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We consider quantum state transfer relative to the Laplacian matrix of agraph. 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 ofquantum state transfer between a pair of twin vertices in a graph when the edgebetween the vertices is perturbed. We find that removal of any set of pairwisenon-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 allLaplacian integral graphs with a pair of twin vertices exhibit LPST when theedge between the vertices is perturbed. In contrast, we conclude that LPST canbe achieved in every complete graph between the end vertices of any number ofsuitably perturbed non-adjacent edges. The results are further generalized toobtain a family of edge perturbed circulant graphs exhibiting Laplacian prettygood state transfer (or LPGST) between twin vertices. A subfamily of which isalso identified to admit LPST at $\frac{\pi}{2}$.

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