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

On The Length Spectra of Simple Regular Periodic Graphs