The first uniformly finite homology group with coefficients in $\mathbb{Z}$ and a characterisation of its vanishing in the transitive case

Planar transitive graphs

We prove that the first homology group of every planar locally transitive finite graph $G$ is a finitely generated ${\rm Aut}(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible.

A Stallings' type theorem for quasi-transitive graphs

We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings' splitting theorem for multi-ended finitely generated groups and indeed it implies this theorem. It will also lead to a characterisation of accessible graphs in terms of tree amalgamations. We obtain applications of our results for hyperbolic graphs, planar graphs and graphs without any thick end. The application for planar graphs answers a question of Mohar in th…

平面中全局刚性边传递图与距离正则图的分类

Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane

本文将Jackson等人关于顶点传递图全局刚性的刻画推广至由自同构群确定的图，利用最小与最大次数分类了平面中的全局刚性边传递图和距离正则图。

Edge-transitive bi-Cayley graphs

A graph $\G$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {\em bi-Cayley graph\/} over $H$. Such a graph $\G$ is called {\em normal\/} if $H$ is normal in the full automorphism group of $\G$, and {\em normal edge-transitive\/} if the normaliser of $H$ in the full automorphism group of $\G$ is transitive on the edges of $\G$. % In this paper, we give a characterisation of normal edge-transitive bi-Cayley graphs, %which form an important subfamily of bi-Cayley graphs, and in particular, we give a detailed description of $2$-arc-…

通过图的 überhomology 对连通支配进行范畴化

Categorifying connected domination via graph überhomology

本文将图的 überhomology 解释为偏序集同调，证明 bold homology 范畴化连通支配多项式，并给出树的消失性与完全图的刻画。

A characterisation of weakly locally projective amalgams related to $A_{16}$ and the sporadic simple groups $M_{24}$ and $He$

A simple undirected graph is weakly $G$-locally projective, for a group of automorphisms $G$, if for each vertex $x$, the stabiliser $G(x)$ induces on the set of vertices adjacent to $x$ a doubly transitive action with socle the projective group $L_{n_x}(q_x)$ for an integer $n_x$ and a prime power $q_x$. It is $G$-locally projective if in addition $G$ is vertex transitive. A theorem of Trofimov reduces the classification of the $G$-locally projective graphs to the case where the distance factors are as in one of the known examples. Although an analogue of Trofimov's result is not yet availabl…

Incidence Homology of Finite Projective Spaces

Let F* be the finite field of q elements and let P(n,q) be the projective space of dimension n-1 over F*. We construct a family H^{n}_{k,i} of combinatorial homology modules associated to P(n,q) over a coefficient field F field of characteristic p_{0}>0 co-prime to q. As FGL(n,q)-representations the modules are obtained from the permutation action of GL(n,q) on the subspaces of F*^n. We prove a branching rule for H^{n}_{k,i} and use this rule to determine these homology representations completely. The main results are a duality theorem and the complete characterisation of H^{n}_{k,i} in terms …

Characterisations of algebraic properties of groups in terms of harmonic functions

We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an infinite, connected, locally finite …

Transitivity in finite general linear groups

It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme …

wreath积中对称群的传递性

Transitivity in wreath products with symmetric groups

研究wreath积G Wr S_n中有限阿贝尔群G的传递子集结构，利用特征理论和关联方案分析，推广Livingstone-Wagner定理，并与正交多项式联系，扩展对称群S_n的研究。

顶点传递图中体积增长、等周性与逃逸概率之间的尖锐关系

Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs

本文在顶点传递图中建立了体积增长、等周性与随机游走逃逸概率之间的尖锐定量关系，给出了 Varopoulos 定理的有限性精细化，并证实了 Benjamini-Kozma 猜想及相关局部性结果。

关于循环图的半传递可图性

On semi-transitive orientability of circulant graphs

本文研究了循环图的半传递可图性，证明了某些k正则循环图的半传递性，并给出了相关表示数的上界。

Circulant association schemes on triples

Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an association scheme on triples (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and…

Semisymmetric elementary abelian covers of the Möbius-Kantor graph

Let $\p_N \colon \tX \to X$ be a regular covering projection of connected graphs with the group of covering transformations isomorphic to $N$. If $N$ is an elementary abelian $p$-group, then the projection $\p_N$ is called $p$-elementary abelian. The projection $\p_N$ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of $\Aut X$ lifts along $\p_N$, and semisymmetric if it is edge- but not vertex-transitive. The projection $\p_N$ is minimal semisymmetric if $p_N$ cannot be written as a composition $\p_N = \p \circ \p_M$ of two (nontrivial) regular cover…

Infinite primitive directed graphs

A group $G$ of permutations of a set $Ω$ is {\em primitive} if it acts transitively on $Ω$, and the only $G$-invariant equivalence relations on $Ω$ are the trivial and universal relations. A graph $Γ$ is {\em primitive} if its automorphism group acts primitively on its vertex set. A graph $Γ$ has {\em connectivity one} if it is connected and there exists a vertex $α$ of $Γ$, such that the induced graph $Γ\setminus \{α\}$ is not connected. If $Γ$ has connectivity one, a {\em block} of $Γ$ is a connected subgraph that is maximal subject to the condition that it does not have connectivity one. Th…