Quantum State Transfer on Neighborhood Corona of Two Graphs

TL;DR

This paper investigates quantum state transfer properties on the neighborhood corona of two graphs, providing conditions for non-periodicity, absence of perfect state transfer, and the presence of pretty good state transfer.

Contribution

It introduces new conditions for quantum state transfer phenomena on neighborhood corona graphs, expanding understanding of their quantum communication capabilities.

Findings

Identifies conditions under which the neighborhood corona is not periodic.

Provides criteria for the absence of perfect state transfer.

Establishes sufficient conditions for pretty good state transfer.

Abstract

Given two graphs $G_{1}$ of order $n_{1}$ and $G_{2}$, the neighborhood corona of $G_{1}$ and $G_{2}$, denoted by $G_{1}\bigstar G_{2}$, is the graph obtained by taking one copy of $G_{1}$ and taking $n_{1}$ copies of $G_{2}$, in the meanwhile, linking all the neighbors of the $i$-th vertex of $G_{1}$ with all vertices of the $i$-th copy of $G_{2}$. In our work, we give some conditions that $G_{1}\bigstar G_{2}$ is not periodic. Furthermore, we demonstrate some sufficient conditions for $G_{1}\bigstar G_{2}$ having no perfect state transfer. Some examples are provided to explain our results. In addition, for the reason that the graph admitting perfect state transfer is rare, we also consider pretty good state transfer on neighborhood corona of two graphs. We show some sufficient conditions for $G_{1}\bigstar G_{2}$ admitting pretty good state transfer.