论文标题
Hypergraph Heilmann- lieb定理
A hypergraph Heilmann--Lieb theorem
论文作者
论文摘要
Heilmann--lieb定理是代数组合学中的基本定理，它提供了匹配图的多项式零零的分布的表征。在本文中，我们建立了Hypergraph Heilmann- lieb定理，如下所示。令$ \ h $为连接的$ k $ -graph，具有最高度$δ\ geq 2 $，让$μ（\ h，x）$为其匹配的多项式。我们表明，在复杂平面的角度$2π/{\ ell} $的旋转下，对于某些正整数$ \ ell $，$ k $是该属性的最大integer，零（\ h，x）$的零（\ h，x）$是不变的。我们进一步证明了$μ（\ h，x）$的所有零的最大模量$λ（\ h）$是$μ（\ h，x）$的简单根，并且满足$$δ^{\ frac {1} {1} {k}} {k}} {k}}} \leqλ（\ h）（\ h）（\ h）（\ h）（\ h）<<\ h）< \ frac {k} {k-1} \ big（（k-1）（δ-1）\ big）^{\ frac {\ frac {1} {k}}} {k}}}。超图。
The Heilmann--Lieb theorem is a fundamental theorem in algebraic combinatorics which provides a characterization of the distribution of the zeros of matching polynomials of graphs. In this paper, we establish a hypergraph Heilmann--Lieb theorem as follows. Let $\h$ be a connected $k$-graph with maximum degree $Δ\geq 2$ and let $μ(\h, x)$ be its matching polynomial. We show that the zeros (with multiplicities) of $μ(\h, x)$ are invariant under a rotation of an angle $2π/{\ell}$ in the complex plane for some positive integer $\ell$ and $k$ is the maximum integer with this property. We further prove that the maximum modulus $λ(\h)$ of all the zeros of $μ(\h, x)$ is a simple root of $μ(\h, x)$ and satisfies $$Δ^{\frac{1}{ k}} \leq λ(\h)< \frac{k}{k-1}\big((k-1)(Δ-1)\big)^{\frac{1}{ k}}.$$ To achieve these, we prove that $μ(\h, x)$ divides the matching polynomial of the $k$-walk-tree of $\h$, which generalizes a classical result due to Godsil from graphs to hypergraphs.
