# ALGEBRAIC COMBINATORICS

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.

Received: 2019-01-06
Revised: 2019-11-27
Accepted: 2020-05-18
Published online: 2020-10-12
DOI: https://doi.org/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.
@article{ALCO_2020__3_5_1099_0,
author = {K\'alm\'an, Tam\'as and T\'othm\'er\'esz, Lilla},
title = {Hypergraph polynomials and the Bernardi process},
journal = {Algebraic Combinatorics},
pages = {1099--1139},
publisher = {MathOA foundation},
volume = {3},
number = {5},
year = {2020},
doi = {10.5802/alco.129},
language = {en},
url = {alco.centre-mersenne.org/item/ALCO_2020__3_5_1099_0/}
}
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. https://alco.centre-mersenne.org/item/ALCO_2020__3_5_1099_0/

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