An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved. (Lucas J. Rusnak) 1 Portions of these results submitted to the 2016 Siemens Competition (regional semi-finalist). 2 Portions of these results appear in 2017 Master's Thesis.

Pathogenic bacteria resistant to many or all antibiotics already exist. With the decline in microbiological research at pharmaceutical companies, the high rate at which resistance has evolved and spread has demanded a novel approach to addressing this critical human health issue. In the present paper, we propose a new paradigm in antibiotic discovery and development, one that applies ecological and evolutionary theory to design antimicrobial drugs that are more difficult and/or more costly to resist. In essence, we propose to simply adopt the strategies invented and applied by bacteria for hundreds of millions of years. Our research focuses on bacteriocins, powerful biological weapons, and their use as alternative therapeutics in human health.