Combinatorial Determinants via Contributor Duality
This thesis gives sentiment to all minors of integer matrix determinant calculations of Laplacian matrices by examining locally signed-graphic behaviors of the associated oriented hypergraph and generalizing the notion of permutations, called contributors. Once contributors are established, we improve on the results of the total-minor polynomial by proving that edge-monicness is sufficient for all contributors. Since signed graphs and ordinary graphs are oriented hypergraphs all their results are subsumed. A study of contributor duality is then the central focus to determine what calculations are universal between both the given oriented hypergraph and its dual. Traditional characteristic polynomial results are then reclaimed as a consequence of contributor duality. These matrices are then placed in a larger universal matrix context to examine their interactions. Finally, the monic condition is adapted to pathing matrices of signed graphs with future work on extending to oriented hypergraphs.
determinants, contributors, mathematics
Dvarishkis, B. (2023). Combinatorial determinants via contributor duality (Unpublished thesis). Texas State University, San Marcos, Texas.