On the Gamma-Vector of Symmetric Edge Polytopes

D’Alì, Alessio; Juhnke-Kubitzke, Martina; Köhne, Daniel; Venturello, Lorenzo

doi:10.1137/22m1492799

Cited by 3 publications

(2 citation statements)

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“…However, one of the major advantages of working with G(n, p) is that it is defined in a way that often allows theoretical results to be obtained. For example, see (9) for the use of the G(n, p) model in the study of Ehrhart theory for symmetric edge polytopes. In this subsection, we prove that for certain values of p, the typical G(n, p) will have facets of a prescribed form.…”

Section: Facets Of G(n P)mentioning

confidence: 99%

“…Symmetric edge polytopes, also called type PV adjacency polytopes, have been the subject of extensive recent study (2,6,7,8,9,10,14,15,16,17,18,20,21,22). One problem of interest is to establish relationships between the combinatorial structure of G and geometric properties of P G , motivated by applications in both pure and applied contexts.…”

Section: Introductionmentioning

confidence: 99%

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Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

Braun,

Bruegge,

Kahle

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

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Section: Facets Of G(n P)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

Braun,

Bruegge,

Kahle

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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On a Generalization of Symmetric Edge Polytopes to Regular Matroids

D’Alì,

Juhnke-Kubitzke,

Koch

2024

International Mathematics Research Notices

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Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we prove that two symmetric edge polytopes are unimodularly equivalent precisely when they share the same graphical matroid. The second goal is to show that one can construct a generalized symmetric edge polytope starting from every regular matroid. Just like in the usual case, we are able to find combinatorial ways to describe the facets and an explicit regular unimodular triangulation of any such polytope. Finally, we show that the Ehrhart theory of the polar of a given generalized symmetric edge polytope is tightly linked to the structure of the lattice of flows of the dual regular matroid.

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