On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Biprimitive Graphs of Smallest Order

A regular and edge-transitive graph which is not vertex-transitive is said to be semisymmetric. Every semisymmetric graph is necessarily bipartite, with the two parts having equal size and the automorphism group acting transitively on each of these parts. A semisymmetric graph is called biprimitive if its automorphism group acts primitively on each part. In this paper biprimitive graphs of smallest order are determined.     

群与图的对称性

首先对平面图形的对称进行分析和利用其对称群进行量化,进而将此推广到考察一般图的广义"对称性"与图自同构群的关系,最后刻画了无平方因子阶局部本原弧传递图的自同构群结构.     

6p阶可解对称图的分类

设ｘ是简单无向图，Ｇ是Ａｕｔ（Ｘ）的一个于群，Ｘ称为Ｇ－对称的，如果Ｇ在ｘ的１－孤（即两相邻顶点构成的有序偶）集合上的作用是传递的；ｘ称为对称图，如果Ｘ是Ａｕｔ（ｘ）－对称的；ｘ称为可解对称的，如果Ａｕｔ（Ｘ）包含可解子群Ｇ，使Ｘ是Ｇ－对称的．本文给出了具有６Ｐ个顶点的可解对称图的一个分类，这里ｐ≥５是素数．     

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.     

4p阶3度点传递图

一个图称为点传递图,如果它的全自同构群在它的顶点集合上作用传递.证明了一个4p(p为素数)阶连通3度点传递图或者是Cayley图,或者同构于下列之一;广义Petersen图P(10,2),正十二面体,Coxeter图,或广义Petersen图P(2p,k),这里k2≡-1(mod 2p).     

On Locally Projective Graphs of Girth 5

Let be a graph and G be a 2-arc transitive automorphism group of . For a vertex x let G(x)^(x) denote the permutation group induced by the stabilizer G(x) of x in G on the set (x) of vertices adjacent to x in . Then is said to be a locally projective graph of type (n,q) if G(x)^(x) contains PSL_n(q) as a normal subgroup in its natural doubly transitive action. Suppose that is a locally projective graph of type (n,q), for some n 3, whose girth (that is, the length of a shortest cycle) is 5 and suppose that G(x) acts faithfully on (x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n=4, q=2, has 506 vertices and , and contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W(n) of which all other graphs for this n are quotients. The graph W(3) satisfies the conditions and has 2²⁰ vertices.     

第二小阶数的双本原半对称图

如果一个正则图是边传递但不是点传递的,那么我们称它是半对称的.每一个半对称图X必定是两部分点数相等的二部图,并且它的自同构群Aut(X)在每一部分上是传递的.如果一个半对称图的自同构群在每一部分上作用是本原的,那么我们称它是双本原的.本文决定了第二小阶数的双本原半对称图.     

Locally Finite-Indicable Groups

A group is locally ?-indicable if every finitely generated subgroup has a nontrivial homomorphism onto a nontrivial ?-group. If ? is a quasi-variety, then the class L(?) of locally ?-indicable groups coincides with the class N(?) of groups which have normal systems with factors in ?. It is not known if ? must be a quasi-variety in order for the equality L(?) = N(?) to hold. We show here that if ? is the class of all finite groups, which is the union of an ascending sequence of quasi-varieties, then L(?) ≠ N(?). Examples of finitely generated groups in L(?)\ N(?) are also constructed.     

pqr阶Cayley图是Hamilton图

本文证明了ｐｑｒ阶连通的Ｃａｙｌｅｙ图是Ｈａｍｉｌｔｏｎ图,这里ｐ,ｑ,ｒ为相异素数．     

Normality of Tetravalent Cayley Graphs of Odd Prime-cube Order and Its Application

Let p be an odd prime. In this paper we prove that all tetravalent connected Cayley graphs of order p^3 are normal. As an application, a classification of tetravalent symmetric graphs of odd prime-cube order is given.     

原始图与三次图的可圈度

本文综述关于原始图与三次图的可圈度的近期结果并提出一些未解决的问题。     

Covers of Point-Hyperplane Graphs

A cover of the non-incident point-hyperplane graph of projective dimension 3 for fields of characteristic 2 is constructed. For fields of even order larger than 2, this leads to an elementary construction of the non-split extension of SL₄( )by ⁶.     

1/2-Transitive Graphs of Order 3p

A graph X is called vertex-transitive, edge-transitive, or arc-transitive, if the automorphism group of X acts transitively on the set of vertices, edges, or arcs of X, respectively. X is said to be 1/2-transitive, if it is vertex-transitive, edge-transitive, but not arc-transitive.In this paper we determine all 1/2-transitive graphs with 3p vertices, where p is an odd prime. (See Theorem 3.4.)     

关于半传递图的若干新结果

我们称无向图Ｘ为半传递的，如果它的自同构群Ａｕｔ Ｘ在Ｘ的顶点集合以及边集合上作用是传递的，但在Ｘ的有序的相邻顶点对的集合上作用非传递。本文综述了自１９９０年以来若干数学家包括作者本人在半传递图方面研究的最新结果，特别地，我们的工到了具有本原自同构群的半传递图，从而肯定地回答了Ｈｏｌｔｏｎ问题，同时还证明了只存在一个４度２７阶的半传递图，解决了Ｈｏｌｔ问题。     

Classifying Arc-Transitive Circulants of Square-Free Order

A circulant is a Cayley graph of a cyclic group. Arc-transitive circulants of square-free order are classified. It is shown that an arc-transitive circulant of square-free order n is one of the following: the lexicographic product , or the deleted lexicographic , where n = bm and is an arc-transitive circulant, or is a normal circulant, that is, Aut has a normal regular cyclic subgroup.     

对称无限布尔方阵的本原指数集的刻划

研究了对称无限布尔方阵,给出了对称无限布尔方阵为本原阵的一个充分必要条件,证明了具有有限直径d的对称无限布尔方阵的本原指数的上确界为2d,最后证明了直径不超过d的全体对称本原无限布尔方阵的本原指数集是E_d={1,2,…,2d}.     

3p阶亚循环群的连通4度Cayley图

研究3p阶(p是大于3的素数)亚循环群的连通4度Cayley图.主要决定了其全自同构群的结构,并由此得到这类图的CI性、正规性和弧传递性.用到单群分类定理.     

Non-Cayley Vertex-Transitive Graphs of Order Twice the Product of Two Odd Primes

For a positive integer n, does there exist a vertex-transitive graph Γ on n vertices which is not a Cayley graph, or, equivalently, a graph Γ on n vertices such that Aut Γ is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, Marusic and Scapellato, and McKay and the second author) has produced answers to this question if n is prime, or divisible by the square of some prime, or if n is the product of two distinct primes. In this paper we consider the simplest unresolved case for even integers, namely for integers of the form n = 2pq, where 2 < q < p, and p and q are primes. We give a new construction of an infinite family of vertex-transitive graphs on 2pq vertices which are not Cayley graphs in the case where p ≡ 1 (mod q). Further, if p ? 1 (mod q), p ≡ q ≡ 3(mod 4), and if every vertex-transitive graph of order pq is a Cayley graph, then it is shown that, either 2pq = 66, or every vertex-transitive graph of order 2pq admitting a transitive imprimitive group of automorphisms is a Cayley graph.     

On Orthogonal Double Covers of Graphs

A transformation which allows us to obtain an orthogonal double cover of a graph G from any permutation of the edge set of G is described. This transformation is used together with existence results for self-orthogonal latin squares, to give a simple proof of a conjecture of Chung and West.