Laplacian State Transfer in Coronas

TL;DR

This paper investigates Laplacian state transfer properties in corona product graphs, proving the absence of perfect state transfer but demonstrating the existence of pretty good state transfer under certain conditions, thus providing new examples of such phenomena.

Contribution

It establishes the first known examples of graphs with Laplacian pretty good state transfer and extends existing results to the corona product setting.

Findings

Corona product graphs lack Laplacian perfect state transfer when the first graph has ≥2 vertices.

Corona product graphs can exhibit Laplacian pretty good state transfer under mild conditions.

Cocktail party graphs with a single vertex exhibit Laplacian pretty good state transfer despite odd cocktail party graphs lacking perfect transfer.

Abstract

We prove that the corona product of two graphs has no Laplacian perfect state transfer whenever the first graph has at least two vertices. This complements a result of Coutinho and Liu who showed that no tree of size greater than two has Laplacian perfect state transfer. In contrast, we prove that the corona product of two graphs exhibits Laplacian pretty good state transfer, under some mild conditions. This provides the first known examples of families of graphs with Laplacian pretty good state transfer. Our result extends of the work of Fan and Godsil on double stars to the Laplacian setting. Moreover, we also show that the corona product of any cocktail party graph with a single vertex graph has Laplacian pretty good state transfer, even though odd cocktail party graphs have no perfect state transfer.