Laplacian State Transfer on Graphs with an Edge Perturbation Between Twin Vertices

TL;DR

This paper explores quantum state transfer on graphs with Laplacian matrices, showing how edge perturbations between twin vertices can induce perfect or pretty good state transfer, with specific results on complete and circulant graphs.

Contribution

It introduces new conditions under which Laplacian perfect and pretty good state transfer occur due to edge perturbations between twin vertices in various graph classes.

Findings

Edge removal in complete graphs with vertices divisible by 4 yields LPST at π/2.

Laplacian integral graphs with twin vertices exhibit LPST upon edge perturbation.

Certain circulant graphs show LPGST and LPST due to edge perturbations.

Abstract

We consider quantum state transfer relative to the Laplacian matrix of a graph. Let $N(u)$ denote the set of all neighbors of a vertex $u$ in a graph $G$. A pair of vertices $u$ and $v$ are called twin vertices of $G$ provided $N(u)\setminus\{v \}=N(v)\setminus\{u \}$. We investigate the existence of quantum state transfer between a pair of twin vertices in a graph when the edge between the vertices is perturbed. We find that removal of any set of pairwise non-adjacent edges from a complete graph with a number of vertices divisible by $4$ results Laplacian perfect state transfer (or LPST) at $\frac{\pi}{2}$ between the end vertices of every edge removed. Further, we show that all Laplacian integral graphs with a pair of twin vertices exhibit LPST when the edge between the vertices is perturbed. In contrast, we conclude that LPST can be achieved in every complete graph between the end vertices of any number of suitably perturbed non-adjacent edges. The results are further generalized to obtain a family of edge perturbed circulant graphs exhibiting Laplacian pretty good state transfer (or LPGST) between twin vertices. A subfamily of which is also identified to admit LPST at $\frac{\pi}{2}$.