The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected $2$-arc-transitive graphs of a fixed valence tends to $0$ as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-$L$ graphs, where $L$ is a fixed quasiprimitive graph-restrictive permutation group.
A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or $p^2$. Xu~[Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified half-arc-transitive graphs of order $p^3$ and valency $4$. In this paper we classify half-arc-transitive graphs of order $p^3$ and valency $6$ or $8$. In particular, the first known infinite family of half-arc-transitive Cayley graphs on non-metacyclic $p$…
A connected graph $Γ=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of $Γ$, and (ii) every minimal normal subgroup of $G$ has at most two orbits on $V$. In her papers [17,18], Praeger proved a connected $2$-arc-transitive graph of valency at least $3$ is a normal cover of some basic $2$-arc-transitive graph, and characterized the group-theoretic structures for basic $2$-arc-transitive graphs. Based on Praeger's theorems on $2$-arc-transiti…
Given integers $k$ and $m$, we construct a $G$-arc-transitive graph of valency $k$ and an $L$-arc-transitive oriented digraph of out-valency $k$ such that $G$ and $L$ both admit blocks of imprimitivity of size $m$.
We prove that, given a finite graph $Σ$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $Σ$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.
In this paper, we discuss the structural information about 2-arc-transitive (non-bipartite and bipartite) graphs of product action type. It is proved that a 2-arc-transitive graph of product action type requires certain restrictions on either the vertex-stabilizers or the valency. Based on the existence of some equidistant linear codes, a construction is given for 2-arc-transitive graphs of non-diagonal product action type, which produces several families of such graphs. Besides, a nontrivial construction is given for 2-arc-transitive bipartite graphs of diagonal product action type
In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $Γ(u) \cup Γ(v)$ is trivial. We find nineteen finitely presented groups with the property that any such group $G$ is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence ${3,4}$ on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup $G$ with $G_{uv}^{[1]} = 1$.
On arc-transitive inner-automorphic Cayley graphs on dihedral groups
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE Beijing Normal University（北京师范大学数学科学学院、数学与复杂系统教育部重点实验室）；Center for Combinatorics, LPMC Nankai University（南开大学组合数学中心、低维空间拓扑及其应用教育部重点实验室）
For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive $k$-circulants. We conjecture that, if $k$ is odd, then a cubic arc-transitive $k$-circulant has order at most $6k^2$. Our main result is a proof of this conjecture when $k$ is squarefree and coprime to $6$.
We show that there exist functions $c$ and $g$ such that, if $k$, $n$ and $d$ are positive integers with $d> g(n)$ and $Γ$ is a $d$-valent $2$-arc-transitive graph of order $kp^n$ with $p$ a prime, then $p\leqslant kc(d)$. In other words, there are only finitely many $d$-valent 2-arc-transitive graphs of order $kp^n$ with $d>g(n)$ and $p$ prime. This generalises a recent result of Conder, Li and Potočnik.
This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.
In this paper we study $G$-arc-transitive graphs $Δ$ where the permutation group $G_x^{Δ(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $Δ(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $Δ$, if $(x,y)$ is an arc of $Δ$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $Δ(x)$ and $Δ(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-l…
A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer $k$, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is $kp$ for some prime $p$, and that if $d\ge 4$, then there exist only finitely many connected $d$-valent 2-arc-transitive graphs whose order is $kp$ or $…
A graph is called cubic and tetravalent if all of its vertices have valency 3 and 4, respectively. It is called vertex-transitive and arc-transitive if its automorphism group acts transitively on its vertex-set and on its arc- set, respectively. In this paper, we combine some new theoretical results with computer calculations to construct all cubic vertex-transitive graphs of order at most 1280. In the process, we also construct all tetravalent arc-transitive graphs of order at most 640.
A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.
A detailed description of the structure of two-ended arc-transitive digraphs is given. It is also shown that several sets of conditions, involving such concepts as Property Z, local quasi-primitivity and prime out-valency, imply that an arc-transitive digraph must be highly-arc-transitive. These are then applied to give a complete classification of two-ended highly-arc-transitive digraphs with prime in- and out-valencies.
In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every finite arc-transitive bicirculant is a normal $r$-cover of an arc-transitive graph that lies in one of eight infinite families or is one of seven sporadic arc-transitive graphs. Moreover, each of these "basic" graphs is either an arc-transitive bicirculant or an arc-transitive circulant, and each graph in the latter case has an arc-transitive bicirculant normal $r$-cover for some integer $r$.
A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a half-arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed.
A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order $12p$.
We study vertex-quasiprimitive $2$-arc-transitive digraphs, reducing the problem of vertex-primitive $2$-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive $2$-arc-transitive digraphs where the action on vertices has O'Nan-Scott type SD or CD.
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.
A reductive characterization of arc-transitive circulants was given independently by Kovacs in 2004 and the first author in 2005. In this paper, we give an explicit characterization of arc-transitive circulants and their automorphism groups. Based on this, we give a proof of the fact that arc-transitive circulants are all CI-digraphs.
We introduce a construction of highly arc transitive digraphs using a layerwise direct product. This product generalizes some known classes of highly arc transitive digraphs but also allows to construct new such. We use the product to obtain counterexamples to a conjecture by Cameron, Praeger and Wormald on the structure of certain highly arc transitive digraphs.
We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc-transitive graphs of PA type that are not standard double covers of $s$-arc-transitive graphs of PA type, answering the existence question for these graphs.
It is known that there are precisely three transitive permutation groups of degree $6$ that admit an invariant partition with three parts of size $2$ such that the kernel of the action on the parts has order $4$; these groups are called $A_4(6)$, $S_4(6d)$ and $S_4(6c)$. For each $L\in \{A_4(6), S_4(6d), S_4(6c)\}$, we construct an infinite family of finite connected $6$-valent graphs $\{Γ_n\}_{n\in \mathbb{N}}$ and arc-transitive groups $G_n \le \rm{Aut}(Γ_n)$ such that the permutation group induced by the action of the vertex-stabiliser $(G_n)_v$ on the neighbourhood of a vertex $v$ is permu…
A general method for constructing sharply $k$-arc-transitive digraphs, i.e. digraphs that are $k$-arc-transitive but not $(k+1)$-arc-transitive, is presented. Using our method it is possible to construct both finite and infinite examples. The infinite examples can have one, two or infinitely many ends. Among the one-ended examples there are also digraphs that have polynomial growth.
Let $Γ$ be a finite $G$-vertex-transitive digraph. The in-local action of $(Γ,G)$ is the permutation group $L_-$ induced by the vertex-stabiliser on the set of in-neighbours of $v$. The out-local action $L_+$ is defined analogously. Note that $L_-$ and $L_+$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_-,L_+)$ are possible. We prove some general results, but pay special attention to the case when $L_-$ and $L_+$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the w…
One version of the polycirculant conjecture states that every vertex-transitive graph has a semiregular automorphism. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open case.
We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we first give a reduction result on the latter family of graphs: let $N$ be a normal subgroup of $G$ which has at least $3$ orbits on vertices. We show that $Γ$ is a cover of its quotient $Γ_N$ modulo the $N$-orbits, and that either $Γ_N$ is $(G/N,3)$-geodesic-transitive of the same girth as $Γ$, or $Γ_N$ is a $(G/N,2)$-arc-transitive strongly regular graph, or $…
We resolve two problems of [Cameron, Praeger, and Wormald -- Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each `building block' is a finite bipartite graph that is not a disjoint union of complete bipartite graphs. This was conjectured impossible in the above paper. We also describe the structure of 2-ended highly arc transitive digraphs in more generality, altho…
The main result of this paper is that, if $Γ$ is a connected 4-valent $G$-arc-transitive graph and $v$ is a vertex of $Γ$, then either $Γ$ is one of a well understood infinite family of graphs, or $|G_v|\leq 2^43^6$ or $2|G_v|\log_2(|G_v|/2)\leq |\VΓ|$ and that this last bound is tight. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs.
We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that $s\leqslant2$ for such digraphs, which provides the first step in determining an upper bound on $s$ for all the vertex-primitive $s$-arc-transitive digraphs.
In 2017, Giudici, Li and the third author constructed the first known family of vertex-primitive $2$-arc-transitive digraphs of valency at least $2$. The smallest digraph in this family admits $\mathrm{PSL}_3(49)$ acting $2$-arc-transitively with vertex-stabilizer $\mathrm{A}_6$ and hence has $30758154560$ vertices. In this paper, we prove that this digraph is the vertex-primitive $2$-arc-transitive digraph of valency at least $2$ with fewest vertices.
An interesting fact is that most of the known connected $2$-arc-transitive nonnormal Cayley graphs of small valency on finite simple groups are $(\mathrm{A}_{n+1},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$. This motivates the study of $2$-arc-transitive Cayley graphs on $\mathrm{A}_n$ for arbitrary valency. In this paper, we characterize the automorphism groups of such graphs. In particular, we show that for a non-complete $(G,2)$-arc-transitive Cayley graph on $\mathrm{A}_n$ with $G$ almost simple, the socle of $G$ is either $\mathrm{A}_{n+1}$ or $\mathrm{A}_{n+2}$. We also construct …
A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than $2$. In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as $B(k,m,n)$, defined for every triple $(k,m,n)$ of integers greater than $1$ with $2^m \equiv 1 \mod n$. In each case, $B(k,m,n)$ is a $2k$-valent vertex- and edge-transitive graph of order $mn^{k-1}$, and Bouwer showed that $B(k,6,9…
Following Alspach and Parsons, a {\em metacirculant graph} is a graph admitting a transitive group generated by two automorphisms $ρ$ and $σ$, where $ρ$ is $(m,n)$-semiregular for some integers $m \geq 1$, $n \geq 2$, and where $σ$ normalizes $ρ$, cyclically permuting the orbits of $ρ$ in such a way that $σ^m$ has at least one fixed vertex. A {\em half-arc-transitive graph} is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explo…
A graph is edge-primitive if its automorphism group acts primitively on the edge set. In this short paper, we prove that a finite 2-arc-transitive edge-primitive graph has almost simple automorphism group if it is neither a cycle nor a complete bipartite graph. We also present two examples of such graphs, which are 3-arc-transitive and have faithful vertex-stabilizers.
We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.
Let $Γ$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of $Γ$ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$. We study the problem of bounding $|G_{uv}|$ in terms of $L$ and the order of $Γ$.
A regular cover of a connected graph is called {\em cyclic} or {\em dihedral} if its transformation group is cyclic or dihedral respectively, and {\em arc-transitive} (or {\em symmetric}) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined.
A graph is edge-primitive if its automorphism group acts primitively on the edge set, and 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs. In this paper, we present a classification for those edge-primitive graphs which are 2-arc-transitive and have soluble edge-stabilizers.
A connected graph whose automorphism group acts transitively on the edges and vertices, but not on the set of ordered pairs of adjacent vertices of the graph is called half-arc-transitive. It is well known that the valence of a half-arc-transitive graph is even and at least four. Several infinite families of half-arc-transitive graphs of valence four are known, however, in all except four of the known specimens, the vertex-stabiliser in the automorphism group is abelian. The first example of a half-arc-transitive graph of valence four and with a non-abelian vertex-stabiliser was described in […
In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let $\Ga$ be such a Cayley graph on a non-abelian simple group $T$. It is proved that either $\Ga$ is a normal Cayley graph or $\Ga$ is $S$-arc-transitive, with $(S,T)=(\A_n,\A_{n-1})$ and $n=7,21,63$ or $84$; and, for each of these four values of $n$, there really exists arc-transitive $7$-valent non-normal Cayley graphs on $\A_{n-1}$ and specific examples are constructed.
It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilisers, each significant for its own reason.
In this paper, we determine the class of finite 2-arc-transitive bicirculants. We show that a connected $2$-arc-transitive bicirculant is one of the following graphs: $C_{2n}$ where $n\geqslant 2$, $\K_{2n}$ where $n\geqslant 2$, $\K_{n,n}$ where $n\geqslant 3$, $ \K_{n,n}-n\K_2$ where $n\geqslant 4$, $B(\PG(d-1,q))$ and $B'(\PG(d-1,q))$ where $d\geq 3$ and $q$ is a prime power, $ X_1(4,q)$ where $q\equiv 3\pmod{4}$ is a prime power, $\K_{q+1}^{2d}$ where $q$ is an odd prime power and $d\geq 2$ dividing $q-1$, $ AT_Q(1+q,2d)$ where $d\mid q-1$ and $d\nmid \frac{1}{2}(q-1)$, $ AT_D(1+q,2d)$ whe…
