ for all positive integers $k$.