# Journal of the Ramanujan Mathematical Society

Volume 35, Issue 2, June 2020 pp. 139–147.

On the length spectra of simple regular periodic graphs

**Authors**:
Chandrasheel Bhagwat and Ayesha Fatima

**Author institution:**Indian Institute of Science Education and Research, Pune, India

**Summary: **
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 ([GIL]). 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 ([BR]). We also prove, in the
context of actions of finitely generated abelian groups on a
graph, that if the adjacency operators ([Clair]) for two actions
of such a group on a graph are similar, then corresponding
periodic graphs are length isospectral.

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