Hypergraph polynomials and the Bernardi process
Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1099-1139.

Bernardi gave a formula for the Tutte polynomial $T\left(x,y\right)$ 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 $T\left(x,1\right)$ 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 Bernardi and further studied by Baker and Wang, between spanning trees and break divisors.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.129
Classification: 05C10, 05C31, 05C50, 05C57, 05C65
Keywords: Hypergraph, bipartite graph, ribbon structure, Tutte polynomial, interior polynomial, embedding activity, root polytope, dissection, shelling order, $h$-vector.
Kálmán, Tamás 1; Tóthmérész, Lilla 2

1 Department of Mathematics Tokyo Institute of Technology H-214, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
2 Cornell University Ithaca New York 14853-4201, USA
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Kálmán, Tamás; Tóthmérész, Lilla. Hypergraph polynomials and the Bernardi process. Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1099-1139. doi : 10.5802/alco.129. http://www.numdam.org/articles/10.5802/alco.129/

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