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 The graph G has perfect state transfer (PST) from a vertex u to another vertex v if there exist such that the uvth entry of has unit modulus. In case when , we say that G is periodic at the vertex u at time . 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 . 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 . Using this we characterize a class of gcd-graphs not exhibiting PST at for all positive integers k.

Keywords:

• Perfect state transfer
• Cayley graph
• Kronecker product of graphs

AMS Subject Classifications:

• 05C12
• 05C50

Acknowledgements

We thank the anonymous reviewer(s) for the useful comments in an earlier manuscript.

Notes

No potential conflict of interest was reported by the authors.