How is a Chordal Graph like a Supersolvable Binary Matroid?

拟阵复形与Orlik-Solomon代数

Matroid complexes and Orlik-Solomon algebras

本文为超可解拟阵的Orlik-Solomon代数构造组合拟自由微分分次模型，推广Kontsevich的容许图cdga，并据此证明其Koszul性。

Graphs whose flow polynomials have only integral roots

We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph. We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots. Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots are the cycle matroids of planar chordal graphs.

Koszul 分次 Möbius 代数与强弦图

Koszul Graded Möbius Algebras and Strongly Chordal Graphs

本文研究了拟阵分次 Möbius 代数何时为 Koszul 代数，基于图的性质刻画了图圈拟阵的 Koszul 分次 Möbius 代数，并通过边排序给出了强弦图的新刻画。

Supersolvability and Freeness for $ψ$-graphical Arrangements

Let $G$ be a simple graph on the vertex set $\{v_1,\dots,v_n\}$ with edge set $E$. Let $K$ be a field. The graphical arrangement $\mathcal{A}_G$ in $K^n$ is the arrangement $x_i-x_j=0, v_iv_j \in E$. An arrangement $\mathcal{A}$ is supersolvable if the intersection lattice $L(c(\mathcal{A}))$ of the cone $c(\mathcal{A})$ contains a maximal chain of modular elements. The second author has shown that a graphical arrangement $\mathcal{A}_G$ is supersolvable if and only if $G$ is a chordal graph. He later considered a generalization of graphical arrangements which are called $ψ$-graphical arrangem…

Modular Construction of Free Hyperplane Arrangements

In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by "gluing" complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrang…

Dirac's theorem on simplicial matroids

We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k=2) and they are a rich new class of simplicial complexes. We show that Dirac's theorem on chordal graphs has a higher-dimensional analogue in which graphs and clique complexes get replaced, respectively, by simplicial matroids and k-hyperclique complexes. We prove also a higher-dimensional analogue of Stanley's reformulation of Dirac's theorem on chordal graphs.

构建集的凸几何

Convex Geometry of Building Sets

证明有限交半格上的构建集构成超可解凸几何，并应用于统一推广嵌套集复形的限制结果。

Chordal and sequentially Cohen-Macaulay clutters

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal non-chordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to…

Bestvina-Brady群的拟阵与同构问题

Matroids and isomorphism problems for Bestvina-Brady groups

本文通过拟阵理论分解Bestvina-Brady群的图同构问题，证明有限表示仅依赖循环拟阵，并构造高连通图使拟阵不同但群同构，且对偶弦图定义的BBG为右角Artin群。

度量图论的一阶逻辑公理化

First-order logic axiomatization of metric graph theory

该研究类比Tarski的欧氏几何公理化，为度量图论中多类核心图建立带介于关系的一阶逻辑公理化，同时证明部分组合/拓扑定义的相关图类无法实现此类公理化。