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On Matrix Product Factorization of Cayley graphs

Allen W. Herman, Bobby Miraftab

TL;DR

This work provides a precise algebraic and combinatorial framework for when Cayley adjacencies factor as matrix products of Cayley adjacencies on the same group. The core criterion, $A(G;U)=A(G;S)A(G;T)$, is equivalent to a unique product representation $U=ST$ and a corresponding group-algebra equation $(\sum_{s\in S}s)(\sum_{t\in T}t)=\sum_{u\in U}u$, with specialized character-theoretic and Sidon-pair interpretations in abelian settings. The authors derive explicit results for abelian and cyclic groups via convolution and mask polynomials, extend factorization theory to dihedral groups through a structural correspondence with $\mathbb{Z}_{2n}$, and discuss automorphism-invariance and equivalence of factorizations. They also demonstrate that not all Cayley graphs admit factorizations and outline broad avenues for generalization to association schemes and related combinatorial structures. Overall, the paper bridges graph-factorization questions with representation theory, convolution, and number-theoretic tools, enriching both theory and constructive methods for Cayley-by-Cayley factorizations.

Abstract

We study when the adjacency matrix of a Cayley graph factors as the product of two adjacency matrices of Cayley graphs. Let $G$ be a finite group and let $U\subseteq G\setminus \{e\}$ be symmetric. Writing $A(G;U)$ for the adjacency matrix of the Cayley graph of $G$ with respect to $U$, we prove that for symmetric subsets $S,T,U$ of $G\setminus \{e\}$, $A(G;U)=A(G;S)\,A(G;T)$ if and only if $U=ST$ and each $u\in U$ has a unique representation $u=st$, equivalently $\bigl(\sum_{s\in S}s\bigr)\bigl(\sum_{t\in T}t\bigr)=\sum_{u\in U}u$ in the group algebra. When $S,T,U$ are unions of conjugacy classes, this is characterized character-theoretically by $χ(U)=χ(S)χ(T)/χ(1)$ for all $χ\in\mathrm{Irr}(G)$. In addition, for abelian groups, we identify $A(G;S)A(G;T)$ with the $0\!-\!1$ convolution $\mathbf{1}_S*\mathbf{1}_T$, so factorability is equivalent to $(S,T)$ being a Sidon pair, i.e., $(S-S)\cap(T-T)=\{0\}$. For cyclic groups, we reformulate factorability via mask polynomials and reduce to prime-power components using the Chinese Remainder Theorem. We also analyze dihedral groups $D_{2n}$, presenting infinite families of factorable generating sets, and give explicit constructions of subsets whose Cayley graphs do and do not admit such factorizations.

On Matrix Product Factorization of Cayley graphs

TL;DR

This work provides a precise algebraic and combinatorial framework for when Cayley adjacencies factor as matrix products of Cayley adjacencies on the same group. The core criterion, , is equivalent to a unique product representation and a corresponding group-algebra equation , with specialized character-theoretic and Sidon-pair interpretations in abelian settings. The authors derive explicit results for abelian and cyclic groups via convolution and mask polynomials, extend factorization theory to dihedral groups through a structural correspondence with , and discuss automorphism-invariance and equivalence of factorizations. They also demonstrate that not all Cayley graphs admit factorizations and outline broad avenues for generalization to association schemes and related combinatorial structures. Overall, the paper bridges graph-factorization questions with representation theory, convolution, and number-theoretic tools, enriching both theory and constructive methods for Cayley-by-Cayley factorizations.

Abstract

We study when the adjacency matrix of a Cayley graph factors as the product of two adjacency matrices of Cayley graphs. Let be a finite group and let be symmetric. Writing for the adjacency matrix of the Cayley graph of with respect to , we prove that for symmetric subsets of , if and only if and each has a unique representation , equivalently in the group algebra. When are unions of conjugacy classes, this is characterized character-theoretically by for all . In addition, for abelian groups, we identify with the convolution , so factorability is equivalent to being a Sidon pair, i.e., . For cyclic groups, we reformulate factorability via mask polynomials and reduce to prime-power components using the Chinese Remainder Theorem. We also analyze dihedral groups , presenting infinite families of factorable generating sets, and give explicit constructions of subsets whose Cayley graphs do and do not admit such factorizations.

Paper Structure

This paper contains 6 sections, 30 theorems, 68 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Cayley Factorization of General Finite Groups
  3. Cayley Factorization on Abelian Groups
  4. Cayley Factorization of Cyclic Groups
  5. Cayley Factorization of Dihedral Groups
  6. Further Research

Key Result

Lemma 1

Let $G,G'$ be finite groups and let $\varphi\colon G\to G'$ be an isomorphism. Let $P_\varphi$ be the $n\times n$ permutation matrix with $(P_\varphi)_{g,\varphi(g)}=1$ for all $g\in G$. Then for every subset $X\subseteq G'$,

Figures (1)

  • Figure 1: The Cayley graph $\mathop{\mathsf{Cay}}(D_8,\{sr^3,s,sr^2,sr\})$ has a factorization to $\mathop{\mathsf{Cay}}(D_{8};\{r,r^3\})$ and $\mathop{\mathsf{Cay}}(D_8;\{s,sr\})$.

Theorems & Definitions (61)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2.1
  • proof
  • ...and 51 more