Shiquan Ren, Chengyuan Wu
A weighted simplicial complex is a simplicial complex with values (called weights) on the vertices. In this paper, we consider weighted simplicial complexes with $\mathbb{R}^2$-valued weights. We study the weighted homology and the weighted analytic torsion for such weighted simplicial complexes.
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We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which combinatorial geometry can be meaningfully applied
Chromatic polynomials of simplicial complexes
我们考虑单纯复形的s-色多项式，图的色多项式的高维类似物.
Signless Laplacian spectral radius of simplicial complexes without holes
研究无洞单纯复形的无符号拉普拉斯谱半径的极值问题，扩展了无环图的经典极值结果到高维单纯复形，同时建立具有指定贝蒂数的单纯复形的无符号拉普拉斯谱半径上界，并推导出超图和单纯复形的图论数上界。
The Moore Bound for Regular Simplicial Complexes
本文推导正则单纯复形的Moore型上界，并基于最小度给出直径的对数下界。
The Fundamental Group of Balanced Simplicial Complexes and Posets
我们建立了一个上界的基数的最小生成集的基本群的一个大家庭的连接，平衡的单纯复形，更一般地说，单纯偏序集。
Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known invariants of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphi…
Latin trades and simplicial complexes
本文引入了拉丁方的交易空间的概念。使用Sage和GAP软件包Simplicial Homology的计算。
Given a finite CAT(0) cubical complex, we define a flag simplicial complex associated to it, called the crossing complex. We show that the crossing complex holds much of the combinatorial information of the original cubical complex: for example, hyperplanes in the cubical complex correspond to vertex links in the crossing complex, and the crossing complex is balanced if and only if the cubical complex is cubically balanced. The most significant result is that the sets of $f$-vectors of CAT(0) cubical complexes and flag simplicial complexes are equal, up to an invertible linear transformation.
We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$'s density. This generalizes a well-known classical result of Erdős and Gallai \cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.
Pure Simplicial and Clique Complexes with a Fixed Number of Facets
本文研究纯单纯复形与团复形的结构和计数问题，给出仅依赖面列表的团复形判定充要条件，证明一类纯团复形可由面邻接矩阵唯一确定，并计数固定面数的纯单纯复形及给出纯团复形数量的上界。
We show that the $h$-vector of a $2$-dimensional PS ear-decomposable simplicial complex is a pure $\mathcal{O}$-sequence. This provides a strengthening of Stanley's conjecture for matroid $h$-vectors in rank $3$. Our approach modifies the approach of combinatorial shifting for arbitrary simplicial complexes to the setting of $2$-dimensional PS ear-decomposable complexes, which allows us to greedily construct a corresponding pure multicomplex.
Cheeger type inequalities for high dimensional simplicial complexes
本文从组合学角度研究高维单纯复形上的离散Cheeger型不等式，将图论中的经典Cheeger不等式推广到高维情形。
Topics in Boolean Representable Simplicial Complexes
本文研究了布尔可表示单纯复形（BRSC）理论，探讨了其上的各种算子，并考察了更高维度的可壳性，同时提出了若干新猜想。
A short simplicial h-vector and the upper bound theorem
我们验证了一类奇维单纯复形的上界猜想，特别是包括所有具有孤立奇点的欧拉单纯复形。该证明依赖于单纯复形的一个新不变量--短单纯h-向量。
Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure strongly shellable complexes. Meanwhile, pure strongly shellable complexes can be characterized by the corresponding codimension one graphs. In addition, we show that the facet ideals of pure strongly shellable complexes have linear quotients.
This article surveys some of the work done in recent years on random simplicial complexes. We mostly consider higher-dimensional analogs of the well known phase transition in $G(n, p)$ theory that occurs at $p = \frac 1n$ . Our main objective is to provide a more streamlined and unified perspective of some of the papers in this area.
Flows on Simplicial Complexes
给定一个图$G$，无处为零的$\ZZ_q$ -流的数量$ϕ_G(q)$已知是$q$中的多项式。我们将无零$\ZZ_q$ -流的定义推广到维数大于1的单纯复形$Δ$，并证明了相应函数$ϕ_Δ(q)$对于某些$q$和单纯复形的某些子类的多项式性.
Simplicial complexes defined on groups
该研究将群上图的研究拓展至单纯复形，定义了多种群自同构保持的单纯复形，重点探究两种独立复形及二者重合的群类结果，为群与组合结构交叉研究提供初步基础。
Denham, Suciu and Panov, Ray computed ranks of homotopy groups and Poincaré series of a moment-angle-complex $\mathcal Z(\mathcal K$) / Davis-Januzskiewicz space $DJ(\mathcal K)$ associated to a flag simplicial complex $\mathcal K$. In this note we revisit these results and interpret them as polynomial bounds on the face numbers of an arbitrary simplicial flag complex.
We introduce the notion of a quasi-matroidal class of ordered simplicial complexes: an approximation to the idea of a matroid cryptomorphism in the landscape of ordered simplicial complexes. A quasi-matroidal class contains pure shifted simplicial complexes and ordered matroid independence complexes. The essential property is that if a fixed simplicial complex belongs to this class for every ordering of its vertex set, then it is a matroid independence complex. Some examples of such classes appear implicitly in the matroid theory literature. We introduce various such classes that highlight dif…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the (lower-)associated independence hypergraph of the random hypergraph in terms of the probability distributions of certain random simplicial complex and certain random independence hypergraph of Erdös-Rényi type. We construct a graded structure of the map algebra explicitly and give algorithms to generate random hypergraphs and random simplicial complexes.
Hopf monoids of ordered simplicial complexes
本文通过引入Hopf类的概念，系统研究有序单纯复形与拟阵相关的Hopf幺半群结构，并在面初始复形和无界有序拟阵两类情形下计算了对极，得到无重数且无抵消的公式。
Lower Bounds for Buchsbaum* Complexes
研究了一类$(d-1)$ -维Buchsbaum* 单纯复形.证明了（完全）平衡Buchsbaum* 单纯复形的秩选择子复形也是Buchsbaum*.利用这一结果，建立了平衡Buchsbaum* 单纯复形的$h$ -数的下界.此外，还导出了标志$m$ -Buchsbaum* 单纯复形的$h$ -数的下界，并对等式的情况进行了处理.
Simplicial Complexes of Triangular Ferrers Boards
我们研究了单纯复形，从非攻击车的位置上的一个子类的费勒斯板，具有$a_i$行的长度$i$，其中$a_i>0$和$i\leq n$的一些正整数$n$。特别是，我们将调查他们的方面，他们的同伦类型和同源性的枚举性质。
Simplicial Complexes Obtained from Qualitative Probability Orders
本文首次研究了定性概率序的初始段--抽象单纯复形。这是一个自然类，包含阈值复合体，并包含在移位复合体中，但两者都不相等。特别地，我们构建了一个定性的概率顺序26个原子，有一个初始段，这是不是一个阈值单纯复形。虽然26可能不是这样的例子存在的最小数目，我们提供了一些证据，它不能小得多。我们证明了这个类的一些必要条件，并对它们的一个特征作了一个猜想。该刻画依赖于合作对策理论的一些思想。
We construct $d$-dimensional pure simplicial complexes and pseudo-manifolds (without boundary) with $n$ vertices whose combinatorial diameter grows as $c_d n^{d-1}$ for a constant $c_d$ depending only on $d$, which is the maximum possible growth. Moreover, the constant $c_d$ is optimal modulo a singly exponential factor in $d$. The pure simplicial complexes improve on a construction of the second author that achieved $c_d n^{2d/3}$. For pseudo-manifolds without boundary, as far as we know, no construction with diameter greater than $n^2$ was previously known.
Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.
We prove that the family of facets of a pure simplicial complex of dimension up to three satisfies the Erdős-Ko-Rado property whenever it is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes.
The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$, $d$, and $(n-d-1)$-vertex-connected. It is also shown that the facet graph of a $d$-cycle cannot be split into more than $s$ connected components by removing at most $s$ vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.
A common recursion for Laplacians of matroids and shifted simplicial complexes
一个递归由于库克表示的拉普拉斯特征值的拟阵M的特征值的删除M-e和压缩M/e的特征值由一个固定的元素e，和一个误差项。我们表明，这个误差项是简单地由拉普拉斯特征值对（M-e，M/e）。我们进一步表明，通过适当的推广删除和收缩到任意单纯复形，拉普拉斯特征值的移位单纯复形满足这个完全相同的递归。我们证明了满足这种递归的单纯复形类在各种自然操作下是封闭的，并且这种递归的几种专门化减少到自然不变量的基本递归。我们还找到了任意一对移位复形的拉普拉斯特征值关于一种广义度序列的简单公式。
Critical groups of simplicial complexes
我们将临界群理论从图推广到单纯复形。具体地说，给定一个单纯复形，我们定义了一个家庭的阿贝尔群的组合拉普拉斯算子，推广的建设的临界群的一个图。我们展示了如何实现这些关键群体明确的cokernels减少拉普拉斯算子，并证明他们是有限的，与单形生成树的加权枚举给出的订单。我们描述了一个复杂的临界组代表沿其面流，并勾勒出另一个潜在的解释类似的周组。
We study ordered matroids and generalized permutohedra from a Hopf theoretic point of view. Our main object is a Hopf monoid in the vector species of extended generalized permutahedra equipped with an order of the coordinates; this monoid extends the Hopf monoid of generalized permutahedra studied by Aguiar and Ardila. Our formula for the antipode is cancellation-free and multiplicity-free, and is supported only on terms that are compatible with the local geometry of the polyhedron. Our result is part of a larger program to understand orderings on ground sets of simplicial complexes (for insta…
The Euler characteristic, thought of as a function that assigns a numerical value to every finite simplicial complex, is locally determined in both a combinatorial sense and a geometric sense. In this note we show that not every function that assigns a numerical value to every finite simplicial complex via a linear combination of the numbers of simplices in each dimension is locally determined in either sense. In particular, the Charney-Davis quantity $λ(L)$ is not locally determined in either sense if it is defined on a set of simplicial complexes that includes all flag spheres of a given odd…
The notion of $f$-ideal is recent and has so far been studied in several papers. In \cite{qfi}, the idea of $f$-ideal is generalized to quasi $f$-ideals, which is much larger class than the class of $f$-ideals. In this paper, we introduce the concept of quasi $f$-simplicial complex and quasi $f$-graph. We give a characterization of quasi $f$-graphs on $n$ vertices. A complete solution of connectedness of quasi $f$-simplicial complexes is described. We have also shown a method of constructing Cohen-Macaulay quasi $f$-graphs.
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, which are simplicial complexes parametrized by graphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.
We give a negative answer to a question proposed in [3], regarding the $h$-vector of ($S_r$) simplicial complexes.
We introduce t-uniform simplicial complexes and we show that the lengths of spheres in such complexes are the terms of certain Lucas sequences. We find optimal constants for the linear isoperimetric inequality in the hyperbolic case.
Conditions for Virtually Cohen--Macaulay Simplicial Complexes
本文给出单纯复形成为虚拟Cohen-Macaulay单纯复形的充分条件，引入虚拟可剥单纯复形这一可剥复形的推广，并证明其具有虚拟Cohen-Macaulay性质，推广了可剥复形是Cohen-Macaulay的结论。
Defined by a single axiom, finite abstract simplicial complexes belong to the simplest constructs of mathematics. We look at a a few theorems.
Simplicial complexes in Macaulay2
本研究聚焦于Macaulay2软件中的SimplicialComplexes包，梳理并介绍了该包的若干核心功能特性，为相关代数与组合数学研究提供工具参考。
On simplicial and cubical complexes with short links
我们考虑闭单纯形和立方$n$ -复形的$(n-2)$ -面的连接.特别是，我们考虑的情况下，当这个链接的大小为3或4，即，每个$(n-2)$ -面包含在3或4个$n$ -面中。这种具有{ \em short}（即长度为3或4）链路的单纯复形完全由它们的{ \em特征划分}来分类。我们还考虑嵌入到超立方体的骨架的单纯形和立方复合体。
On Simplicial Complexes with Extremal Total Betti number and Total Bigraded Betti Number
本文研究了在每一维给定顶点数的情况下，确定哪些单纯复形的总Betti数和总双阶Betti数达到最大或最小值。
Alexander's conjecture for infinite simplicial complexes
将Alexander猜想从有限单纯复形推广到无限单纯复形，证明任意两个具有共同子划分的无限三角剖分必有共同星形子划分。
Dirac's theorem on simplicial matroids
我们引入k-超团复形的概念，即，最大的单纯复形上的集合[n]与一个固定的k-骨架。这些单纯复形是团复形（或旗复形）的高维类似物（情况k=2），它们是一类丰富的新的单纯复形。我们表明，狄拉克定理弦图有一个更高的维模拟图和团复合体得到取代，分别由单纯拟阵和k-超团复合体。我们还证明了一个高维模拟斯坦利的重新制定的狄拉克定理弦图。
We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.
Topology of medial regime random simplicial complexes
该研究分析中间 regime 随机单纯复形的拓扑，证明其具高度连通性与迭代悬挂同伦型，提出更灵活的高连通性组合判据，确定其拓扑复杂度上界为2且齐次类几乎必然等于2。
We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces. In particular, we provide sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length equal to its codimension). We also show that all balanced simplicial complexes are virtually Cohen-Macaulay.
In this manuscript, we define and study probabilistic values for cooperative games on simplicial complexes. Inspired by the work of Weber "Probabilistic values for games", we establish the new theory step by step, following the classical axiomatization, i.e. using the linearity axiom, the dummy axiom, etc. Furthermore, we define Shapley values on simplicial complexes generalizing the classical notion in literature. Remarkably, the traditional axiomatization of Shapley values can be extended to this general setting for a rather interesting class of complexes that generalize the notion of vertex…
Necessary Conditions for Geometric Realizability of Simplicial Complexes
我们与任何单纯复形$\K$和任何整数$m$的线性方程组和不等式系统。如果$\K$在$\R^m$中有单纯嵌入，则系统有整数解。这一结果推广了I.诺维克（2000）。
Composition of simplicial complexes, polytopes and multigraded Betti numbers
对于m个顶点上的单纯复形K和单纯复形K1，.，Km是复合单纯复形K（K1，.，Km）的。这种构造推广了A. Bahri，M. Bendersky，F. R. Cohen和S. Gitler，并允许描述多面体P（P1，...，Pm）由G.阿格纳森该复合定义了一组有限单纯复形上的操作数的结构。我们证明了以下内容：（1）一个合成复形K（K1，.，Km）是单形球当且仅当K是单形球，Ki是单形的边界;（2）一类球面神经复合体在复合运算下是闭的（3）最后，我们给出了K（K1，…，Km）表示为K，K1，.的多重贝蒂数，Km使用生成函数的组合。
We correct the proofs of the main theorems in our paper "Limit theorems for Betti numbers of random simplicial complexes".
We suggest a new possible high dimensional analogue to metric distortion. We then show a possible method for providing lower bounds to this distortion and use this method to prove a "Bourgain-type" distortion theorem for Linial-Meshulam random complexes.
Glicci simplicial complexes
联络论中的一个主要问题是：多项式环中的齐次Cohen-Macaulay理想是否都是glicci，即是否在完全交的G-联络类中。对于由弱顶点可分解的单纯复形定义的Stanley-Reisner理想，我们给出了肯定的回答.这类复形分别包括拟阵、移位复形和Gorenstein复形。此外，我们还构造了一个单纯复形，证明了它的Glicci性质依赖于基场的性质。作为我们的方法的应用，我们建立了新的证据，斯坦利的两个aesturtures可分割的配合物和斯坦利分解。
Weighted simplicial complexes (WSCs) are powerful tools for describing weighted cloud data or networks with weighted nodes. In this paper, we propose a novel approach to study WSCs via the concept of polarization. Polarization of a WSC allows one to construct a new (unweighted) simplicial complex which coincides with an object called the mixed wreath product. This new construction preserves several properties and invariants of the underlying simplicial complex of a WSC. Our main focus is to analyze WSCs through their underlying simplicial complexes and mixed wreath products. Combinatorially, w…
We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal Cohen-Macaulay. We show that many interesting examples of Cohen-Macaulay complexes in combinatorics are minimal, including Rudin's ball, Ziegler's ball, the dunce hat, and recently discovered non-partitionable Cohen-Macaulay complexes. We further provide various ways to construct such complexes.
We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that rank selected subcomplexes of balanced simplicial complexes satisfying Serre's condition $(S_{\ell})$ retain $(S_{\ell})$. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial compex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.
Colorings of simplicial complexes and vector bundles over Davis-Januszkiewicz spaces
证明了单纯复形K的染色性质可以通过Davis-Januszkiewicz空间上丛的分裂性质来反映，Davis-Januszkiewicz空间的Chern类由K的Stanley-Reisner代数的生成元中的初等对称多项式给出.
Enriched homology and cohomology modules of simplicial complexes
对于{1，2，.}上的单纯复形X，n}我们定义了丰富的同调和上同调模。它们是k[x_1，..]上的分次模，x_n]，其秩等于约化同调群和上同调群的维数。我们用丰富模刻画了Cohen-Macaulay，l-Cohen-Macaulay，Buchsbaum和Gorenstein* 复形X以及可定向流形.本文引入了单纯复形围长的概念，并提出了一个关于围长与单纯复形不变量之间关系的猜想。我们还对增广的同调模施加了强消失条件，并刻画了由此得到的单纯复形。它们是区组设计，包括Steiner系统S（c，d，n）和偶数维的循环多面体。
A relative simplicial complex is a collection of sets of the form $Δ\setminus Γ$, where $Γ\subset Δ$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of $f$-vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homog…
Line graphs of simplicial complexes
本文研究纯单纯复形的线图，证明可基于其线图的组合结构计算面理想的第二阶分次Betti数，并刻画了线图为完全（二部）图的纯单纯复形及线图应满足的条件。
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