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Zeon and Idem-Clifford Formulations of Hypergraph Problems

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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. 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.

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Notes

  1. A hypergraph whose edges consist of vertex pairs is commonly called a graph.

  2. Recall the notation \([rn]=\{1, \ldots , rn\}\).

  3. Note that Proposition 3.21 treats the case of 0-intersecting matchings.

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  1. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA

    Samuel Ewing & G. Stacey Staples

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  1. Samuel Ewing
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Ewing, S., Staples, G.S. Zeon and Idem-Clifford Formulations of Hypergraph Problems. Adv. Appl. Clifford Algebras 32, 61 (2022). https://doi.org/10.1007/s00006-022-01242-y

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