Lucas Rusnak received his Ph.D from Binghamton University in 2010. His research is in oriented hypergraphs and combinatorial matrix theory. He is particularly interested in generalizations of the structural and matroidal properties of signed and gain graphs, as well as the roles frustration and balance play in oriented hypergraphs.
Dr. Rusnak has introduced an oriented hypergraphic generalization of Harary's theorem for balance equivalence via closed weak walks, provided a characterization of the matroid circuits of balanced oriented hypergraphs, and extended the familiar algebraic graph theory results relating the degree, adjacency, incidence, and Laplacian matrices to any matrix by providing a unifying combinatorial interpretation of their entries in terms of weak walks.
Dr. Rusnak is currently investigating: (1) hypergraphic generalizations of frustration and its possible relationship to Ising spins and/or cognitive modeling, (2) a characterization of the minors of the oriented hypergraphic Laplacian via weak walks and graphical Stirling numbers, and (3) classifying the matroid circuits of balanceable and unbalanceable oriented hypergraphs.