The signless Laplacian state transfer in Q-graph

TL;DR

This paper investigates the conditions under which the $ ext{Q}$-graph of a regular graph exhibits signless Laplacian state transfer phenomena, revealing that integer eigenvalues prevent perfect state transfer and providing conditions for pretty good state transfer.

Contribution

It introduces new criteria for signless Laplacian state transfer in $ ext{Q}$-graphs, linking eigenvalue properties of the original graph to transfer phenomena in the derived graph.

Findings

Regular graphs with integer eigenvalues lack perfect state transfer in their $ ext{Q}$-graphs.

Sufficient conditions are provided for pretty good state transfer in $ ext{Q}$-graphs.

The study connects spectral properties of the original graph to quantum state transfer in derived graphs.

Abstract

The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and adding a new edge between two new vertices which lie on adjacent edges of $G$. In this paper, we consider to study the existence of the signless Laplacian perfect state transfer and signless Laplacian pretty good state transfer in $\mathcal{Q}$-graphs of graphs. We show that, if all the signless Laplacian eigenvalues of a regular graph $G$ are integers, then the $\mathcal{Q}$-graph of $G$ has no signless Laplacian perfect state transfer. We also give a sufficient condition that the $\mathcal{Q}$-graph of a regular graph has signless Laplacian pretty good state transfer when $G$ has signless Laplacian perfect state transfer between two specific vertices.