Non-isomorphic Cayley Graphs with Same Random Walk Distributions

TL;DR

This work constructs infinite families of triples $(G,S_1,S_2)$ where ${\tt Cay}(G,S_1)$ and ${\tt Cay}(G,S_2)$ are non-isomorphic yet yield identical simple random-walk distributions at each step under a vertex correspondence. The key idea is a common covering graph $Y$ built from sliding block puzzles on theta graphs $\theta_{a,b}$, with $X(G,S_i)$ realized as quotients of $Y$ by subgroups generated by automorphisms $\rho$ and $\psi$. Random-walk equality follows from a path-bijection induced by the covering structure, while spectral distinctness is captured through a detailed abelian-covering zeta-function analysis, which also yields a two-way splitting of the spectrum into $A$ and $B_i$ with $B_2=-B_1$. Collectively, the results demonstrate that non-isomorphic Cayley graphs can share identical random-walk dynamics while exhibiting structured, mirrored spectral sets, offering a new lens on graph indistinguishability via stochastic processes and zeta-function techniques.

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.

Figures (9)

• Figure 1: Cayley graphs the symmetric group ${\mathfrak S}_3$ generated by $S_1$ (left) and $S_2$ (right)
• Figure 2: Another example of the Cayley graphs simple random walks on which have the same total variation distance from the uniform distribution. All edges are bi-directed.
• Figure 3: The graph $\theta_0$
• Figure 4: $\theta_{2,3}$
• Figure 5: Images of $\rho$ (left) and $\psi$ (right) for $\theta_{2,3}$

Theorems & Definitions (29)

• Theorem 1
• Definition 1
• Lemma 1
• proof
• Theorem 2
• proof
• Remark 1
• Example 1
• Lemma 2
• Lemma 3