相关论文: Angle sums on polytopes and polytopal complexes

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes…

A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph $G$ is the minimum…

The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler…

In this paper, we employ methods of contour integration and residue calculus to investigate the parity of two classes of cyclotomic Euler-type sums. One class involves products of cyclotomic harmonic numbers, while the other involves…

Several recent papers have addressed the problem of characterizing the $f$-vectors of cubical polytopes. This is largely motivated by the complete characterization of the $f$-vectors of simplicial polytopes given by Stanley, Billera, and…

For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Gamma(\Theta)$ (dependent on the sequence of subdivisions) whose $f$-vector is the…

The angle defect, which is the standard way to measure curvature at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the…

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and…

We find families of simplicial complexes where the simplicial chromatic polynomials defined by Cooper--de Silva--Sazdanovic \cite{CdSS} are Hilbert series of Stanley--Reisner rings of auxiliary simplicial complexes. As a result, such…

In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general…

In this paper we continue our recent analysis [K. Haydukivska et al., J. Mol. Liq., 2021, 328, 115456] of complex molecules with two branching points at both ends of the linear backbone with $f_1$ and $f_2$ side arms starting from them,…

We discuss some formulae which express the Alexander polynomial (and thus the zeta-function of the classical monodromy transformation) of a plane curve singularity in terms of the ring of functions on the curve. One of them describes the…

A triangulation of a simplicial complex $\Delta$ is called uniform if the $f$-vector of its restriction to a face of $\Delta$ depends only on the dimension of that face. This paper proves that the entries of the $h$-vector of a uniform…

Let $\delta(\Pc) = (\delta_0, \delta_1,..., \delta_d)$ be the $\delta$-vector of an integral polytope $\Pc \subset \RR^N$ of dimension $d$. Following the previous work of characterizing the $\delta$-vectors with $\sum_{i=0}^d \delta_i \leq…

The $f$-vector of a $d$-dimensional polytope $P$ stores the number of faces of each dimension. When $P$ is simplicial the Dehn--Sommerville relations condense the $f$-vector into the $g$-vector, which has length…

We study the vertices of the polytopes of all affine maps (a.k.a. hom-polytopes) between higher dimensional simplices, cubes, and crosspolytopes. Systematic study of general hom-polytopes was initiated in [3]. The study of such vertices is…

In this paper, we discuss f- and flag-vectors of 4-dimensional convex polytopes and cellular 3-spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if there are many…

We recall first Gallai-simplicial complex $\Delta_{\Gamma}(G)$ associated to Gallai graph $\Gamma(G)$ of a planar graph $G$. The Euler characteristic is a very useful topological and homotopic invariant to classify surfaces. In Theorems 3.2…

In this paper, we define extended trigonometric functions via series and employ the method of contour integration to investigate the parity of certain cyclotomic Euler sums and multiple polylogarithm function. We can provide the statement…

We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power…