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arXiv · 1810.00812

Hypergraph polynomials and the Bernardi process

Tamás Kálmán·Lilla Tóthmérész
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Abstract

Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses) constructed by Baker and Wang between spanning trees and break divisors.

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Tamás Kálmán, Lilla Tóthmérész. 2018-10-01. Hypergraph polynomials and the Bernardi process. https://arxiv.org/abs/1810.00812

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