Gareth A. Jones
Building on earlier work of Biggs, James, Wilson and the author, and using the Graver-Watkins description of the 14 classes of edge-transitive maps, we complete the classification of the edge-transitive embeddings of complete graphs.
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A construction of 2-arc-transitive non-solvable covers of complete graphs
本文提出一种构造完全图2-弧传递非可解覆盖的方法，解决了该类覆盖的存在性问题。
Edge-Transitive Homogeneous Factorisations of Complete Graphs
本文研究了具有边传递因子的完全图的齐次因子分解。完全图$K_n$的因子分解是将其边划分为不相交的类。$K_n$的因子分解中的每一类边对应于称为因子的生成子图。如果所有的因子彼此同构，则$K_n$的因子分解称为同构因子分解。完全图的齐次分解是同构分解，其中存在传递置换因子的群$G$和$G$的正规子群$M$，使得每个因子是$M$ -顶点传递的。如果$M$也对每个因子起边传递作用，则$K_n$的齐次因子分解称为边传递齐次因子分解。本文的目的是研究$K_n$的边传递齐次分解。除了$G$是$Z_p^R \rtimes G_0$形式的仿射2-齐次群的情况外，我们实现了几乎完全的显式分类，其中$G_0 \leq ΓL(1,p^R)$。在这种情况下，我们获得了必要和充分的算术条件，某些参数，这样的因式分解存在，并给出了一个通用的建设，指定齐次因式分解完全，给定的条件，参数。此外，我们给出了两个无限族的例子，我们明确指定的参数的建设。在第二无限族中，弧传递因子是Peisert在2001年构造的某些弧传递自补图的概括。
New families of completely transitive codes and distance transitive graphs
本文提出了新的线性二元完全传递码的无限族。它们具有覆盖半径$ρ= 3$和4，并且分别是长度为$n=2^m-1$和$2^m$的二进制汉明码和二进制扩展汉明码的一半，其中$m$是偶数。从这些新的完全传递码，以通常的方式，即，作为陪集图，构造了直径分别为3和4的距离传递陪集图的无限族的新表示。
We prove the following result. Let q be a power of an odd prime and let Sp(2m,q) denote the symplectic group of degree 2m over F_q. Then if q=1 mod 4, no solvable subgroup of Sp(2m,q) acts transitively on a complete symplectic spread defined on the underlying vector space of dimension 2m, unless m=1 and q=5. The solvable group Sp(2,3) of order 24 acts transitively on a complete symplectic spread defined on a two-dimensional vector space over F_5. By contrast, when q=3 mod 4, there is a metacyclic group of order 2m(q^m+1) that acts transitively on a complete symplectic spread.
Transitive path decompositions of Cartesian products of complete graphs
本文研究完全图笛卡尔积 $K_n \Box K_n$ 的传递路径分解，当 $n$ 为奇素数时构造了路径长度相对于顶点数较大的分解，支持了 Gallai 猜想和 Ringel 猜想的扩展版本。
Quotient-complete arc-transitive latin square graphs from groups
我们考虑给定有限群$H$的Cayley表的拉丁方图$Γ= \rm{LSG}(H)$.我们刻画了所有对$(Γ,G)$，其中$G$是$H$的Cayley表的自同构子群，使得$G$对$Γ$弧传递，并且$Γ$的所有非平凡$G$ -正规商图是完备的.证明了$H$必是初等阿贝尔的，并确定了完全正规代数的个数$k$.这就产生了直径为2的弧传递图的新的无限族，它们具有$k = 1$或$k = 2$ .
Fissioned triangular schemes via sharply 3-transitive groups
n [D. de Caen，E.R.凡·达姆。通过交比的分裂三角形方案，{欧洲。J. Combin.}，22（2001）297-301]，de Caen和van Dam在$\PG(1,q)$上构造了一个三角格式的分裂格式$\FT(q+1)$.这种裂变方案来自于$\PGL(2,q)$对$\PG(1,q)$的2元素子集的自然诱导作用。群$\PGL(2,q)$是两个有限的3-传递群的无限族之一。另一个这样的家族$\Mq(q)$是$\PGL(2,q)$的“扭曲”版本，其中$q$是奇素数的偶次幂。$\PSL(2,q)$组是$\PGL(2,q)$和$\Mq(q)$的交集。本文分别研究了由$\PSL(2,q)$、$\Mq(q)$和$\PML(2,q)$的作用产生的关联方案。通过在[H.D.L.]中引入的圆锥模型，Hollmann，Q.翔从$\PGL(2, q) $固定一个非奇异二次曲线的行动的关联方案，{J.代数学Combin.}，24（2006）157-193]，我们引入了将$\PML(2,q)$嵌入到$\PML(3,q)$中。对于上面提到的三个群中的每一个，这种嵌入产生了两个同构的结合方案：一个在三维正交几何中的双曲线上，另一个在双曲点上（通过零奇偶校验）。这种嵌入使我们能够同时处理这三个同构结合方案。
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…
Maps, simple groups, and arc-transitive graphs
本文确定了有限几乎单群的因子分解条件，并分类了允许无核弧传递子群的映射嵌入图，除特殊情形外仅包括完全图、Johnson图及14个额外图，特殊情形下构造无穷多个嵌入。
We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than $48$, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than $48$. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focussing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list…
A note on some embedding problems for oriented graphs
我们猜想，每个定向图$G$在$n$的顶点上都含有$δ^+ (G) , δ^- (G) \geq 5n/12$，它包含一个Hamilton圈的平方。我们还给出了在定向图中传递三角形的最小半度的一个几何界。拉姆齐数和传递竞赛图的完美包装之间的联系也被认为是。
A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two
给出了最多1000个顶点的所有连通的弧传递的不对称有向图的一个完整列表。作为副产品，一个完整的列表，所有连接的4-valve图承认一个半弧-传递群的自同构上最多1000个顶点。几个图论性质的元素，我们的普查计算和讨论。
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
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…
Arc-transitive maps with coprime Euler characteristic and edge number -- II
作为系列论文第二篇，本文完成不可解自同构群的弧传递地图分类，结合首篇成果完整刻画欧拉示性数与边数互素的弧传递地图，分类过程构建了若干新颖反向地图。
Normal Edge-Transitive Cayley Graphs of Frobenius Groups
一个群$G$的Cayley图称为正规边传递的，如果它允许$G$的全形（$\operatorname{Sym}(G)$中$G$的正则副本的正规化子）的某个子群的边传递作用.通过对两个素数的乘积为a阶的弗罗贝尼乌斯群的Cayley图的研究，完成了这类图的分类.我们确定这些图的自同构群，特别证明有一个独特的顶点本原的例子，即Fano平面的旗图。
The complete classification of triply-transitive strongly regular graphs
本文通过证明两类极图的三重传递性，完成了三重传递强正则图的完全分类，并以Terwilliger代数等式刻画其局部对称性。
Strongly regular edge-transitive graphs
本文利用正规商约化方法研究了点传递和边传递强正则图的结构。证明了这类不可约图具有拟本原自同构群，并利用有限单群的分类证明了这类不可约图不具有全纯单群.我们还发现了一些限制的参数，在这个家庭的图，减少到完全图。
A Classification of the flag-transitive $2$-$(v,k,2)$ designs
本文分类了仿射型旗传递自同构群作用下的 $2$-$(v,k,2)$ 设计，构造了七类新设计族，并结合已有结果给出除半线性 $1$ 维情形外的完整分类。
We first characterize all faithful arc-transitive covers of complete graphs and we give a general construction of such covers. For a counterpart of cover, we say a graph $Γ$ is a pseudocover of its quotient $Σ$ if they have the same valency and $Γ$ is not a cover of $Σ$.As the second result of this paper, we prove that the complete graph $\K_n$ has a connected arc-transitive pseudocover if and only if $n-1$ is not a prime.
In this paper, we complete the classification of transitive ovoids of finite Hermitian polar spaces.
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 $…
Highly regular vertex-transitive graphs are globally rigid
该研究证明了高度规则的顶点传递图在任何维度下都是全局刚性的，并展示了规则常数的最优性。
A classification of graphs whose subdivision graphs are locally $G$-distance transitive
连通图$Σ$的细分图$S(Σ)$是通过在每条边的中间添加一个顶点来构造的。在之前与Cheryl E. Praeger，我们刻画了图$Σ$，使得$S(Σ)$对于$s\leq 2\, diam(Σ)-1$和某些$G\leq Aut(Σ)$是局部$(G,s)$ -距离传递的.在本文中，我们解决了其余的情况下，通过分类所有的图$Σ$，使得细分图是局部$(G,s)$ -距离传递的$s\geq 2\, diam(Σ)$和一些$G\leq Aut(Σ)$。特别地，除完全图外，它们的剖分图都是局部$G$ -距离传递的.
Locally $s$-distance transitive graphs
对于每一个正整数$s$，我们给出了一个统一的方法来分析一类有限连通图，它包含所有的距离传递图以及直径至少为$s$的局部$s$ -弧传递图.一个图是在类中，如果它是连通的，如果，对于每个顶点$v$，固定$v$的自同构子群传递作用在距离$i$到$v$的顶点集上，对于每个$i$从1到$s$。我们证明了这个类在形成正规映射下是闭的。这类图中的几个图被称为退化图，如果这类图中的一个非退化图的所有非平凡正规子都是退化的，则称这类图为基本图。我们证明了，对于$s\geq 2$，类中的非退化非基本图要么是完全多部图，要么是基本图的正规覆盖。进一步证明了除了完全二部图外，每个基本图在其（1或2）个顶点轨道上都有忠实的拟本原作用，或有双拟本原作用.这些结果邀请详细的额外的分析的基本图形，使用准本原置换群的理论。
Cubic vertex-transitive non-Cayley graphs of order 12p
一个图称为{ \em顶点传递的非凯莱图}，如果它的全自同构群在它的顶点上作用是传递的，并且不包含正则作用在它的顶点上的子群.本文给出了阶为$12p$（其中$p$是素数）的三次点传递非Cayley图的一个完全分类.因此，有$11$零星和一个无限族这样的图，其中零星的发生时，$p=5$，$7$或$17$，和无限族存在当且仅当$p\equiv1\ (\mod 4)$，在这个家庭有一个唯一的图为给定的顺序。
On arc-transitive inner-automorphic Cayley graphs on dihedral groups
本文研究二面体群上的弧传递内自同构赛雷图，分析其结构并给出存在条件，构造无限例并完成2-距离传递赛雷图的分类。
Finite 2-geodesic transitive graphs of prime valency
我们对满足自同构群在弧集和$2$ -测地线集上都是传递的性质的非完全素价图进行了分类。证明了$Γ$是2-弧传递的，或者$p$的价满足$p\equiv 1\pmod 4$，并且对于每个素数，都存在唯一的具有这个性质的图：它是完全图$K_{p+1}$的非二部对极双覆盖，自同构群为$PSL(2,p)\times Z_2$，直径为3.
In an earlier paper the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositi…
A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of $2$-geodesics. In this paper, a complete classification of $2$-geodesic transitive graphs of order $p^n$ is given for each prime $p$ and $n\leq 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph $K_{4,4}$ of order $8$, the Schläfli graph of order $27$ and its complement, and fourteen infinite families: the cycles $C_p,…
A classification of two-distance-transitive Cayley graphs over the generalized quaternion groups
本文推广了经典距离传递图的概念，定义了2-距离传递图，并完全确定了广义四元数群上2-距离传递Cayley图的分类。
On the Classification of Extremal Doubly Even Self-Dual Codes with 2-Transitive Automorphism Group
本文完成了具有2-传递自同构群的极值双偶自对偶码的分类。
Transitive factorizations of permutations and geometry
我们给出了一个帐户，我们的工作传递因式分解的排列。这项工作对数学的其他领域产生了影响，如图嵌入的枚举、随机矩阵、分支覆盖和曲线的模空间。从代数组合学的角度来看，这些看似不相关的领域的各个方面在统一的观点中被认为是相关的。在某些方面，这部作品与理查德·斯坦利的作品有着重要的联系。
On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs
一个置换群称为半本原群，如果它的每个正规子群是可迁的或半正则的。给定非平凡有限可迁置换群$L_1$和$L_2$（其中$L_1$不是半本原的），构造了一类秩为2的置换型$[L_1,L_2]$和博雷尔子群的严格递增秩的混杂群.作为一个应用，我们证明了局部$[L_1,L_2]$图的边稳定子的阶没有界。我们还考虑了相应的问题，汞合金等级$k\geq 3$。我们完全解决了这一问题，证明了Borel子群的顺序是有界的置换类型$[L_1,...,L_k]$只有在平凡的情况下，每个$L_1,...,L_k$是正规的。
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。
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.
Line graphs and $2$-geodesic transitivity
对图$Γ$、正整数$s$和子群$G\leq \Aut(Γ)$，证明了$G$在$Γ$的$s$ -弧集上传递的充要条件是$Γ$的围长至少为$2(s-1)$，且$G$在其线图的$(s-1)$ -测地线集上传递.作为应用，我们首先证明了不完全局部循环$2$ -测地传递图是完全多部图$K_{3[2]}$和二十面体。其次，我们对4度围长3的2-测地传递图进行了分类，并确定了哪些图是测地传递的。
In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $d\leq 20$ or $d$ is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $d\geq4$. Even for this problem, it was only solved f…
Classification of line-transitive point-imprimitive linear spaces with line size at most 12
本文完成了线长不超过12的有限线性空间$\cS$的一个分类，其中允许自同构的线传递点非本原子群。这些例子是$4,7, 9$和11阶的Deskiesian射影平面，91个点上的两个设计，线尺寸为6，以及729个点上的467个设计，线尺寸为8。
Quasi-Transitive Mixed Graphs and Undirected Squares of Oriented Graphs
本文将拟传递性推广至混合图，完整刻画了最大度为3及围长至少为4的有向图无向平方，并证明其识别问题在度受限情形下的复杂度分界。
Transitive graphs in counterexamples to Karp's conjecture
卡普指出，所有非平凡单调图的性质都是回避的。这是证明了n一个素数的权力，和n=6，其中n是图的顶点数，由卡恩，萨克斯，和Sturtevant。我们给出了一个完整的描述，其中传递图包含在一个可能的反例时，n=10。
Classification of two-distance-transitive Cayley graphs of the semi-dihedral groups
本文聚焦半二面体群的二距离传递凯莱图分类问题，通过特定方法，给出了该类图的完整分类，为代数图论相关研究提供了重要成果。
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 graph $Γ$ is $G$-symmetric if $G$ is a group of automorphisms of $Γ$ which is transitive on the set of ordered pairs of adjacent vertices of $Γ$. If $V(Γ)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $Γ_{\cal B}$ of $Γ$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then $Γ$ is called an almost multicover of $Γ_{\cal B}$. In this case an incidence structure with point set ${\cal B}$ arises naturally, and it is a $(G, 2)$-point-transitive and $G$-block-transitive 2-design if in addit…
Given a finite transitive permutation group $G\leq \operatorname{Sym}(Ω)$, with $|Ω|\geq 2$, the derangement graph $Γ_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, {\it J. Combin. Theory Ser. A}, 180:105390, 2021] recently proved that $\operatorname{Sym}(2)$ acting on $\{1,2\}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also sh…
Tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, that is a subgroup acting transitively on its vertices and its edges but not on its arcs, are investigated. One of the most fruitful approaches for the study of structural properties of such graphs is the well known paradigm of alternating cycles and their intersections which was introduced by Marušič 20 years ago. In this paper a new parameter for such graphs, giving a further insight into their structure, is introduced. Various properties of this parameter are given and the parameter is completely determined for th…
We study point-block incidence structures $(\mathcal{P},\mathcal{B})$ for which the point set $\mathcal{P}$ is an $m\times n$ grid. Cameron and the fourth author showed that each block $B$ may be viewed as a subgraph of a complete bipartite graph $\mathbf{K}_{m,n}$ with bipartite parts (biparts) of sizes $m, n$. In the case where $\mathcal{B}$ consists of all the subgraphs isomorphic to $B$, under automorphisms of $\mathbf{K}_{m,n}$ fixing the two biparts, they obtained necessary and sufficient conditions for $(\mathcal{P},\mathcal{B})$ to be a $2$-design, and to be a $3$-design. We first re-i…
Edge-Transitive Graphs
如果一个图的自同构群作用在它的边上是传递的，则称它是边传递的。已知边传递图是点传递图或二部图。本文给出了顶点数小于或等于$20$的连通边传递图的一个完全分类。然后，我们提出了一个无限族的边传递的二部图的建设，并使用此建设表明，存在一个非平凡的二部子图的$K_{m,n}$是连接和边传递时，$gcd(m,n)>2$。此外，我们调查的充要条件的连接$(r,2)$双正则子图的$K_{m,n}$，以及唯一性，并使用这些结果来解决的情况下，$gcd(m,n)=2$。然后，我们在点传递图中提出了无限族的边传递图，包括几类循环图。特别地，我们给出了某些循环图边传递的必要条件和充分条件。
Geodesic-transitive graphs with large diameter
本文研究了直径大于4的图的测地传递性质，发现除了少数例外情况外，这些图具有清晰的几何结构。同时，作者给出了距离传递但非测地传递的图的例子，包括直径为3的无限族和直径为3、4或7的孤立例子。
Transitive and Gallai colorings
本文将Gallai着色与传递着色推广到拟阵、Coxeter系统和交换代数，证明最大颜色数等于拟阵的秩，并得到A型根系着色计数函数的对称性与Schur正性。
Cubic vertex-transitive graphs of girth six
本文完整分类了围长为6的有限简单三次点传递图，证明除Desargues图外，它们均可由环面六边形平铺、闭双曲曲面弧传递三角剖分的截断或6-正则图的弧传递二面体方案截断得到。
On vertex-transitive distance-regular covers of complete graphs with an extremal smallest eigenvalue
本文研究完全图的阿贝尔距离正则覆盖，利用秩至多3的置换群分类，刻画了具有极值最小特征值且自同构群顶点传递、弧集上至多两个轨道的覆盖族。
Cubic edge-transitive bi-Cayley graphs over inner-abelian p-groups
一个图称为群H上的双Cayley图，如果它允许H是一个具有两个轨道的半正则作用在其顶点上的自同构自同构群.一个非阿贝尔群称为内阿贝尔群，如果它的所有真子群都是阿贝尔群.本文完成了奇素数p的内交换p-群上连通三次边传递双Cayley图的分类。
A Ladder Tournament
阶梯锦标赛广泛用于现实世界组织和游戏中的个人排名。然而，它们的数学性质仍然知之甚少。我们正式的排名规则产生的阶梯锦标赛，我们表明，它既不是完整的，也不传递一般。如果它是完全的，那么它是传递的，它的非对称分量是传递竞赛图的有限并。我们还研究了一个人的排名和他的表现之间的关系，作为衡量的频率，他是关键。我们证明了个体的稳定性是其等级的弱增函数。
A note on the structure of locally finite planar quasi-transitive graphs
本研究基于Hamann的工作，证明了3-连通局部有限平面拟传递图的典范树分解结构定理，给出了Hamann等人2022年相关结果的新证明，是拟传递平面图的Droms型结构定理。
Edge-transitive bi-Cayley graphs
一个图$\G$允许一个自同构群$H$以恰好两个轨道半正则地作用在顶点上，称为$H$上的{ \em bi-Cayley图\/}.如果$H$在$\G$的全自同构群中是正规的，则这样的图$\G$称为{ \em正规\/}，如果$H$在$\G$的全自同构群中的正规化子在$\G$的边上是传递的，则称为{ \em正规边传递\/}。本文给出了正规边传递双Cayley图的一个特征，特别是$2$ -弧传递正规双Cayley图.利用这一点，我们研究了三类双Cayley图，即阿贝尔群，二面体群和亚循环$p$ -群上的双Cayley图。我们发现，在一定条件下，这三类图的“正规边传递”与“正规”是相同的。作为一个副产品，我们得到了一个完整的分类，所有连通的三价边传递图的围长最多$6$，并回答了一些开放的问题，从文献中的$2$ -弧传递，半弧传递和半对称图。
Chordal graphs are important in the structural and algorithmic graph theory. A digraph analogue of chordal graphs was introduced by Haskin and Rose in 1973 but has not been a subject of active studies until recently when a characterization of semicomplete chordal digraphs in terms of forbidden subdigraphs was found by Meister and Telle. Locally semicomplete digraphs, quasi-transitive digraphs, and extended semi-complete digraphs are amongst the most popular generalizations of semicomplete digraphs. We extend the forbidden subdigraph characterization of semicomplete chordal digraphs to locally …
For a non-complete graph $Γ$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u\neq w$ and $u,w$ are not adjacent. Then $Γ$ is said to be $2$-geodesic transitive if its automorphism group is transitive on both arcs and 2-geodesics. In previous work the author showed that if a $2$-geodesic transitive graph $Γ$ is locally disconnected and its automorphism group $\Aut(Γ)$ has a non-trivial normal subgroup which is intransitive on the vertex set of $Γ$, then $Γ$ is a cover of a smaller 2-geodesic transitive graph. Thus the `basic' graphs to study are th…
Transitivity conditions in infinite graphs
本文研究了具有多个端点的图的传递性。我们对距离传递图进行了完全分类，并对所有的$k \geq 3$，对$k$ -CS传递图进行了完全分类.
A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$ exist for $1 \leq i < j \leq t$. Recognizing semi-transitive orientability of a graph is an NP-complete problem. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Semi-transitive orientability of spit graphs was recently studied in the literature. The main result in this paper is proving that recognition of …
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