Michael Giudici, Primoz Potocnik, Gabriel Verret
The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On the order of semiregular automorphisms of cubic vertex-transitive graphs
本文证明了有限连通三次顶点传递图要么存在阶至少为6的半正则自同构，要么其顶点数受绝对常数上界控制。
The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non) existence of $G$-invariant partitions of $X$. Conditions presented in this paper are derived by studying spectral properties of associated $G$-invariant digraphs. As an essential tool, irreducible complex characters of $H$ are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht, Graver …
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 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.
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known semiregular automorphism of a graph. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers. Let $p\geq 5$ be a prime and $Γ$ a connected symmetric graph of valency $p$ admitting a quasi-semiregular automorphism. In this paper, we first prov…
We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.
In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph X does not tend to infinity as the number of vertices of X tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author. However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when X is either a Cayley graph or an arc-transitive graph.
Symmetries of the Woolly Hat graphs
本文研究允许含三个轨道的半正则自同构的四价边传递图（即Woolly Hat图），证明不存在边传递的该类图，并完成了顶点传递的Woolly Hat图的分类。
In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices. We analyze the structure of the automorphism group, focusing in particular on polytopes with two kinds of regular facets occurring in an "alternating" fashion. In particular we use group amalgamations to prove that given two compatible n-polytopes P and Q, there exists a universal abstract semiregu…
A connected graph of order $n$ admitting a semiregular automorphism of order $n/k$ is called a $k$-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper we study the broader class of cubic vertex-transitive graphs of order $n$ admitting an automorphism of order $n/3$ or larger that may not be semiregular. In particular, we show that any such graph is either a $k$-multicirculant for some $k \leq 3$, or it belongs to an infinite family of graphs …
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 $6$.
Non-trivial $2$-$(k^{2},k,λ)$ designs, with $λ\mid k$, admitting a flag-transitive almost simple automorphism group are classified.
Sporadic simple groups as flag-transitive automorphism groups of symmetric designs
本文分类了以散在单群为基座且旗传递的对称设计，证明此类设计恰有七个，其中一个点非本原、其余点本原。
On symmetric 2-designs of prime order with almost simple flag-transitive automorphism groups
本文对允许旗传递几乎单自同构群的素数阶对称2-设计进行了完全分类，基于先前点本原性结果，给出了此类设计的完整刻画。
Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.
An infinite family of counterexamples to the Polycirculant Conjecture
反驳多循环猜想，证明存在无半正则自同构的顶点传递图。采用Chen等人方法构造度为16464的难以捉摸群\(7^6.\mathrm{PSU}_3(3)\)，其是2 - 闭的，此例扩展出多循环猜想无穷反例及相关顶点传递图。
The main result of this paper is that, if $Γ$ is a finite connected $4$-valent arc-transitive graph, then either $Γ$ is part of a well-understood family of graphs, or every non-identity automorphism of $Γ$ fixes at most $1/3$ of the vertices. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs. Based on these results we propose a conjecture on the number of fixed points of non-identity automorphisms of vertex-transitive graphs of bounded valency.
2-(v,k,3) designs admitting an almost simple, flag-transitive automorphism group with socle PSL(2,q)
本文研究了允许特定自同构群的2-(v,k,3)设计的分类问题，重点在于其socle结构为PSL(2,q)的情况。
A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.
In this paper, we present a classification of $2$-designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,λ)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $AΓL_{1}(q)$.
On automorphism groups of half-arc-transitive tetravalent graphs
本文刻画了存在子群M<H使得图是M-半弧传递且H-弧传递的连通四价图，构造了各类例子，包括对Rivera和Šparl 2019年问题的反例以及首个非交换单群上的四价正规边传递非正规Cayley图。
Mathieu groups as flag-transitive automorphism groups of block designs
本文研究基座为 Mathieu 群的旗传递几乎单自同构群作用下的 2-设计，完整分类并显式构造，新得 101 个旗传递 2-设计。
On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency
本文证明了有限连通3-价顶点传递图或4-价顶点与边传递图的非恒等自同构最多固定1/3的边，回答了Potočnik等提出的问题。
In this article, we study $2$-designs with prime replication number admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine $2$-designs with prime replication number admitting an almost simple automorphism group.
We give a construction of a family of designs with a specified point-partition, and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric $2$-$(1408,336,80)$ design with automorphism group $2^{12}:((3\cdot\mathrm{M}_{22}):2)$, and a construction of one of the families of the symplectic designs (the designs $S^-(n)$) exhibit…
In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $Γ$, $Γ\not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\la…
Flag-transitive automorphism groups of $2$-designs with $λ\geq (r,λ)^2$ are not product type
本文证明满足 $λ\geq (r,λ)^2$ 的非平凡 $2$-设计的旗传递自同构群不是乘积作用型，从而得出其为本原的仿射型或几乎单型群。
A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite family of rank two amalgams of permutation type $[L_1,L_2]$ and Borel subgroups of strictly increasing order. As an application, we show that there is no bound on the order of edge-stabilisers in locally $[L_1,L_2]$ graphs. We also consider the corresponding question for amalgams of rank $k\geq 3$. We completely resolve this by showing that the order of the Bo…
The theory of voltage graphs has become a standard tool in the study graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphism that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertextransitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with a…
Alternating groups as flag-transitive automorphism groups of 2-designs with block size seven
该研究刻画了区组大小为7且自同构群为交错群型几乎单群的旗传递2-设计，证明其仅有两类可能情形。
Transitive and Co-Transitive Caps
PG（r，q）中的cap是一组点，其中没有三个点是共线的。一个帽子被称为传递的，如果它的自同构群在PGammaL（r+1，q）中传递地作用在帽子上，而共传递的，如果自同构群传递地作用在PG（r，q）中帽子的补上。传递的、共传递的帽被表征为以下之一：PG（3，q）中的椭圆二次曲面; PG（3，q）中的Suzuki-Tits卵形体; PG（2，4）中的超卵形; PG（4，3）中大小为11的帽; PG（r，2）中超平面的补;或PG（r，q）中Singer轨道的并，其自同构群来自GammaL（1，q ^{r+1}）的子群。
For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class. Extending earlier results of Siran, Tucker and Watkins, these are used to determine which symmetric groups $S_n$ can arise in this way for each class. Similar results are obtained for all finite simple groups, building on work of Leemans and Liebeck, Nuzhin and others on generating sets for such groups. It is also shown that each edge-transitive class realises…
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.
Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs
本文研究了容许旗传递且点本原仿射自同构群的对称设计，证明了当λ为素数时自同构群的结构，并给出了此类对称设计的完整分类。
A graph with a semiregular group of automorphisms can be thought of as the derived cover arising from a voltage graph. Since its inception, the theory of voltage graphs and their derived covers has been a powerful tool used in the study of graphs with a significant degree of symmetry. We generalise this theory to graphs with a group of automorphisms that is not necessarily semiregular, and we generalise several well-known results of the classical theory of voltage graphs to this broader setting.
A graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertices with two orbits. A non-abelian group is called an inner-abelian group if all of its proper subgroups are abelian. In this paper, we complete the classification of connected cubic edge-transitive bi-Cayley graphs over inner-abelian p-groups for an odd prime p.
The symmetric $2$-$(v,k,λ)$ designs, with $k>λ\left(λ-3 \right)/2$, admitting a flag-transitive, point-imprimitive automorphism group are completely classified: they are the known $2$-designs with parameters $(16,6,2),(45,12,3),(15,8,4)$ or $(96,20,4)$.
Almost simple groups as flag-transitive automorphism groups of 2-designs with λ = 2
本文研究λ=2的2-设计的旗传递几乎单自同构群，证明基群为有限李型例外单群时此类设计不存在，并完成相关设计的分类，明确其属于特定无限族或有限参数集的设计。
A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679--693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called $0$-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism g…
Block-transitive, point-primitive Steiner 3-designs
本文研究了容许块传递自同构群的 Steiner 3-设计的归约问题，证明了非平凡 Steiner 3-设计的点本原且块传递自同构群必为仿射型或几乎单型。
In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs.
$2$-designs admitting a flag-transitive automorphism group with socle $PSL(2,q)$
研究了具有基座为\(PSL(2,q)\)（\(q = p^{f}\geq 4\)）的旗传递自同构群的\(2 -\)设计，分点本原和点非本原情况，后者获完全分类，前者除特定线性空间外还分类出\(48\)个零星例子。
In this note, we complete the classification of extremal doubly even self-dual codes with 2-transitive automorphism groups.
In this paper, we classify the $m$-ovoids of finite classical polar spaces that admit a transitive automorphism group acting irreducibly on the ambient vector space. In particular, we obtain several new infinite families of transitive $m$-ovoids.
Vertex-transitive strongly regular graphs in the switching class of doubly transitive two-graphs
本文对关联二图源于PΣL(2,q)的顶点传递强正则图进行分类，得到了一类新的此类图族。
On $m$-partite oriented semiregular representations of finite groups
本文研究了有限群的$m$-部有向半则表示，通过扩展有向图的概念，提出了$m$-部有向半则表示的定义，并给出了$m\geq2$时无$m$-部Haar有向表示或$m$-部有向半则表示的有限群的完整分类。
In this article, we study $2$-designs with $\gcd(r, λ)=1$ admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a $2$-design with $\gcd(r, λ)=1$ and $G$ is a flag-transitive almost simple automorphism group of $\mathcal{D}$ whose socle is $X=\mathrm{PSU}(n, q)$ with $(n, q)\neq (3, 2)$ and prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a design …
A positive integer $n$ is a Cayley number if every vertex-transitive graph of order $n$ is a Cayley graph. In 1983, Dragan Marušič posed the problem of determining the Cayley numbers. In this paper we give an infinite set $S$ of primes such that every finite product of distinct elements from $S$ is a Cayley number. This answers a 1996 outstanding question of Brendan McKay and Cheryl Praeger, which they "believe to be the key unresolved question" on Cayley numbers. We also show that, for every finite product $n$ of distinct elements from $S$, every transitive group of degree $n$ contains a semi…
The pairs $(\mathcal{D},G)$, where $\mathcal{D}$ is a non-trivial $2$-$(k^{2},k,λ)$ design, with $λ\mid k$, and $G$ is a flag-transitive automorphism group of $\mathcal{D}$ of affine type such that $G \nleq A ΓL_{1}(k^{2})$, are classified.
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.
Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a nowhere-zero 3-flow.
On quartic half-arc-transitive metacirculants
根据Alspach和Parsons，{ \em metacirculant graph}是一个允许由两个自同构$ρ$和$σ$生成的传递群的图，其中$ρ$对于某些整数$m \geq 1$，$n \geq 2$是$(m,n)$ -半正则的，并且其中$σ$规范化$ρ$，循环置换$ρ$的轨道，使得$σ^m$至少有一个固定顶点。一个{ \em半弧传递图}是一个点和边传递但不是弧传递图。在这篇文章中，四次半弧传递亚循环图和他们的所谓的紧连接四次半弧传递图的连接进行了探讨。证明了非紧附着的四次半弧传递亚循环线存在三种不同的可能性。这些图被广泛研究，并构造了一些这样的图的无限族。
We determine all finite primitive groups that are automorphism groups of edge-transitive hypergraphs. This gives an answer to a problem proposed by Babai and Cameron
A graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime p, we call a bi-Cayley graph over a metacyclic p-group a bi-p-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-p-metacirculant is characterized for an odd prime p, and the result reveals that a connected cubic edge-transitive bi-p-metacirculant exists only when p=3. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic 3-…
Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order $2$. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with ve…
Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight
1991年，方卫东和李慧玲证明了对于给定的gcd（k，r），只有1000个非平凡线性空间允许一个线传递的点-非本原群作用，其中k是线的长度，r是点上的线的个数。本文的目的是使这一结果有效。我们得到了所有具有此性质的线性空间的分类，gcd（k，r）至多为8。为了实现这一点，我们收集现有的理论，并证明了额外的理论限制的组合和群论性质。这些被组织成一系列算法，对于gcd（k，r）直到给定的最大值，返回候选参数值和候选组的列表。我们详细研究了这些算法返回的gcd（k，r）最多为8的可能性，并在这种情况下完成分类。
On the size of spheres of relations with a transitive group of automorphisms
设$Γ=(V,E)$是点传递自反关系。设$v\in V$和$r=|Γ(v)|.$也假设为$Γ^j(v)\cap Γ^{-}(v)=\{v\}$。然后$$ |Γ^{j} (v)\setminus Γ^{j-1} (v)| \ge r-1.$$特别是我们有$ |Γ^{j} (v)| \ge 1+(r-1)j.$最后的结果证实了最近的猜想西摩的情况下点传递图。对点传递图的Caccetta-Häggkvist猜想的有效性给出了一个简短的证明，推广了Shepherdson的一个可加性结果.
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 $…
Flag-transitive $2$-$(v,k,λ)$ designs with $λ\ge (r,λ)^2$
本文针对允许旗传递自同构群作用的满足λ≥(r,λ)²的2-设计，完成了n≥3时基座为PSLₙ(q)或PSUₙ(q)的几乎单群对应的满足λ≥(r,λ)²>1的该类设计的分类。
Vertex-primitive s-arc-transitive digraphs of symplectic groups
本文针对有限辛群证实了2018年提出的猜想，即有限点本原s-弧传递有向图中的$s$至多为2。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。