On The Length Spectra of Simple Regular Periodic Graphs

One can define the notion of primitive length spectrum for a simple regular periodic graph via counting the orbits of closed reduced primitive cycles under an action of a discrete group of automorphisms. We prove that this primitive length spectrum satisfies an analogue of the `Multiplicity one' property. We show that if all but finitely many primitive cycles in two simple regular periodic graphs have equal lengths, then all the primitive cycles have equal lengths. This is a graph-theoretic analogue of a similar theorem in the context of geodesics on hyperbolic spaces. We also prove, in the context of actions of finitely generated abelian groups on a graph, that if the adjacency operators for two actions of such a group on a graph are similar, then corresponding periodic graphs are length isospectral.