Eigenvalues of K-Uniform Hypergraphs

Abstract

We de ne two separate attempts to generalize the de nition of eigenvalues to hypergraphs and show several results related to each. The rst approach is rooted in 2-dimensional matrices and allows for the generalization of many results from graph theory. The second approach covered is more sophisticated and may only be applied to k-uniform hypergraphs. We include the development of a sound algorithm using the resultant of polynomials that can be used for any k-uniform hypergraph. Speci c examples are provided to demonstrate the power of the algorithm. Further, we show that certain results hold for the eigenvalues and associated eigenvectors of k-uniform hypergraphs and those hypergraphs obtained from combinatorial designs such as Steiner triple systems.