In this paper, we give a criterion of the Gorenstein property Ehrhart ring stable set polytope an h-perfect graph: graph $G$ is if and only (1) sizes maximal cliques are constant (say $n$) (2) (a) $n=1$, (b) $n=2$ there no odd cycle without chord length at least 7 or (c) $n\geq 3$ 5.

Noncompact hyperbolic Coxeter n-polytopes of finite volume and having n+ 3 facets are studied in this paper. Unlike the spherical and parabolic cases, no complete classification exists as yet for hyperbolic Coxeter polytopes of finite volume. It has been shown that the dimension of a bounded Coxeter polytope is at most 29 (Vinberg, 1984), while an upper estimate in the unbounded case is 995 (Pr...

A Gaussian polytope is the convex hull of normally distributed random points in a Euclidean space. We give an improved error bound for the expected number of facets of a Gaussian polytope when the dimension of the space is fixed and the number of points tends to infinity. The proof applies the theory of the asymptotic distribution of the top order statistic of a collection of independently dist...

This work is set in the field of combinatorial topology, a mathematical field of research in the intersection of the fields of topology, geometry, polytope theory and combinatorics. This work investigates polyhedral manifolds as subcomplexes of the boundary complex of a regular polytope. Such a subcomplex is called k-Hamiltonian, if it contains the full k-skeleton of the polytope. Since the cas...

If P ⊂ R is a rational polytope, then iP (n) := #(nP ∩Z ) is a quasipolynomial in n, called the Ehrhart quasi-polynomial of P . The period of iP (n) must divide D(P ) = min{n ∈ Z>0 : nP is an integral polytope}. Few examples are known where the period is not exactly D(P ). We show that for any D, there is a 2-dimensional triangle P such that D(P ) = D but such that the period of iP (n) is 1, th...

The space of summands (with respect to vector addition) of a convex polytope in n dimensions is studied. This space is shown to be isomorphic to a convex pointed cone in Euclidean space. The extreme rays of this cone correspond to similarity classes of indecomposable polytopes. The decomposition of a polytope is described and a bound is given for the number of indecomposable summands needed. A ...

On the basis of empirical evidence from molecular dynamics simulations, molecular conformational space can be described by means of a partition of central conical regions (cells) characterized by the dominance relations between cartesian coordinates. This work presents a geometric and combinatorial description of the cell arrangement which is polar to a 3x(N-1)-dimensional polytope. Conformatio...

The volume of the n-dimensional polytope Π n (x) := {y ∈ R n : y i ≥ 0 and y 1 + · · · + y i ≤ x 1 + · · · + x i for all 1 ≤ i ≤ n} for arbitrary x := (x 1 ,. .. , x n) with x i > 0 for all i defines a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as th...

The stable set polytope of a graph is the convex hull of the 01 vectors corresponding to stable sets of vertices. To any nontrivial facet ∑n i=1 aixi ≤ b of this polytope we associate an integer δ, called the defect of the facet, by δ = ∑n i=1 ai − 2b. We show that for any fixed δ there is a finite collection of graphs (called “basis”) such that any graph with a facet of defect δ contains a sub...