群与图及相关课题的研究

本综述了把群与图综合起来进行研究的一些工作,简述了与之相关的一些课题,并提出了有待解决的一些问题。     

Bubble-sort图和Modified Bubble-sort图的自同构群

Bubble-Sort图和Modified Bubble-Sort图是两类特殊的Cayley图,由于其在网络构建中的应用而受到广泛关注.本文完全确定了这两类图的自同构群.     

三度边正则图的三个无限族

图X称为边正则图，若X的自同构群Aut(X)在X的边集上的作用是正则的．本文考察了三度边正则图与四度Cayley图的关系，给出了一个由四度Cayley图构造三度边正则图的方法，并且构造了边正则图的三个无限族．     

4p~2阶4度半传递图

如果图X的全自同构群Aut(X)作用在其顶点集V(X)和边集E(X)上都是传递的,但作用在弧集Arc(X)上非传递,则称X是半传递图.研究了4p~2(p3且p≡-1(mod4))阶4度半传递图,确定了4p~2阶4度半传递图的连通性及其自同构群的阶.     

有限素数度弧正则图

如果一个图的全自同构群在其弧集上正则,则称此图为弧正则图.本文刻画素数度的立方自由阶弧正则图,证明任何素数度2倍奇立方自由阶弧正则图都是正规或二部正规Cayley图,且不存在任意素数度4倍奇立方自由阶的弧正则图,推广了一些已知的结果,得到阶为8倍奇平方自由阶素数度弧正则图的分类,并发现新的弧正则图类.此外,基于所得的结果,我们提出一个猜想和有待后续研究的一些问题.     

6p阶2度有向Cayley图的正规性

群G的Cayley有向图X=Cay(G,S)叫做正规的,如果G的右正则表示R(G)在X的全自同构群Aut(X)中正规.决定了6p(p素数)阶2度有向Cayley图的正规性,发现了一个新的2度非正规Cayley有向图.     

2pq阶Cayley图是Hamilton图

一、引言对Cayley图的Hamilton性的研究近几年有所突破[1]现最好的结果是[2]的主要定理:若群G上的换位子群C′是p~n(p是素数,n是正整数)阶循环群时,G上的每个Cayley图皆为Hamilton图。1987年D.Marusic还证明了2p~2(p是素数)阶Cayley图为Hamilton图[4]。本文用群的构造理论证明:2pq(p,q是素数)阶Cayley图是Hamilton图。本文中所提到的群G皆指有限群;群的有关术语和记号同于文献[3];图的有关术     

偶错位图的张量幂的最大独立集和自同构群

若A_n 是X := {1, 2,..., n} 上的偶置换构成的交错群, E_n 是X 上的偶错位集, 则Cayley 图AΓ_n := Γ(A_n, E_n) 称为偶错位图. 令AΓ_n^q 为q 个AΓ_n 的张量幂. 在本文中, 我们研究了AΓ_n^q 的连通性、直径、独立数、团数、色数和最大独立集等性质. 利用AΓ_n^q 最大独立集的结果, 我们完全确定了AΓ_n^q 的自同构群的结构.     

关于交换群上的Cayley有向图的正规性

Cayley有向图X=Cay(G,S)叫做正规的,如果G的右正则表示R(G)在X的全自同构群Aut(X)中正规,我们定出了交换群上的小度数的非正规的Cayley有向图, 并给出了一个猜想．应用这个结果,给出了pn(n≤2)个点上的度数不超过3的有向对称图的分类,这里p是一个奇素数．     

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．     

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.     

Tetravalent vertex‐transitive graphs of order 4p

A graph is vertex‐transitive if its automorphism group acts transitively on vertices of the graph. A vertex‐transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this article, the tetravalent vertex‐transitive non‐Cayley graphs of order 4p are classified for each prime p. As a result, there are one sporadic and five infinite families of such graphs, of which the sporadic one has order 20, and one infinite family exists for every prime p>3, two families exist if and only if p≡1 (mod 8) and the other two families exist if and only if p≡1 (mod 4). For each family there is a unique graph for a given order. © 2011 Wiley Periodicals, Inc.     

一类Cayley图的可扩性

设 Sn是那个对称群 .让〈n〉 ={ 1,2 ,… ,n} ,B*表示 Sn中所有对换的集合和 B B* .关于 B的对换图 Wn 被定义为 V(Wn) =〈n〉,E(Wn) ={ [uv]:(uv)∈ B} .如果 Wn是一棵树 ,则这个对换图称为一棵对换树 Tn.Tn 是 Sn 的一个极小生成集 .在这篇文章里 ,我们研究了 Cayley图 Cay(Sn,Tn)的性质 .证明了Cay(Sn,Tn)是 (n - 2 ) -可扩的 ,即 ,Cay(Sn,Tn)的可扩性达到最大 .     

The Spectrum and Automorphism Group of the Set-Inclusion Graph

Let n,k and l be integers with 1 ≤ k ＜ l ≤ n-1.The set-inclusion graph G(n,k,l) is the graph whose vertex set consists of all k-andl-subsets of[n]={1,2,...,n},where two distinct vertices are adjacent if one of them is contained in the other.In this paper,we determine the spectrum and automorphism group of G(n,k,l).     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Dihedral Hamiltonian Cycle Systems of the Cocktail Party Graph

The existence problem for a Hamiltonian cycle decomposition of (the so called cocktail party graph) with a dihedral automorphism group acting sharply transitively on the vertices is completely solved. Such Hamiltonian cycle decompositions exist for all even n while, for n odd, they exist if and only if the following conditions hold: (i) n is not a prime power; (ii) there is a suitable ? such that (mod 2^?) for all prime factors p of n and the number of the prime factors (counted with their respective multiplicities) such that (mod ) is even. Thus in particular one has a dihedral Hamiltonian cycle decomposition of the cocktail party graph on 8k vertices for all k, while it is known that no such cyclic Hamiltonian cycle decomposition exists. Finally, this paper touches on a recently introduced symmetry requirement by proving that there exists a dihedral and symmetric Hamiltonian cycle system of if and only if (mod 4).