On Matrix Product Factorization of Cayley graphs

TL;DR

This paper characterizes when the adjacency matrix of a Cayley graph factors into the product of two such matrices, linking algebraic, character-theoretic, and combinatorial conditions, with specific results for abelian and dihedral groups.

Contribution

It provides a comprehensive characterization of matrix factorization for Cayley graphs, connecting group algebra, character theory, and combinatorics, including new results for abelian and dihedral groups.

Findings

Factorization occurs iff $U=ST$ with unique representations in the group algebra.

Character-theoretic condition: $ ext{chi}(U)= ext{chi}(S) ext{chi}(T)/ ext{chi}(1)$ for all irreducible characters.

For abelian groups, factorability is equivalent to the pair being a Sidon pair.

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 $\chi(U)=\chi(S)\chi(T)/\chi(1)$ for all $\chi\in\mathrm{Irr}(G)$. 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.