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Zeon and Idem-Clifford Formulations of Hypergraph Problems

Samuel Ewing, G. Stacey Staples

TL;DR

This work extends zeon ($\mathfrak{Z}$) and idem-Clifford ($\mathfrak{I}$) algebras from graphs to finite hypergraphs to address core combinatorial problems. It introduces the hypergraph nilpotent adjacency matrix $\Omega = XZ$ and constructs representations $\Gamma_H$, $\Phi_H$, and $\sigma$ to count walks, trails, independent sets, (weak/strong) cliques, matchings, and minimum transversals, using symbolic calculations. The authors provide concrete algebraic formulations for $k$-paths, $k$-matchings, $j$-intersecting matchings, and minimum transversals, and reformulate Ryser's and Frankl's conjectures within this framework, illustrating powerful open-problem translations. The approach enables symbolic-computation workflows and points to future work on hypergraph colorings and connections to zeon-based Laplacians, broadening the applicability of algebraic enumeration to complex relational structures.

Abstract

Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. In contrast to an ordinary graph, in which each edge connects exactly two vertices, an edge (or, "hyperedge") can join any number of vertices in a hypergraph. In game theory, hypergraphs are called simple games. Hypergraphs have been used for problems in biology, chemistry, image processing, wireless networks, and more. In the current work, zeon ("nil-Clifford") and "idem-Clifford" graph-theoretic methods are generalized to hypergraphs. In particular, zeon and idem-Clifford methods are used to enumerate paths, trails, independent sets, cliques, and matchings in hypergraphs. An approach for finding minimum hypergraph transversals is developed, and zeon formulations of some open hypergraph problems are presented.

Zeon and Idem-Clifford Formulations of Hypergraph Problems

TL;DR

This work extends zeon () and idem-Clifford () algebras from graphs to finite hypergraphs to address core combinatorial problems. It introduces the hypergraph nilpotent adjacency matrix and constructs representations , , and to count walks, trails, independent sets, (weak/strong) cliques, matchings, and minimum transversals, using symbolic calculations. The authors provide concrete algebraic formulations for -paths, -matchings, -intersecting matchings, and minimum transversals, and reformulate Ryser's and Frankl's conjectures within this framework, illustrating powerful open-problem translations. The approach enables symbolic-computation workflows and points to future work on hypergraph colorings and connections to zeon-based Laplacians, broadening the applicability of algebraic enumeration to complex relational structures.

Abstract

Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. In contrast to an ordinary graph, in which each edge connects exactly two vertices, an edge (or, "hyperedge") can join any number of vertices in a hypergraph. In game theory, hypergraphs are called simple games. Hypergraphs have been used for problems in biology, chemistry, image processing, wireless networks, and more. In the current work, zeon ("nil-Clifford") and "idem-Clifford" graph-theoretic methods are generalized to hypergraphs. In particular, zeon and idem-Clifford methods are used to enumerate paths, trails, independent sets, cliques, and matchings in hypergraphs. An approach for finding minimum hypergraph transversals is developed, and zeon formulations of some open hypergraph problems are presented.
Paper Structure (17 sections, 7 theorems, 52 equations, 3 figures)

This paper contains 17 sections, 7 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Graph and Hypergraph Theory
  4. Zeon and idem-Clifford Algebras
  5. Zeon Algebras
  6. Generalized Zeon Algebras
  7. Idem-Clifford Algebras
  8. Infinite-Dimensional Algebras
  9. Hypergraph Enumeration Problems
  10. Walks on Hypergraphs
  11. The hypergraph nilpotent adjacency matrix
  12. Cliques and Independent Sets
  13. Matchings
  14. $j$-intersecting matchings
  15. Minimal Vertex Coverings and Transversals
  16. ...and 2 more sections

Key Result

Theorem 3.4

Let $H=(V,E)$ be a hypergraph with $n$ vertices and $m$ hyperedges and let $\Omega$ be the nilpotent hypergraph adjacency matrix of $H$. Then for $k \in \mathbb{N}$ and $1 \leq i \neq j \leq n$, we have where $\omega_I$ is the number of $k$-paths from $v_i$ to $v_j$ in $H$ on vertex set $I$ and hyperedge set $J$. Further, when $i=j$ and $k \geq 2$, we have where $\omega_I$ denotes the number of

Figures (3)

  • Figure 1: A hypergraph $H$
  • Figure 2: The incidence matrix of $H$
  • Figure 3: The bipartite representation of $H$.

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Remark 2.9
  • Definition 3.1
  • ...and 31 more