Total Minor Polynomials of Oriented Hypergraphs

 dc.contributor.author Reynes, Josephine Elizabeth Anne dc.contributor.author Rusnak, Lucas dc.date.accessioned 2020-06-17T15:55:08Z dc.date.available 2020-06-17T15:55:08Z dc.date.issued 2019-05 dc.description.abstract Concepts of graph theory can be generalized to integer matrices through the use of oriented hypergraphs. An oriented hypergraph is an incidence structure consisting of vertices, edges, and incidences, equipped with three functions: a vertex incidence function, an edge incidence function, and an incidence orientation function. This thesis provides a unifying generalization of Seth Chaiken’s All-Minors Matrix-Tree Theorem and Sachs’ Coefficient Theorem to all integer adjacency and Laplacian matrices – extending the results of Rusnak, Robinson et. al. – by introducing a polynomial in |V|2 indeterminants indexed by minor order whose monomial coefficients are the minors. The coefficients are determined by embedding the oriented hypergraph into the smallest uniform hypergraph that contains it and summing over a class of sub-monic mappings of paths of length one relative to the original oriented hypergraph. It is known that the non-cancellative mappings associated to each degree-1 monomials are in one-to-one correspondence with Tuttes Matrix-Tree Theorem. This is extended to Tuttes k-arborescence decomposition via the degree-k monomials. dc.description.department Honors College dc.format Text dc.format.extent 35 pages dc.format.medium 1 file (.pdf) dc.identifier.citation Reynes, J. (2019). Total minor polynomials of oriented hypergraphs (Unpublished thesis). Texas State University, San Marcos, Texas. dc.identifier.uri https://hdl.handle.net/10877/11844 dc.language.iso en dc.subject Laplacian dc.subject hypergraph dc.subject signed graph dc.subject characteristic dc.subject polynomial dc.subject matrix-tree theorem dc.subject combinations dc.subject Honors College dc.title Total Minor Polynomials of Oriented Hypergraphs thesis.degree.department Honors College thesis.degree.discipline Mathematics thesis.degree.grantor Texas State University txstate.documenttype Honors Thesis

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