Real state transfer on edge perturbed graphs with generalized clusters

Abstract: We study the existence of real state transfer in edge-perturbed graphs containing generalized clusters, where the Hamiltonian is taken to be either the adjacency matrix, the Laplacian matrix, or the signless Laplacian matrix of an associated weighted graph. This framework provides a unified approach for constructing new graphs that exhibit perfect real state transfer, building on known examples with this property. A central observation is that the evolution of certain quantum states depends solely on the local structure of the underlying graph. In particular, we construct an infinite family of graphs with maximum valency five that exhibit perfect pair state transfer-under each of the aforementioned matrices-between the same pair of states at the same time, despite being non-regular. Additionally, we identify instances of perfect pair state transfer in edge-perturbed graphs, including complete graphs, complete bipartite graphs, blow-up graphs, and related structures. We also examine various graph operations-such as the sequential join, complement, Cartesian product, lexicographic product, and corona product-that generate new families of graphs exhibiting perfect real state transfer with respect to all three choices of the Hamiltonian.