Perfect state transfer, integral circulants and join of graphs

TL;DR

This paper introduces new graph families that demonstrate quantum perfect state transfer, utilizing join operations, circulant generalizations, and Cartesian products, expanding the known classes of such graphs with specific constructions and answering open questions.

Contribution

The paper constructs new integral circulant and regular graphs with perfect state transfer, including non-periodic double-cone graphs, based on join operations and generalizations, extending previous results.

Findings

Integral circulant ICG_{n} has perfect state transfer for specific parameters.

Double-cone graphs exhibit perfect state transfer despite being non-periodic.

New constructions answer open questions in quantum graph theory.

Abstract

We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Ba\v{s}i\'{c} et al \cite{bps09,bp09} and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $\textsc{ICG}_{n}(\{2,n/2^{b}\} \cup Q)$ has perfect state transfer, where $b \in \{1,2\}$, $n$ is a multiple of 16 and $Q$ is a subset of the odd divisors of $n$. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil \cite{godsil08}.