$q$-Laplacian State Transfer on Graphs with Involutions

We study the existence of state transfer with respect to the $q$-Laplacian matrix of a graph equipped with a non-trivial involution. We show that the occurrence of perfect state transfer between certain pair (or plus) states in such a graph is equivalent to the existence of vertex state transfer in a subgraph induced by the involution with potentials. This yields infinite families of trees with potentials and unicyclic graphs of maximum degree three that exhibit perfect pair state transfer. In particular, we investigate vertex and pair state transfer in edge-perturbed complete bipartite graphs, cycles, and paths with potentials only at the end vertices.