Schreier图：传递性与覆盖

格罗-陈覆盖的群由有界自动机生成的施雷jer图

Galois coverings of Schreier graphs of groups generated by bounded automata

本文研究了由有界自动机生成的群的施雷jer图的格罗-陈覆盖特性，并探讨了几个特定群的施雷jer图的齐次函数和L函数。

Counting dimer coverings on self-similar Schreier graphs

We study partition functions for the dimer model on families of finite graphs converging to infinite self-similar graphs and forming approximation sequences to certain well-known fractals. The graphs that we consider are provided by actions of finitely generated groups by automorphisms on rooted trees, and thus their edges are naturally labeled by the generators of the group. It is thus natural to consider weight functions on these graphs taking different values according to the labeling. We study in detail the well-known example of the Hanoi Towers group $H^{(3)}$, closely related to the Sier…

On groups and simplicial complexes

The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this paper is to begin to develop such a framework for $k$-regular simplicial complexes of general dimension $d$. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of $k$-regular simplicial complexes as quotients of one universal object: the $k$-regular $d$-dimensional arboreal complex, which is its…

稀疏图及其Benjamini-Schramm极限：一次谱游历

Sparse graphs and their Benjamini-Schramm limits: a spectral tour

本文综述了有界平均度稀疏图的Benjamini-Schramm收敛的谱性质及其应用，重点关注随机Schreier图和覆盖图，并提出了强收敛在顶点间典型图距离上的新应用。

对所有 r 均具有独立精确 r-覆盖的图

Graphs with Independent Exact $r$-covers for all $r$

本文构造了对所有 r ≤ d 均含独立精确 r-覆盖的有限 d-正则图，回答了 Gray 和 Johnson 的问题，并利用公共覆盖与 Schreier 陪集图理论给出了 d=3 至 6 时阶数最小的构造。

On the Classification of Binary Completely Transitive Codes with Almost-Simple Top-Group

A code $C$ in the Hamming metric, that is, is a subset of the vertex set $V\varGamma$ of the Hamming graph $\varGamma=H(m,q)$, gives rise to a natural distance partition $\{C,C_1,\ldots,C_ρ\}$, where $ρ$ is the covering radius of $C$. Such a code $C$ is called completely transitive if the automorphism group $\rm{Aut}(C)$ acts transitively on each of the sets $C$, $C_1$, \ldots, $C_ρ$. A code $C$ is called $2$-neighbour-transitive if $ρ\geq 2$ and $\rm{Aut}(C)$ acts transitively on each of $C$, $C_1$ and $C_2$. Let $C$ be a completely transitive code in a binary ($q=2$) Hamming graph having ful…