一个超图Tutte多项式
A Hypergraph Tutte Polynomial
AI总结：
研究引入超图的Tutte多项式$T_{\mathrm{HG}}$及相关的$T_k$，证明$T_{\mathrm{HG}}$有Tutte型性质，$T_k$有通用性定理等，还关联其与其他模型，比较它和$\mathcal{T}_P$，表明两者区分能力不可比。
AI中文摘要：
我们引入了超图的Tutte多项式$T_{\mathrm{HG}}$以及与$k$-多拟阵相关的Tutte多项式$T_k$。这两个不变量都有在各自类别内的删除-收缩递归关系，并且在任何$(k + 1)$-均匀超图的关联多拟阵上，$T_{\mathrm{HG}}$特化为$T_k$。我们证明$T_{\mathrm{HG}}$满足几个理想的Tutte型性质，$T_k$还具有一个通用性定理和一个卷积积公式。在均匀超图设置中，这些结果特化为$T_{\mathrm{HG}}$。我们还将$T_{\mathrm{HG}}$与Potts和随机簇模型的超图扩展相关联。最后，我们比较$T_{\mathrm{HG}}$与Bernardi、K'alm'an和Postnikov的多拟阵Tutte多项式$\mathcal{T}_P$，表明两者在区分能力上不可比。
英文摘要：
We introduce a Tutte polynomial for hypergraphs, $T_{\mathrm{HG}}$, together with $T_k$, a related Tutte polynomial for $k$-polymatroids. Both invariants admit deletion--contraction recursions that remain within their respective classes, and they are linked by the fact that $T_{\mathrm{HG}}$ specializes to $T_k$ on the associated polymatroid of any $(k+1)$-uniform hypergraph. We show that $T_{\mathrm{HG}}$ satisfies several desirable Tutte type properties, including multiplicativity and duality, while $T_k$ further admits a universality theorem, as well as a convolution product formula. In the uniform hypergraph setting, these latter results specialize back to $T_{\mathrm{HG}}$. We also relate $T_{\mathrm{HG}}$ to hypergraph extensions of the Potts and random cluster models. In particular, we study degree dependent random cluster and Potts partition functions, as well as Grimmett's many body Potts model, and compare their relationship with $T_{\mathrm{HG}}$ in both the general and uniform settings. Finally, we compare $T_{\mathrm{HG}}$ with the polymatroid Tutte polynomial $\mathcal{T}_P$ of Bernardi, K'alm'an, and Postnikov, showing that the two are incomparable in distinguishing power. As a consequence, we answer negatively a question raised by these authors by proving that the characteristic polynomial is not, in general, a specialization of $\mathcal{T}_P$.
