Edge Polytopes and Edge Rings

Herzog, Jürgen; Hibi, Takayuki; Ohsugi, Hidefumi

doi:10.1007/978-3-319-95349-6_5

Cited by 3 publications

(3 citation statements)

References 27 publications

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“…However, G Q establishes a link to a different viewpoint. The quasi-independence ideals are also studied under the title of toric ideals of edge rings [33] and many combinatorial and algebraic aspects of these ideals are known [11]. Additionally the so-called edge polytopes, whose vertices are the columns of the incidence matrix of G Q , are introduced in [8] and their faces are characterized in terms of certain subgraphs of…”

Section: Preliminariesmentioning

confidence: 99%

“…Similar to the toric fiber product, we show that the Gröbner basis of I × Q J can be computed by lifting polynomials from the Gröbner bases of ideals I and J and adding in extra binomials which come from the monomial map φ Q . The Gröbner bases of the ideals I Q = ker(φ Q ) are completely known and have been studied in algebraic statistics [3,5] and commutative algebra as the toric ideals of edge rings of bipartite graphs [8,11,18,33]. Unlike the toric fiber product though, we allow for φ Q to be any monomial homomorphism which arises as the parameterization of a quasi-independence model and do not require that I and J are homogeneous with respect to multigradings.…”

Section: Introductionmentioning

confidence: 99%

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Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing

Hollering¹,

Johnson²,

Portakal³

et al. 2022

Preprint

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Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing

Hollering¹,

Johnson²,

Portakal³

et al. 2022

Preprint

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“…Over decades, several lattice polytopes arising from graphs have been studied, extensively. Prominent examples include matching polytopes, cut polytopes, edge polytopes, adjacency polytopes of several types, among which are symmetric edge polytopes (see e.g., [23,4,19,21,28,10]). Following this line of research, in 2017, Braun and Meyer [6] initiated the study of Laplacian simplices that are defined as the convex hull of the columns of the classical Laplacian matrix of a simple graph (see also [24,3]).…”

Section: Introductionmentioning

confidence: 99%

Laplacian polytopes of simplicial complexes

Juhnke-Kubitzke¹,

Köhne²

2023

Preprint

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Given a (finite) simplicial complex, we define its i-th Laplacian polytope as the convex hull of the columns of its i-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of these polytopes, we focus on the d-th Laplacian polytope of the boundary of a pd `1q-simplex Bpσ d`1 q. If d is odd, then as for graphs, the d-th Laplacian polytope turns out to be a pd `1qsimplex in this case. If d is even, we show that the d-th Laplacian polytope of Bpσ d`1 q is combinatorially equivalent to a d-dimensional cyclic polytope on d `2 vertices. Moreover, we provide an explicit regular unimodular triangulation for the d-th Laplacian polytope of Bpσ d`1 q. This enables us to to compute the normalized volume and to show that the h ˚-polynomial is real-rooted and unimodal, if d is odd and even, respectively.

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