A Harary-Sachs Theorem for Hypergraphs.

We generalize the Harary-Sachs theorem to $k$-uniform hypergraphs: the codegree-$d$ coefficient of the characteristic polynomial of a uniform hypergraph ${\cal H}$ can be expressed as a weighted sum of subgraph counts over certain multi-hypergraphs with $d$ edges. This includes a detailed description of said multi-hypergraphs and a formula for their corresponding weights, which we use to describe the low codegree coefficients of the characteristic polynomial of a 3-uniform hypergraph. We further provide explicit and asymptotic formulas for the contribution from $k$-uniform simplices and conclude by showing that the Harary-Sachs theorem for graphs is indeed a special case of our main theorem.