Hypergraph polynomials and the Bernardi process

Kálmán, Tamás; Tóthmérész, Lilla

doi:10.5802/alco.129

Cited by 5 publications

(33 citation statements)

References 10 publications

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“…We call the component of 𝑇 − 𝜀 containing 𝑏 0 the base component. The following property of fundamental cuts of Jaeger trees (an updated version of [13,Lemma 6.14]) makes them very useful to us. Lemma 5.5.…”

Section: F I G U R E 6 Dissection Induced By a Ribbon Structurementioning

confidence: 99%

“…Remark 10.3. The case of bipartite graphs 𝐺 with each indegree in 𝑊 equal to 2 and each outdegree in 𝑊 equal to 0 (which is just a standard orientation) plays an important role in the preliminary papers [9,11,13]. In those works, bipartite graphs are thought of as hypergraphs with 𝑈 corresponding to vertices and 𝑊 corresponding to hyperedges.…”

Section: A C K N O W L E D G E M E N T Smentioning

confidence: 99%

“…This paper grew out of efforts to establish ‘signed versions’ of our recent results [13] regarding the interplay between root polytopes and ribbon structures of bipartite graphs (indeed, certain arguments extend almost word‐by‐word), but it ended up achieving quite a bit more. We found significant simplifications and new connections, rounding eventually into a theory of a wide class of signed bipartite graphs and their ‘interior’ polynomial invariants.…”

Section: Introductionmentioning

confidence: 99%

“…We establish the dissection in a new way which is rather easier than our previous proof in the case of the standard orientation [13, Section 4]. It involves a natural order on the set of all spanning trees of a directed ribbon graph.…”

Section: Introductionmentioning

confidence: 99%

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Root polytopes and Jaeger‐type dissections for directed graphs

Kálmán

Tóthmérész

2022

Mathematika

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We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid to what we call the semi‐balanced case, that is, when each cycle has the same number of edges pointing in the two directions. Given a ribbon structure, we identify a natural class of spanning trees and show that, in the semi‐balanced case, they induce a shellable dissection of the root polytope into maximal simplices. This allows for a computation of the h∗$h^*$‐vector of the polytope and for showing some properties of this new graph invariant, such as a product formula and that in the planar case, the h∗$h^*$‐vector is equivalent to the greedoid polynomial of the dual graph. We obtain a general recursion relation as well. We also work out the case of layer‐complete directed graphs, where our method recovers a previously known triangulation. Indeed our dissection is often but not always a triangulation; we address this with a series of examples.

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Section: F I G U R E 6 Dissection Induced By a Ribbon Structurementioning

confidence: 99%

Section: A C K N O W L E D G E M E N T Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Root polytopes and Jaeger‐type dissections for directed graphs

Kálmán

Tóthmérész

2022

Mathematika

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“…In [4,5], the authors established a relation between the top of the HOMFLY polynomial of any special alternating link and the interior polynomial of the Seifert graph (which is a bipartite graph) of the link. Recently, Kato [7] introduced the signed interior polynomial of signed bipartite graphs and extended the relation to any oriented links, and in [6] Kálmán and Tóthmérész defined two one-variable generating functions I and X using two 'embedding activities' and proved that the generating function of internal embedding activities coincides with the interior polynomial.…”

Section: Introductionmentioning

confidence: 99%

On coefficients of the interior and exterior polynomials

Guan¹,

X²

2022

Preprint

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The interior polynomial and the exterior polynomial are generalizations of valuations on (1/ξ, 1) and (1, 1/η) of the Tutte polynomial T G (x, y) of graphs to hypergraphs, respectively. The pair of hypergraphs induced by a connected bipartite graph are abstract duals and are proved to have the same interior polynomial, but may have different exterior polynomials. The top of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the pair of hypergraphs induced by the Seifert graph of the link. Let G = (V ∪ E, ε) be a connected bipartite graph. In this paper, we mainly study the coefficients of the interior and exterior polynomials. We prove that the interior polynomial of a connected bipartite graph is interpolating. We strengthen the known result on the degree of the interior polynomial for connected bipartite graphs with 2-vertex cuts in V or E. We prove that interior polynomials for a family of balanced bipartite graphs are monic and the interior polynomial of any connected bipartite graph can be written as a linear combination of interior polynomials of connected balanced bipartite graphs. The exterior polynomial of a hypergraph is also proved to be interpolating. It is known that the coefficient of the linear term of the interior polynomial is the nullity of the bipartite graph, we obtain a 'dual' result on the coefficient of the linear term of the exterior polynomial: if G − e is connected for each e ∈ E, then the coefficient of the linear term of the exterior polynomial is |V | − 1. Interior and exterior polynomials for some families of bipartite graphs are computed.

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The Sandpile Group of a Trinity and a Canonical Definition for the Planar Bernardi Action

Kálmán¹,

Lee²,

Tóthmérész³

2022

Combinatorica

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Baker and Wang define the so-called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but it is independent of the base vertex if and only if the ribbon structure is planar, moreover, in this case the Bernardi action is compatible with planar duality. Earlier, Chan, Church and Grochow and Chan, Glass, Macauley, Perkinson, Werner and Yang proved analogous results about the rotor-routing action. Baker and Wang moreover showed that the Bernardi and rotor-routing actions coincide for plane graphs.We clarify this still confounding picture by giving a canonical definition for the planar Bernardi/rotor-routing action, and also a canonical isomorphism between sandpile groups of planar dual graphs. Our canonical definition implies the compatibility with planar duality via an extremely short argument. We also show hidden symmetries of the problem by proving our results in the slightly more general setting of balanced plane digraphs.Any balanced plane digraph gives rise to a trinity, i.e., a triangulation of the sphere with a three-coloring of the 0-simplices. Our most important tool is a group associated to trinities, introduced by Cavenagh and Wanless, and a result of a subset of the authors characterizing the Bernardi bijection in terms of a dissection of a root polytope.

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Hypergraph polynomials and the Bernardi process

Cited by 5 publications

References 10 publications

Root polytopes and Jaeger‐type dissections for directed graphs

Root polytopes and Jaeger‐type dissections for directed graphs

On coefficients of the interior and exterior polynomials

The Sandpile Group of a Trinity and a Canonical Definition for the Planar Bernardi Action

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