Non-isomorphic Cayley Graphs with Same Random Walk Distributions

TL;DR

This paper constructs infinite families of non-isomorphic Cayley graphs that have identical random walk distributions and specific spectral properties, challenging assumptions about graph isomorphism and spectral invariants.

Contribution

It introduces a novel method to generate non-isomorphic Cayley graphs with identical random walk distributions and particular spectral decompositions.

Findings

Constructed infinite families of such Cayley graphs.

Demonstrated non-isomorphism despite identical random walk distributions.

Showed spectral decompositions with symmetric properties.

Abstract

We construct an infinite family of triples (G,S1, S2) each consisting of a group G and a pair (S1, S2) of distinct subsets of G with the following properties. i The two Cayley graphs Cay(G, S1) and Cay(G,S2) are non-isomorphic. ii The distributions of the simple random walks on Cay(G,S1) and Cay(G,S2) are the same if one takes an appropriate correspondence between the two vertex sets at each step. iii The spectral set of Cay(G, Si) is decomposed into a disjoint union of two subsets A and B_i of the equal size which satisfies B1 = -B2.