CONNECTIVITIES OF RANDOM CIRCULANT DIGRAPHS

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THE NORMALITY OF CAYLEY GRAPHS OF SIMPLE GROUP A5 WITH SMALL VALENCIES

Let G be a finite group and S a subset of G not containing the identity element 1. We define the Cayley (di)graph X = Cay(G, S) of G with respect to S by V(X) = G,E(X) = {(g, sg) [ g ∈ G, s ∈ S}. A Cayley (di)graph X = Cay(G, S) is called normal if GR A = Aut(X). In this paper we prove that if S = {a, b, c} is a 3-generating subset of G = A5 not containing the identity 1, then X = Cay(G, S) is a normal Cayley digraph.     

HAMILTONIAN DECOMPOSITION OF CAYLEY GRAPHS OF ORDERS p~2 AND pq

1. IntroductionLet G be a finite group and S a subset of G such that S--1 ~ S, and 1 f S. The Cayleygraph Cay (G, S) is defined as the simple graph with V ~ G, and E = {glgZ I g,'g, or g,'g,6 S, gi, gi E G}. Cay (G, S) is vertex-transitive, and it is connected if and only if (S) = G,i.e. S is a generating set of G[1]. If G = Zn, then Cay (Zn, S) is called a circulant graph. Ithas been proved that any connected Cayley graph on a finite abelian group is hamiltonianl2].Furthermore, …     

一类非正规Cayley有向图

本文研究了2p2(p奇素数)阶非交换群上两度Cayley有向图的正规性,发现 了一无限族非正规的Cayley有向图.     

A Family of Nonnormal Cayley Digraphs

We call a Cayley digraph Γ = Cay(G, S) normal for G if G _R , the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p ² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found. Received February 23, 1998, Revised September 25, 1998, Accepted October 27, 1998     

线有向图的幂敛指数

设G是有向图，LG表示c的线有向图，本文得到了它们的幂敛指数及周期之间的关系：k（G）－1≤（LG）≤k（G）＋1,p（LG）=p（G）特别地，当G为本原图时有：k（LG）＝k（G）＋1     

对称群上Cayley图的DNA计算

有一类图称为Cayley图或群图.猜想每个Cayley图都是Hamilton图.求Cayley图和有向Cayley图中的Hamilton圈和路自然产生在计算科学里.这篇文章研究了对称群上Cayley图的DNA计算和给出了求它的Hamilton圈的DNA算法.     

On the Normality of Cayley Digraphs of Valency 2 on Non-Abelian Group of Odd Order

In this paper, we prove that the Cayley digraph = Cay(G, S) of valency 2 on non-abelian group G of odd order is normal if the automorphism group of A(), a graph constructed from by using the method presented in the paper, is primitive on the vertices set V(A(). We also prove that the Cayley digraphs of valency 2 on non-abelian group of order pq² are normal, where p and q are distinct odd primes.AMS Subject Classification (2000) 05C25 20B25Supported by the National Natural Science Foundation of China (Grant no. 19971086) and the Doctoral Program Foundation of the National Education Department of China.     

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.     

On non‐Hamiltonian circulant digraphs of outdegree three

We construct infinitely many connected, circulant digraphs of outdegree three that have no Hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 319–331, 1999     

On Arc-Regular Permutation Groups Using Latin Squares

For a given a permutation group G, the problem of determining which regular digraphs admit G as an arc-regular group of automorphism is considered. Groups which admit such a representation can be characterized in terms of generating sets satisfying certain properties, and a procedure to manufacture such groups is presented. The technique is based on constructing appropriate factorizations of (smaller) regular line digraphs by means of Latin squares. Using this approach, all possible representations of transitive groups of degree up to seven as arc-regular groups of digraphs of some degree is presented.Partially supported by the Comissionat per a Universitats i Recerca of the Generalitat de Catalunya under Grant 1997FI-693, and through a European Community Marie Curie Fellowship under contract HPMF-CT-2001-01211.     

Cayley Digraphs of Finite Simple Groups with Small Out-Valency

Abstract

For a group G and a subset S of G which does not contain the identity of G, the Cayley digraph Cay(G, S) is called normal if R(G) is normal in Aut(Γ). In this paper, we investigate the normality of Cayley digraphs of finite simple groups with out-valency 2 and 3. We give several sufficient conditions for such Cayley digraphs to be normal. By using this result, we consider the digraphical regular representations of finite simple groups.     

凯莱-海森堡群上的格林函数

本文研究凯莱-海森保群上的格林函数.利用凯莱-海森堡群上热核的解析表达式,导出了一阶凯莱-海森堡群上的格林函数的有理分式表示的公式.     

On the Classification of Arc-transitive Circulant Digraphs of Order Odd-Prime-Squared

A Cayley graph F = Cay（G, S） of a group G with respect to S is called a circulant digraph of order pk if G is a cyclic group of the same order. Investigated in this paper are the normality conditions for arc-transitive circulant （di）graphs of order p^2 and the classification of all such graphs. It is proved that any connected arc-transitive circulant digraph of order p^2 is, up to a graph isomorphism, either Kp2, G（p^2,r）, or G（p,r）[pK1], where r｜p- 1.     

A family of weakly distance-regular digraphs of girth 2

A family of commutative weakly distance-regular digraphs of girth 2 was classified in [K. Wang, Commutative weakly distance-regular digraphs of girth 2, European J. Combin. 25 (2004) 363-375]. In this paper, we classify this family of digraphs without the assumption of commutativity.     

对称本原图的集指数与本原简单图的广义上指数的极图

一个有向图D称为本原的，如果存在某个正整数k，使得对于D中的任一点x到任一点y都有长为k的途径，这样的正整数k中的最小者称为D的本原指数，作为本原指数概念的推广，R．A．Brualdi和柳柏濂于1990年引入了本原有向图的广义本原指数的新概念，本文给出了对称本原图的集指数的一些性质，并对本原简单图的广义上指数的极图进行了完全刻划。     

Automorphism Groups of 2-Valent Connected Cayley Digraphs on Regular <Emphasis Type="Italic">p</Emphasis>-Groups

 Let X=Cay(G,S) be a 2-valent connected Cayley digraph of a regular p-group G and let G _R be the right regular representation of G. It is proved that if G _R is not normal in Aut(X) then X≅[2K ₁ ] with n>1, Aut(X) ≅Z ₂ wrZ _2n , and either G=Z _2n+1 =<a> and S={a,a ²ⁿ⁺¹ }, or G=Z _2n ×Z ₂ =<a>×<b> and S={a,ab}. Received: May 26, 1999 Final version received: June 19, 2000     

Further results on the enumeration of hamilton paths in Cayley digraphs on semidirect products of cyclic groups

First, let m and n be positive integers such that n is odd and gcd(m,n)=1. Let G be the semidirect product of cyclic groups given by . Then the number of hamilton paths in Cay(G:x,y) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order are (0,0), (4mn²+2n,16m²n), (n,4m), and (0,8m). Second, let m and n be positive integers such that n is odd. Let G be the semidirect product of cyclic groups given by . Then the number of hamilton paths in Cay(G:x,y) (with initial vertex 1) is (3m-1)n+m⌊(n+1)/3⌋+1.     

Cayley digraphs and lexicographic product

In this paper, we prove that a Cayley digraph Γ = Cay(G, S) is a nontrivial lexicographical product if and only if there is a nontrivial subgroup H of G such that S∖H is a union of some double cosets of H in G.