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Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this fact states that every polytope is…

We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…

For every finite simple connected graph $G = (V,E)$, we introduce an invariant, its blowup-polynomial $p_G(\{ n_v : v \in V \})$. This is obtained by dividing the determinant of the distance matrix of its blowup graph $G[{\bf n}]$…

This paper is concerned with the distribution in the complex plane of the roots of a polynomial sequence $\{W_n(x)\}_{n\ge0}$ given by a recursion $W_n(x)=aW_{n-1}(x)+(bx+c)W_{n-2}(x)$, with $W_0(x)=1$ and $W_1(x)=t(x-r)$, where $a>0$,…

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally):…

The convex hull $\psi_{n,n}$ of certain $(n!)^2$ tensors was considered recently in connection with graph isomorphism. We consider the convex hull $\psi_n$ of the $n!$ diagonals among these tensors. We show: 1. The polytope $\psi_n$ is a…

Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${\cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that…

Every generic linear functional $f$ on a convex polytope $P$ induces an orientation on the graph of $P$. From the resulting directed graph one can define a notion of $f$-arborescence and $f$-monotone path on $P$, as well as a natural graph…

Let $G$ be a graph and $x_1, x_2, \ldots, x_k$ be distinct vertices of $G$. We say $(G,x_1x_2\ldots x_k)$ has a $C_k$-minor or $G$ has a $C_k$-minor rooted at $x_1x_2\ldots x_k$, if there exist pairwise disjoint sets $X_1, X_2, \ldots,…

Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles.…

A path $P = v_1, ..., v_t$ is a {\em triangle path} (respectively, {\em monophonic path}) of $G$ if no edges exist joining vertices $v_i$ and $v_j$ of $P$ such that $|j - i| > 2$; (respectively, $|j - i| > 1$). A set of vertices $S$ is {\em…

Toric topology assigns to each $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets an $(m+n)$-dimensional moment-angle manifold $\mathcal{Z}_P$ with an action of a compact torus $T^m$ such that $\mathcal{Z}_P/T^m$ is a…

Given a graph $G=(V,E)$ and a set $S \subseteq \binom{V}{2}$ of terminal pairs, the minimum multicut problem asks for a minimum edge set $\delta \subseteq E$ such that there is no $s$-$t$-path in $G -\delta$ for any $\{s,t\}\in S$. For…

An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and…

Given a directed graph D = (N, A) and a sequence of positive integers 1 <= c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that are defined as the convex hulls of simple paths and cycles of D of cardinality c_p for…

The symmetric edge polytope ($\mathrm{SEP}$) of a finite simple graph $G$ is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. Among the information encoded by these polytopes are the symmetries of…

The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of…

Let $G=(V,E)$ be a simple connected graph. A connected edge cover of $G$ is a subset $S\subseteq E$ such that every vertex of $G$ is incident with at least one edge in $S$ and the subgraph induced by $S$ is connected. The connected edge…

Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from…

Using knot theory, we construct a linear representation of the CGW algebra of type $D_n$. This representation has degree $n^2-n$, the number of positive roots of a root system of type $D_n$. We show that the representation is generically…