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On the Isomorphism Problem of Cayley Graphs of Graph Products

Marjory Mwanza

TL;DR

This work studies when Cayley graphs of graph products $ ext{Cay}(G_ ,S)$ are isomorphic to those of another graph product $ ext{Cay}(H_ ,T)$ by reducing the problem to vertex-group data. Using Green's Normal Form for graph products and a vertex-wise Cayley-graph isomorphism $f_v$ for each vertex, the authors construct a global isomorphism and show $ ext{Cay}(G_ ,S)\u2227cong ext{Cay}(H_ ,T)$ when $S=igcup S_v$ and $T=igcup T_v$. They also prove that finite-graph products with vertex groups of equal orders have isomorphic Cayley graphs even when the groups themselves are not isomorphic, and they illustrate non-isomorphic pairs that share isomorphic Cayley graphs using maximal finite-subgroup analysis. The results contribute a precise isomorphism criterion for Cayley graphs of graph products and demonstrate nontrivial examples where isomorphic Cayley graphs do not force group isomorphism, highlighting both rigidity and flexibility in this setting.

Abstract

We investigate Cayley graphs of graph products by showing that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs.

On the Isomorphism Problem of Cayley Graphs of Graph Products

TL;DR

This work studies when Cayley graphs of graph products are isomorphic to those of another graph product by reducing the problem to vertex-group data. Using Green's Normal Form for graph products and a vertex-wise Cayley-graph isomorphism for each vertex, the authors construct a global isomorphism and show when and . They also prove that finite-graph products with vertex groups of equal orders have isomorphic Cayley graphs even when the groups themselves are not isomorphic, and they illustrate non-isomorphic pairs that share isomorphic Cayley graphs using maximal finite-subgroup analysis. The results contribute a precise isomorphism criterion for Cayley graphs of graph products and demonstrate nontrivial examples where isomorphic Cayley graphs do not force group isomorphism, highlighting both rigidity and flexibility in this setting.

Abstract

We investigate Cayley graphs of graph products by showing that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs.

Paper Structure

This paper contains 3 sections, 8 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graph products
  3. Cayley graphs

Key Result

Corollary 1.1

Let $\Gamma$ be a finite graph and let $G_\Gamma$ and $H_\Gamma$ be graph products of finite groups $G_v$ and $H_v$, with $|G_v|=|H_v|$. Then for suitable choices of vertex groups, $G_{\Gamma}\not\cong H_{\Gamma}$, while $\mathop{\mathrm{Cay}}\nolimits(G_\Gamma,S_1)\cong \mathop{\mathrm{Cay}}\nolimi

Figures (1)

  • Figure 1: Cayley graph of $\mathbb{Z}_4 * S_3$ (The blue edges represent $S_3$ while the red ones represent $\mathbb{Z}_4$.)

Theorems & Definitions (16)

  • Corollary 1.1
  • Theorem 2.1: The Normal Form Theorem for Graph Products
  • Lemma 2.2
  • proof
  • Remark 3.1: Elementary properties of Cayley graphs
  • Example 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 6 more