Perfect state transfer on graphs with clusters

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.