Perfect state transfer on graphs with clusters

TL;DR

This paper introduces a unified method for constructing graphs with perfect state transfer between specific states, applicable to various matrices and graph types, including non-regular graphs, and explores conditions and graph families exhibiting this property.

Contribution

It presents a unified approach for perfect state transfer on graphs with clusters across different matrices, including non-regular graphs, and constructs infinite families with this property.

Findings

Existence of infinitely many connected graphs with maximum valency k≥5 exhibiting perfect state transfer.

Sufficient conditions for pair state transfer in edge-perturbed graphs like complete and bipartite graphs.

Generation of new graph families with perfect state transfer using graph products.

Abstract

Using graphs with clusters, we provide a unified approach for constructing graphs with pair state transfer-relative to the adjacency, Laplacian, and signless Laplacian matrix-between the same pair of states at the same time, despite being non-regular. We show that for each $k\geq 5$, there are infinitely many connected graphs with maximum valency $k$ admitting this property. This framework also aids in establishing sufficient conditions for pair state transfer in edge-perturbed graphs, including complete graphs and complete bipartite graphs. Furthermore, we utilize graph products to generate new infinite families of graphs with the above property.