Zeon and Idem-Clifford Formulations of Hypergraph Problems

TL;DR

This paper extends zeon and idem-Clifford algebraic methods from graphs to hypergraphs, enabling enumeration of hypergraph structures and addressing problems like minimum transversals.

Contribution

It introduces generalized zeon and idem-Clifford formulations for hypergraphs, providing new algebraic tools for hypergraph enumeration and problem-solving.

Findings

Zeon and idem-Clifford methods successfully enumerate hypergraph structures.

New formulations for minimum hypergraph transversals are developed.

Presented zeon formulations for open hypergraph problems.

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.