On matrix product factorization of Cayley graphs

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⊆ G \ 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 \, $A(G;U)=A(G;S)\,A(G;T)$ if and only if $U=ST$ and each u∈ U has a unique representation $u=st$, equivalently (∑s∈ Ss)(∑t∈ Tt)=∑u∈ Uu 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(G). In addition, for abelian groups, we identify $A(G;S)A(G;T)$ with the $0\!-\!1$ convolution 1S*1T, so factorability is equivalent to $(S,T)$ being a Sidon pair, i.e., (S-S)∩(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 D₂ₙ, presenting infinite families of factorable generating sets, and give explicit constructions of subsets whose Cayley graphs do and do not admit such factorizations.