Laplacian Pair State Transfer on Total Graphs

TL;DR

This paper studies Laplacian pair state transfer properties on total graphs derived from regular graphs, establishing conditions under which perfect or pretty good transfer occurs or fails, and providing infinite examples of such graphs.

Contribution

It characterizes when total graphs of regular graphs exhibit Laplacian perfect or pretty good pair state transfer, introducing new conditions and infinite classes of graphs with these properties.

Findings

Total graphs of certain regular graphs do not have Laplacian perfect pair state transfer.

Complete graphs with more than three vertices have total graphs that lack perfect pair state transfer.

Under mild conditions, total graphs can exhibit Laplacian pretty good pair state transfer.

Abstract

The total graph of a graph $G$, denoted $\mathcal{T}(G)$, is defined as the graph whose vertex set is the union of the vertex set of $G$ and the edge set of $G$ such that two vertices of $\mathcal{T}(G)$ are adjacent if the corresponding elements of $G$ are adjacent or incident. In this paper, we investigate Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on $\mathcal{T}(G)$, where $G$ is an $r$-regular graph. We prove that if $r>2$ and $r+1$ is not a Laplacian eigenvalue of $G$, then $\mathcal{T}(G)$ fails to exhibit Laplacian perfect pair state transfer. We also prove that if $G$ is a complete graph on more than three vertices, then $\mathcal{T}(G)$ fails to exhibit Laplacian perfect pair state transfer. Further, we prove that under some mild conditions, $\mathcal{T}(G)$ exhibits Laplacian pretty good pair state transfer, where $G$ is an $r$-regular graph such that $r>2$ and $r+1$ is not a Laplacian eigenvalue of $G$. We use these conditions to obtain infinitely many total graphs exhibiting Laplacian pretty good pair state transfer.