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dc.contributor.advisorRusnak, Lucas
dc.contributor.authorReynes, Josephine Elizabeth Anne ( Orcid Icon 0000-0002-3862-9176 )
dc.identifier.citationReynes, J. E. A. (2021). Applications of hypergraphic matrix minors via contributors (Unpublished thesis). Texas State University, San Marcos, Texas.

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.

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.

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.

dc.format.extent93 pages
dc.format.medium1 file (.pdf)
dc.subjectHadamard matrix
dc.subjectIncidence hypergraph
dc.subjectOriented hypergraph
dc.subjectSigned graph
dc.titleApplications of Hypergraphic Matrix Minors via Contributors
dc.contributor.committeeMemberCurtin, Eugene
dc.contributor.committeeMemberShen, Jian State University of Science



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