Applications of Hypergraphic Matrix Minors via Contributors
Date
2021-05
Authors
Reynes, Josephine Elizabeth Anne
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Abstract
<p>Hypergrahic matrix-minors via contributors can be utilized in a variety of ways. Specifically, this thesis illustrates that they are useful in extending Kirchhoff-type Laws to signed graphs and to reinterpret Hadamard's maximum determinant problem.</p>
<p>First, we discuss how the incidence-oriented structures of bidirected graphs allow for a generalization of transpedances which enables the extension of Kirchhoff-type laws to signed graphs. Reduced incidence-based cycle covers, or contributors, form Boolean classes, and the single-element classes are equivalent to Tutte's 2-arborescences. When using entire Boolean classes, which naturally cancel in a graph, a generalized contributor-transpedance is introduced and graph conservation is shown to be a property of the trivial Boolean classes. These contributor-transpedances on signed graphs produce non-conservative Kirchhoff-type Laws based on each contributor having a unique source-sink path. Additionally, the signless Laplacian is used to calculate the maximum value of a contributor-transpedance.</p>
<p>Second, we discuss how hypergraphic matrix-minors via contributors can be used to calculate the determinant of a given {±1}-matrix. This is done by examining classes of contributors that have multiple symmetries. The oriented hypergraphic Laplacian and the incidence-based notion of cycle-covers allow for this analysis. If a family of these cycle-covers is non-edge-monic, it will sum to zero in every determinant which means the only remaining, n! edge-monic families are counted. Also, any one of them can be utilized to determine the absolute value of the determinant. Hadamard's maximum determinant problem is equivalent to optimizing the number of locally signed circles of a specified sign in an edge-monic families or across all edge-monic families. Theta-subgraphs have different fundamental circles that yield various symmetries regarding the orthogonality condition, which are equivalent to {0,+1}-matrices.</p>
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Keywords
Hadamard matrix, Incidence hypergraph, Oriented hypergraph, Laplacian, Signed graph, Arborescence, Transpedance, Kirchhoff
Citation
Reynes, J. E. A. (2021). <i>Applications of hypergraphic matrix minors via contributors</i> (Unpublished thesis). Texas State University, San Marcos, Texas.