Threshold graphs, shifted complexes, and graphical complexes

Simplicial matrix-tree theorems

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $Δ$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $Δ$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $Δ$ and replacing the entries …

A Conjectural Brouwer Inequality for Higher-Dimensional Laplacian Spectra

We present a generalization of Brouwer's conjectural family of inequalities -- a popular family of inequalities in spectral graph theory bounding the partial sum of the Laplacian eigenvalues of graphs -- for the case of abstract simplicial complexes of any dimension. We prove that this family of inequalities holds for shifted simplicial complexes, which generalize threshold graphs, and give tighter bounds (linear in the dimension of the complexes) for simplicial trees. We prove that the conjecture holds for the the first, second, and last partial sums for all simplicial complexes, generalizing…

Flow-Firing 的三区域定理

A Three-Regime Theorem for Flow-Firing

本文研究 2 维胞腔复形上的流重路由点火系统，借鉴前人工作，回答了该过程是否终止及终止是否唯一的问题，提出了完全判定系统终止性的三区域定理。

平方循环图的全割复形的同伦型

Homotopy Type of Total Cut Complexes of Squared Cycle Graphs

本文研究了平方循环图的全割复形的同伦型，证明了Bayer等人对k=3时的猜想，并确认了n=3k+1和3k+2的一般情形。

Cellular resolutions of cointerval ideals

Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d=2), and strictly contains the class of `strongly stable' hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decomposit…

映射空间的图示组合：配置空间类型的模之间的映射空间

Graphical composition of mapping spaces between modules of configuration-space-type

本文提出了一种组合图示方法，用于描述配置空间类型模之间的映射空间的组合结构，并探讨了余诱导函子在相关余模上的作用。

度序列实现图的最大哈密顿性

Maximal Hamiltonicity of realization graphs of degree sequences

本文证明度序列实现图是最大哈密顿的，通过归纳切割和移位族分类，解决了 Mütze 和 Barrus 的开放问题，并在 Lean 4 中形式化验证。

Spanning Simplicial Ccomplexes of Uni-Cyclic Graphs

In this paper, we introduce the concept of spanning simplicial complexes $Δ_s(G)$ associated to a simple finite connected graph G. We give the characterization of all spanning trees of the uni-cyclic graph $U_{n,m}$. In particular, we give the formula for computing the Hilbert series and h-vector of the Stanley-Riesner ring k[Δ_s(U_{n,m})]. Finally, we prove that the spanning simplicial complex $Δ_s(U_{n,m})$ is shifted hence $Δ_s(U_{n,m})$ is shellable.