One can define the notion of length spectrum for a simple regular periodic graph via counting the orbits of closed reduced cycles under an action of a discrete group of automorphisms. We prove that this length spectrum satisfies an analogue of the Multiplicity one property. We show that if all but finitely many cycles in two simple regular periodic graphs have equal lengths, then all the 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. 查看全文>>