Abstract
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 Bernardi and further studied by Baker and Wang, between spanning trees and break divisors.
Highlights
A few years ago a pair of polynomial invariants of hypergraphs was introduced [9], which generalize the valuations T (x, 1) and T (1, y) of the two-variable graph invariant T (x, y) due to Tutte [16]
This leads to the definition of a natural order among Bernardi/Jaeger trees and we prove, as a key step to our main theorem, that this order is a shelling order of the dissection
For triangulations of QG by maximal simplices, all their h-vectors are the same, too [11, Theorem 3.10].) [11, Equation (5.1)] states that the same coefficient sequence gives the interior polynomial of both hypergraphs (V, E) and (E, V ) that are induced by G: Theorem 3.3
Summary
A few years ago a pair of polynomial invariants of hypergraphs was introduced [9], which generalize the valuations T (x, 1) and T (1, y) of the two-variable graph invariant T (x, y) due to Tutte [16]. Hypertrees were introduced in [14] (and so named in [9]) They generalize characteristic vectors of spanning trees of a graph, preserving some nice polyhedral properties. Both for graphs and hypergraphs, the computation of individual activities requires fixing an order of the set of edges or hyperedges, respectively, albeit temporarily, because the aggregate polynomials do not depend on it. For a given spanning tree, he traced the boundary of the neighborhood of the tree and numbered the edges of the graph along the way He used this order to compute the internal and external activities of the tree. Hypergraph, bipartite graph, ribbon structure, Tutte polynomial, interior polynomial, embedding activity, root polytope, dissection, shelling order, h-vector
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