The edge-transitive tetravalent Cayley graphs of square-free order

A classification is given for connected edge-transitive tetravalent Cayley graphs of square-free order. The classification shows that, with a few exceptions, a connected edge-transitive tetravalent Cayley graph of square-free order is either arc-regular or edge-regular. It thus provides a generic construction of half-transitive graphs of valency 4.     

Finite edge-transitive dihedrant graphs

In this paper, we first prove that each biquasiprimitive permutation group containing a regular dihedral subgroup is biprimitive, and then give a classification of such groups. The classification is then used to classify vertex-quasiprimitive and vertex-biquasiprimitive edge-transitive dihedrants. Moreover, a characterization of valencies of normal edge-transitive dihedrants is obtained, and some classes of examples with certain valences are constructed.     

Tetravalent edge-transitive graphs of order p 2 q

A graph is called edge-transitive if its full automorphism group acts transitively on its edge set.In this paper,by using classification of finite simple groups,we classify tetravalent edge-transitive graphs of order p2q with p,q distinct odd primes.The result generalizes certain previous results.In particular,it shows that such graphs are normal Cayley graphs with only a few exceptions of small orders.     

Vertex‐Transitive Cubic Graphs of Square‐Free Order

A classification of connected vertex‐transitive cubic graphs of square‐free order is provided. It is shown that such graphs are well‐characterized metacirculants (including dihedrants, generalized Petersen graphs, Möbius bands), or Tutte's 8‐cage, or graphs arisen from simple groups PSL(2, p).     

Edge-transitive dihedral or cyclic covers of cubic symmetric graphs of order 2P

A regular cover of a connected graph is called dihedral or cyclic if its transformation group is dihedral or cyclic, respectively. Let X be a connected cubic symmetric graph of order 2p for a prime p. Several publications have investigated the classification of edge-transitive dihedral or cyclic covers of X for specific p. The edge-transitive dihedral covers of X have been classified for p=2 and the edge-transitive cyclic covers of X have been classified for p≤5. In this paper an extension of the above results to an arbitrary prime p is presented.     

Locally Transitive Graphs Admitting a Group with Cyclic Sylow Subgroups

All graphs are finite simple undirected and of no isolated vertices in this paper. Using the theory of coset graphs and permutation groups, it is completed that a classification of locally transitive graphs admitting a non-Abelian group with cyclic Sylow subgroups. They are either the union of the family of arc-transitive graphs, or the union of the family of bipartite edge-transitive graphs.     

Edge-transitive products

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edge-transitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a strong product is edge-transitive if and only if all factors are complete graphs. In addition, a connected, infinite non-trivial Cartesian product graph G is edge-transitive if and only if it is vertex-transitive and if G is a finite weak Cartesian power of a connected, edge- and vertex-transitive graph H, or if G is the weak Cartesian power of a connected, bipartite, edge-transitive graph H that is not vertex-transitive.     

A classification of semisymmetric graphs of order 2pq

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. This paper gives a classification of semisymmetric graphs of order 2pq where p and q are distinct primes. It is shown that there are 143 examples of such graphs, 131 of which are biprimitive.     

素数立方阶群局部传递的图

目的是研究局部传递图的性质和分类.运用置换群和陪集图的理论,获得了关于素数立方阶群局部传递图的完全分类,证明了这些图是一些互不相交的关于素数立方阶群边传递图的并.     

Semiregular automorphisms of edge-transitive graphs

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism: a non-trivial 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.     

一类点传递但边不传递的正则图及其覆盖图

利用轮子图构造出一类图,证明了这类图都是点传递但边不传递的正则图,并证明了通过覆盖的方法,可以使一类2m²(m>3,m为正整数)阶非边传递图变成对称图,这类对称图实际上是亚循环图.     

Classifying cubic edge-transitive graphs of order 8<Emphasis Type="Italic">p</Emphasis>

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime. It was shown by Folkman (J. Combin. Theory 3 (1967) 215–232) that a regular edge-transitive graph of order 2p or 2p ² is necessarily vertex-transitive. In this paper, an extension of his result in the case of cubic graphs is given. It is proved that, every cubic edge-transitive graph of order 8p is symmetric, and then all such graphs are classified.     

Core-free,rank two coset geometries from edge-transitive bipartite graphs

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge-transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers.     

Finite primitive groups and edge-transitive hypergraphs

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.     

Semisymmetric prime-valent graphs of order $$2p^3$$ 2 p 3

Journal of Algebraic Combinatorics - A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime and $$\Gamma $$ a...     

Cubic edge-transitive graphs of order 2p3

Let p be a prime. It was shown by Folkman (J. Combin. Theory 3 (1967) 215) that a regular edge-transitive graph of order 2p or 2p² is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that, with the exception of the Gray graph on 54 vertices, every cubic edge-transitive graph of order 2p³ is vertex-transitive.     

Circular regular dessins

Journal of Algebraic Combinatorics - A dessin is called circular if the boundary of each face is a circuit (a simple cycle). In this paper, we address the problem of characterizing edge-transitive...     

On edge-transitive bi-Frobenius-metacirculants

In this paper, we classify a family of edge-transitive bi-Cayley graphs on Frobenius metacyclic groups. This provides a new construction of an infinite family of half-arc-regular bi-Cayley graphs on metacyclic groups.     

2p~n阶连通p~2度的半对称图

称一个有限简单无向图X是半对称图,如果图X是正则的且边传递但非点传递.本文主要利用仿射几何构造了一类2p~n阶连通p~2度的半对称图的无限族,其中p≥n≥5.     

Cubic Semisymmetric Graphs of Order 2qp2

Acta Mathematicae Applicatae Sinica, English Series - A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. Let p be a prime. By Folkman [J. Combin. Theory 3...