Symmetric edge polytopes and matching generating polynomials

Ohsugi, Hidefumi; Tsuchiya, Asato

doi:10.5070/c61055371

Cited by 7 publications

(4 citation statements)

References 27 publications

(33 reference statements)

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“…See, e.g., [18, p. 27]. Moreover, for n ≥ 3, it is known [16,Exam. 4.5] that, γ (n, x) is the γ -polynomial of the PV-type adjacency polytope ∇ PV W n of W n .…”

Section: Wheel Graphsmentioning

confidence: 99%

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PQ-Type Adjacency Polytopes of Join Graphs

Ohsugi

Tsuchiya

2022

Discrete Comput Geom

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PQ-type adjacency polytopes $$\nabla ^PQ _G$$ ∇ P Q G are lattice polytopes arising from finite graphs G. There is a connection between $$\nabla ^PQ _G$$ ∇ P Q G and the engineering problem known as power-flow study, which models the balance of electric power on a network of power generation. In particular, the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G plays a central role. In the present paper, we focus on the case where G is a join graph. In particular, formulas of the $$h^*$$ h ∗ -polynomial and the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G of a join graph G are presented. Moreover, we give explicit formulas of the $$h^*$$ h ∗ -polynomial and the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G when G is a complete multipartite graph or a wheel graph.

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“…See, e.g., [18, p. 27]. Moreover, for n ≥ 3, it is known [16,Exam. 4.5] that, γ (n, x) is the γ -polynomial of the PV-type adjacency polytope ∇ PV W n of W n .…”

Section: Wheel Graphsmentioning

confidence: 99%

“…4.5] that, γ (n, x) is the γ -polynomial of the PV-type adjacency polytope ∇ PV W n of W n . (Note that ∇ PV W n is called the symmetric edge polytope of type A of W n in [16].) The h * -polynomial of ∇ PQ W n is described by this function as follows.…”

Section: Wheel Graphsmentioning

confidence: 99%

PQ-Type Adjacency Polytopes of Join Graphs

Ohsugi

Tsuchiya

2022

Discrete Comput Geom

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“…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%

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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“…Symmetric edge polytopes, also called type PV adjacency polytopes, have been the subject of extensive recent study [2,6,7,8,9,10,13,14,15,16,17,19,20,21]. 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%

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

Braun¹,

Bruegge²,

Kahle³

2022

Preprint

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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 Erdős-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 Watts-Strogatz clustering coefficient and the number of facets for a graph 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 Watts-Strogatz clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

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Symmetric edge polytopes and matching generating polynomials

Cited by 7 publications

References 27 publications

PQ-Type Adjacency Polytopes of Join Graphs

PQ-Type Adjacency Polytopes of Join Graphs

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

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

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