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

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) \ {v} = N (v) \ {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 (LPST) at π 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 be- tween 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 (LPGST) between twin vertices.