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We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…
We classify graphs and, more generally, finite relational structures that are identified by C2, that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a…
Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs…
We introduce a graded homology theory for graded \'etale groupoids. For $\mathbb Z$-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we…
In this paper, we study orthogonal representations of simple graphs $G$ in $\mathbb{R}^d$ from an algebraic perspective in case $d = 2$. Orthogonal representations of graphs, introduced by Lov\'asz, are maps from the vertex set to…
Very recently, Galashin, Postnikov, and Williams introduced the notion of higher secondary polytopes, generalizing the secondary polytope of Gelfand, Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in…
A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the…
We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and…
We give a complete combinatorial characterization of weakly $d$-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in $\mathbb…
In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show the close relationship between lifted digraphs of voltage digraphs and line…
In this paper, we study the $d$-dimensional rectilinear drawings of the complete $d$-uniform hypergraph $K_{2d}^d$. Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove…
We consider the generating series of oriented and non-oriented hypermaps with controlled degrees of vertices, hyperedges and faces. It is well known that these series have natural expansions in terms of Schur and Zonal symmetric functions,…
A bicirculant is a regular graph that admits a semi-regular automorphism with two vertex-orbits of the same size. By $m$ we denote the size of vertex-orbits and by $d$ the valence of a bicirculant. Furthermore, we denote by $s$ the valence…
In this work, for a given oriented graph $D$, we study its interval and hull numbers, respectively, in the oriented geodetic, P3 and P3* convexities. This last one, we believe to be formally defined and first studied in this paper, although…
We construct a large class of indecomposable positive linear maps from the $2\times 2$ matrix algebra into the $4\times 4$ matrix algebra, which generate exposed extreme rays of the convex cone of all positive maps. We show that extreme…
We study the deformation complex of the dg wheeled properad of $\mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the…
This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional…
We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $\Gamma$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there…
Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2, \ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<\cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${\rm GK}(G)$, is defined as follows: its vertex set is…
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of…