A hypergraph Heilmann–Lieb theorem

The Heilmann–Lieb theorem is a fundamental theorem in algebraic combinatorics which provides a characterization of the distribution of the zeros of matching polynomials of graphs. In this paper, we establish a hypergraph Heilmann–Lieb theorem as follows. Let H \mathcal {H} be a connected k k -graph with maximum degree Δ ≥ 2 {\Delta }\geq 2 and let μ ( H , x ) \mu (\mathcal {H}, x) be its matching polynomial. We show that the zeros (with multiplicities) of μ ( H , x ) \mu (\mathcal {H}, x) are invariant under a rotation of an angle 2 π / ℓ 2\pi /{\ell } in the complex plane for some positive integer ℓ \ell and k k is the maximum integer with this property. We further prove that the maximum modulus λ ( H ) \lambda (\mathcal {H}) of all the zeros of μ ( H , x ) \mu (\mathcal {H}, x) is a simple root of μ ( H