Pretty Good State Transfer on Some NEPS

TL;DR

This paper investigates pretty good state transfer (PGST) in NEPS graphs, especially those based on a path of three vertices, and explores conditions under which PGST occurs or fails, including Cartesian products.

Contribution

It characterizes when NEPS of a 3-vertex path exhibit PGST and shows that certain NEPS do not have perfect state transfer but do admit PGST, also analyzing PGST in Cartesian products.

Findings

NEPS of P3 with certain basis do not have perfect state transfer

These NEPS admit PGST under specific conditions

Graphs can have PGST from one vertex to multiple vertices

Abstract

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H_{A}(t):=\exp{(-itA)},\;t\in\Rl$. We say that the graph $G$ admits perfect state transfer between the verteices $u$ and $v$ at $\tau\in\Rl$ if the $uv$-th entry of $H_{A}(\tau)$ has unit modulus. Perfect state transfer is a rare phenomena so we consider an approximation called pretty good state transfer. We find that NEPS (Non-complete Extended P-Sum) of the path on three vertices with basis containing tuples with hamming weights of both parities do not exhibit perfect state transfer. But these NEPS admit pretty good state transfer with an additional condition. Further we investigate pretty good state transfer on Cartesian product of graphs and we find that a graph can have PGST from a vertex $u$ to two different vertices $v$ and $w$.