我们证明了局部有限拟传递连通图可及性的两个刻画。首先，我们证明任意这样的图 $G$ 是可及的，当且仅当其有限阶分离的集合是一个 ${\ m Aut}(G)$-有限生成的半环。第二个刻画表明，$G$ 是可及的，当且仅当每一个以树融合形式进行的分裂过程都在有限步后停止。
英文摘要：
We prove two characterisations of accessibility of locally finite quasi-transitive connected graphs. First, we prove that any such graph $G$ is accessible if and only if its set of separations of finite order is an ${\rm Aut}(G)$-finitely generated semiring. The second characterisation says that $G$ is accessible if and only if every process of splittings in terms of tree amalgamations stops after finitely many steps.
We introduce the notion of an \emph{$n$-dimensional mixed dihedral group}, a general class of groups for which we give a graph theoretic characterisation. In particular, if $H$ is an $n$-dimensional mixed dihedral group then the we construct an edge-transitive Cayley graph $Γ$ of $H$ such that the clique graph $Σ$ of $Γ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducing a particular \emph{edge-affine} action on $\K_{2^n,2^n}$. Conversely, we prove that if $Σ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducin…
In his doctoral thesis, Snover proved that any binary $(m,256,δ)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,δ)=(16,6)$ or $(15,5)$ respectively. We prove that these codes are also characterised as \emph{completely regular} binary codes with $(m,δ)=(16,6)$ or $(15,5)$, and moreover, that they are \emph{completely transitive}. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular cod…
We study the first uniformly finite homology group of Block and Weinberger for uniformly locally finite graphs, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_2$. When the graph is a tree, or coefficients are in $\mathbb{Z}_2$, a characterisation of the group is obtained. In the general case, we describe three phenomena that entail non-vanishing of the group; their disjunction is shown to also be necessary for non-vanishing in the case of transitive graphs.
This short paper presents characterisations of normal arc-transitive circulants and arc-transitive normal circulants, that is, for a connected arc-transitive circulant $Γ=\Cay(C,S)$, it is shown that 1. Aut(C,S) is transitive on S if and only if each element of S has order n; 2. $AutΓ\rhd C$ if and only if S does not contain a coset of any subgroup. This completes the classification of arc-transitive circulants given by Li-Xia-Zhou.
We consider codes of length $m$ over an alphabet of size $q$ as subsets of the vertex set of the Hamming graph $Γ=H(m,q)$. A code for which there exists an automorphism group $X\leq Aut(Γ)$ that acts transitively on the code and on its set of neighbours is said to be neighbour transitive, and were introduced by the authors as a group theoretic analogue to the assumption that single errors are equally likely over a noisy channel. Examples of neighbour transitive codes include the Hamming codes, various Golay codes, certain Hadamard codes, the Nordstrom Robinson codes, certain permutation codes …
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…
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-…
We prove that the cut space of any transitive graph $G$ is a finitely generated ${\rm Aut}(G)$-module if the same is true for its cycle space. This confirms a conjecture of Diestel which says that every locally finite transitive graph whose cycle space is generated by cycles of bounded length is accessible. In addition, it implies Dunwoody's conjecture that locally finite hyperbolic transitive graphs are accessible. As a further application, we obtain a combinatorial proof of Dunwoody's accessibility theorem of finitely presented groups.
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.
We give a new characterisation of virtually free groups using graph minors. Namely, we prove that a finitely generated, infinite group is virtually free if and only if for any finite generating set, the corresponding Cayley graph is minor excluded. This answers a question of Ostrovskii and Rosenthal. The proof relies on showing that a finitely generated group that is minor excluded with respect to every finite generating set is accessible, using a graph-theoretic characterisation of accessibility due to Thomassen and Woess.
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…
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 …
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 …
We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papsoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess.
