Dissertation/Thesis Abstract

Oriented hypergraphs
by Rusnak, Lucas J., Ph.D., State University of New York at Binghamton, 2010, 201; 3409134
Abstract (Summary)

The column dependencies of {0,±1}-matrices which contain at most two non-zero entries in each column have been characterized using orientations of graphs and signed graphs. We introduce a hypergraphic model of {0,±1}-matrices, called oriented hypergraphs, to investigate the structure of the column dependencies of any {0,±1}-matrix.

In this work we will see: (1) The introduction of a new oriented incidence-based approach to hypergraphs as an extension of signed graphs. (2) The development of new hypergraphic structures and operations that are essential to the classification of the minimal dependencies of {0,±1}-matrices. (3) The adaptation and generalization of the theory of balanced matrices to oriented hypergraphs, and the identification of the oriented hypergraphic obstructions to balanceability. (4) Construction and identification of minimal dependencies of oriented hypergraphs based on varying degrees of balance. (5) A conjectured complete classification of the minimal column dependencies of {0,±1}-matrices using oriented hypergraphs. (6) An outlined extension of the theory of oriented hypergraphs to model any matrix over any field with the purpose to obtain a complete circuit classification of representable matroids.

Indexing (document details)
Advisor: Zaslavsky, Thomas
Commitee: Anderson, Laura, Cornuejolls, Gerard P., Mazur, Marcin
School: State University of New York at Binghamton
Department: Mathematical Sciences
School Location: United States -- New York
Source: DAI-B 71/07, Dissertation Abstracts International
Subjects: Mathematics
Keywords: Balanced matrices, Graph theory, Matroids, Oriented hypergraphs, Signed graphs, {0,+/-1}-matrix
Publication Number: 3409134
ISBN: 9781124062075
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