Applications of the Harary-Sachs Theorem for Hypergraphs

The Harary-Sachs theorem for $k$-uniform hypergraphs equates the codegree-$d$ coefficient of the adjacency characteristic polynomial of a uniform hypergraph with a weighted sum of subgraph counts over certain multi-hypergraphs with $d$ edges. We begin by showing that the classical Harary-Sachs theorem for graphs is indeed a special case of this general theorem. To this end we apply the generalized Harary-Sachs theorem to the leading coefficients of the characteristic polynomial of various hypergraphs. In particular, we provide explicit and asymptotic formulas for the contribution of the $k$-uniform simplex to the codegree-$d$ coefficient. Moreover, we provide an explicit formula for the leading terms of the characteristic polynomial of a 3-uniform hypergraph and further show how this can be used to determine the complete spectrum of a hypergraph. We conclude with a conjecture concerning the multiplicity of the zero-eigenvalue of a hypergraph.