The Polytope of $k$-Star Densities

This paper describes the polytope $\mathbf{P}_{k;N}$ of $i$-star counts, for all $i\le k$, for graphs on $N$ nodes. The vertices correspond to graphs that are regular or as regular as possible. For even $N$ the polytope is a cyclic polytope, and for odd $N$ the polytope is well-approximated by a cyclic polytope. As $N$ goes to infinity, $\mathbf{P}_{k;N}$ approaches the convex hull of the moment curve. The affine symmetry group of $\mathbf{P}_{k;N}$ contains just a single non-trivial element, which corresponds to forming the complement of a graph.The results generalize to the polytope $\mathbf{P}_{I;N}$ of $i$-star counts, for $i$ in some set $I$ of non-consecutive integers. In this case, $\mathbf{P}_{I;N}$ can still be approximated by a cyclic polytope, but it is usually not a cyclic polytope itself.Polytopes of subgraph statistics characterize corresponding exponential random graph models. The elongated shape of the $k$-star polytope gives a qualitative explanation of some of the degeneracies found in such random graph models.

NORMAL CYCLIC POLYTOPES AND CYCLIC POLYTOPES THAT ARE NOT VERY AMPLE

Let $d$ and $n$ be positive integers such that $n\geq d+ 1$ and ${\tau }_{1} , \ldots , {\tau }_{n} $ integers such that ${\tau }_{1} \lt \cdots \lt {\tau }_{n} $. Let ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )\subset { \mathbb{R} }^{d} $ denote the cyclic polytope of dimension $d$ with $n$ vertices $({\tau }_{1} , { \tau }_{1}^{2} , \ldots , { \tau }_{1}^{d} ), \ldots , ({\tau }_{n} , { \tau }_{n}^{2} , \ldots , { \tau }_{n}^{d} )$. We are interested in finding the smallest integer ${\gamma }_{d} $ such that if ${\tau }_{i+ 1} - {\tau }_{i} \geq {\gamma }_{d} $ for $1\leq i\lt n$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is normal. One of the known results is ${\gamma }_{d} \leq d(d+ 1)$. In the present paper a new inequality ${\gamma }_{d} \leq {d}^{2} - 1$ is proved. Moreover, it is shown that if $d\geq 4$ with ${\tau }_{3} - {\tau }_{2} = 1$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is not very ample.

The Cyclic Sieving Phenomenon for Faces of Cyclic Polytopes

A cyclic polytope of dimension $d$ with $n$ vertices is a convex polytope combinatorially equivalent to the convex hull of $n$ distinct points on a moment curve in ${\Bbb R}^d$. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-Stanton-White, for faces of an even-dimensional cyclic polytope, under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytopes, we enumerate the faces that are invariant under an automorphism that reverses the order of the vertices and an automorphism that interchanges the two end vertices, according to the order on the curve. In particular, for $n=d+2$, we give instances of the phenomenon under the groups that cyclically translate the odd-positioned and even-positioned vertices, respectively.

On Linked Spatial Representations

What is the smallest positive integer m=m(L) such that every linear spatial representation of the complete graph with n vertices, n≥m contain cycles isotopic to link L? In this paper, we show that [Formula: see text]. The proof uses the well-known cyclic polytope and its combinatorial description in terms of oriented matroids.