Related papers: On Vietoris--Rips complexes of hypercube graphs

We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For…

Let $Q_k$ denote the $k$-dimensional hypercube on $2^k$ vertices. A vertex in a subgraph of $Q_k$ is {\em full} if its degree is $k$. We apply the Kruskal-Katona Theorem to compute the maximum number of full vertices an induced subgraph on…

Kendall's Similarity Shape Theory for constellations of N points in the carrier space $\mathbb{R}^d$ as quotiented by the similarity group was developed for use in Probability and Statistics. It was subsequently shown to reside within…

We study combinatorial connectivity for two models of random geometric complexes. These two models - \v{C}ech and Vietoris-Rips complexes - are built on a homogeneous Poisson point process of intensity $n$ on a $d$-dimensional torus using…

Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $\aut_G(S(V)^*)$ denote…

We prove that the complex of proper non-trivial non-degenerate subspaces of a finite-dimensional vector space endowed with a non-degenerate sesquilinear form is homotopy equivalent to a wedge of spheres. Additionally, we show that the same…

For integers n\geq 1, k\geq 0, the stable Kneser graph SG_{n,k} (also called the Schrijver graph) has as vertex set the stable n-subsets of [2n+k] and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not…

The cut polytope ${\rm CUT}(n)$ is the convex hull of the cut vectors in a complete graph with vertex set $\{1,\ldots,n\}$. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation…

We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\Omega(n/\log n)$. This establishes a strong form of a claim by Thurston, for which the construction and proof…

Let $\mathcal{H}$ be a hypergraph on a finite set $V$. A {\em cover} of $\mathcal{H}$ is a set of vertices that meets all edges of $\mathcal{H}$. If $W$ is not a cover of $\mathcal{H}$, then $W$ is said to be a {\em noncover} of…

A ternary graph is a graph with no induced cycles of length $0$ modulo $3$. It was recently shown that, if the independence complex of a ternary graph is not contractible, then it is homotopy equivalent to a sphere. When a ternary graph…

The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…

$Q_{n,p}$, the random subgraph of the $n$-vertex hypercube $Q_n$, is obtained by independently retaining each edge of $Q_n$ with probability $p$. We give precise values for the cover time of $Q_{n,p}$ above the connectivity threshold.

Combinatorially and stochastically defined simplicial complexes often have the homotopy type of a wedge of spheres. A prominent conjecture of Kahle quantifies this precisely for the case of random flag complexes. We explore whether such…

We prove and organize some results on the normal forms of Hermitian operators composed with the Veronese map. We apply this general framework to prove two specific theorems in CR geometry. First, extending a theorem of Faran, we classify…

In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^3$-actions on rational spheres and complex projective spaces, and provide some properties of $S^1$-actions on a general rational complex.

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively…

A Hypercube $Q_n$ is a graph in which the vertices are all binary vectors of length n, and two vertices are adjacent if and only if their components differ in exactly one place. A galaxy or a star forest is a union of vertex disjoint stars.…

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to…

We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the…