On Matrix Product Factorization of Cayley graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908721780621312 |
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| author | Herman, Allen W. Miraftab, Bobby |
| author_facet | Herman, Allen W. Miraftab, Bobby |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17110 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Matrix Product Factorization of Cayley graphs Herman, Allen W. Miraftab, Bobby Combinatorics Group Theory 20K01, 05C50 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. |
| title | On Matrix Product Factorization of Cayley graphs |
| topic | Combinatorics Group Theory 20K01, 05C50 |
| url | https://arxiv.org/abs/2512.17110 |