Induced cayley graphs and d0ouble covers

Motivated by a construction of highly expanding simple Cayley graphs of dihedral groups derived from-or induced by-highly expanding Cayley digraphs fo cyclic groups presented by F. R. K. chung, constructions of simple Cayley graphs on a semidirect product of groups and Cayley digraphs of one of its factors are suggested. In case of the non-normal factor being the cyclic group of order 2. a condition is given to derive spectral bounds of the Cayley graph of the product from those of the Cayley graph of the normal factor.     

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 automorphism groups of quasiprimitive 2-arc transitive graphs

We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs. This work forms part of an ARC grant project and is supported by a QEII Fellowship.     

An explicit characterization of cubic symmetric bi-Cayley graphs on nonabelian simple groups

One of the remarkable contributions in the study of symmetric Cayley graphs on nonabelian simple groups is the complete classification of such graphs that are cubic and nonnormal. This naturally motivates the study of cubic (normal and nonnormal) symmetric bi-Cayley graphs on nonabelian simple groups. In this paper, the full automorphism groups of these graphs are determined, and necessary and sufficient conditions are given for a graph being a cubic normal symmetric Cayley or bi-Cayley graph on a nonabelian simple group (one may then find many examples). As an application, we also prove that cubic symmetric Cayley graphs on nonabelian simple groups are stable.     

Cayley Snarks and Almost Simple Groups

A Cayley snark is a cubic Cayley graph which is not 3-edge-colourable. In the paper we discuss the problem of the existence of Cayley snarks. This problem is closely related to the problem of the existence of non-hamiltonian Cayley graphs and to the question whether every Cayley graph admits a nowhere-zero 4-flow. So far, no Cayley snarks have been found. On the other hand, we prove that the smallest example of a Cayley snark, if it exists, comes either from a non-abelian simple group or from a group which has a single non-trivial proper normal subgroup. The subgroup must have index two and must be either non-abelian simple or the direct product of two isomorphic non-abelian simple groups. Received January 18, 2000 Research partially supported by VEGA grant 1/3213/96 Research partially supported by VEGA grants 1/3213/96 and 1/4318/97     

Isomorphisms of finite semi-Cayley graphs

Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph(Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph(also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits(of equal size). In this paper, we introduce the concept of SCI-graph(semi-Cayley isomorphism)and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.     

关于图的强乘积的可迁性的研究（英文）

Since many large graphs are composed from some existing smaller graphs by using graph operations, say, the Cartesian product, the Lexicographic product and the Strong product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the Strong product of vertex-transitive graphs. In particular, we show that the Strong product of Cayley graphs is still a Cayley graph.     

点传递图有向连通度的下界

本文研究点传递有向图与定向留连通度的下界，对达到此下界的Ｃｈｙｌｅｙ有向图与定向图进行了刻划。     

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Γ=Cay(G,S)is said to be normal if G is normal in Aut Γ.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Γ of finite nonabelian simple groups G,where the vertex stabilizer Av is soluble for A=Aut Γ and v ∈ VΓ.We prove that either Γ is normal or G=A5,A10,A54,A274,A549 or A1099.Further,11-valent symmetric nonnormal Cayley graphs of A5,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.     

On restricted edge-connectivity of replacement product graphs

This paper considers the edge-connectivity and the restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, we construct a Cayley graph with degree d whose restricted edge-connectivity is equal to d + s for given odd integer d and integer s with d 5 and 1 s d- 3, which answers a problem proposed ten years ago.     

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.     

Sur la separation dans les graphes de Cayley abeliens

In this note, we prove that Cayley digraphs on Abelian groups have some separation properties similar to undirected graphs.     

Fundamental groupoids of digraphs and graphs

We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.     

Circulant graph imbeddings

An algebraic characterization is given for those Cayley graphs for cyclic groups in which the neighborhood of any vertex is a cycle. A triangular imbedding is obtained for each such graph, either in the sphere, the torus, or the Klein bottle. The automorphism groups of these graphs are determined, and necessary and sufficient conditions are given for two such graphs to be isomorphic.     

Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two suffcient and necessary conditions for such graphs to be 1- or 2-arc-regular are given and based on the conditions, several infinite families of 1-or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.     

On the Cayley Graphs of Brandt Semigroups

In this article, the Cayley graphs of Brandt semigroups are investigated. The basic structures and properties of this kind of Cayley graphs are given, and a necessary and sufficient condition is given for the components of Cayley graphs of Brandt semigroups to be strongly regular. As an application, the generalized Petersen graph and k-partite graph, which cannot be obtained from the Cayley graphs of groups, can be constructed as a component of the Cayley graphs of Brandt semigroups.     

Relative difference sets fixed by inversion and Cayley graphs

Using graph theoretical technique, we present a construction of a (30,2,29,14)-relative difference set fixed by inversion in the smallest finite simple group—the alternating group A₅. To our knowledge this is the first example known of relative difference sets in the finite simple groups with a non-trivial forbidden subgroup. A connection is then established between some relative difference sets fixed by inversion and certain antipodal distance-regular Cayley graphs. With the connection, several families of antipodal distance-regular Cayley graphs which are coverings of complete graphs are presented.     

On normal 2-geodesic transitive Cayley graphs

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T,S). We first prove that if Cay(T,S) is neither cyclic nor K_4[2], then 〈a〉?{1}??S for all a∈S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2-arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.     

On the Cayley graphs of completely simple semigroups

In this paper, the Cayley graphs of completely simple semigroups are investigated. The basic structure and properties of this kind of Cayley graph are given, and a condition is given for a Cayley graph of a completely simple semigroup to be a disjoint union of complete graphs. We also describe all pairs (S,A) such that S is a completely simple semigroup, A⊆S, and Cay (S,A) is a strongly connected bipartite Cayley graph.     

二部图的单特征值

一个图的特征值通常指的是它的邻接矩阵的特征值,在图的所有特征值中,重数为1的特征值即所谓的单特征值具有特殊的重要性.确定一个图的单特征值是一个比较困难的问题,主要是没有一个通用的方法.1969年,Petersdorf和Sachs给出了点传递图单特征值的取值范围,但是对于具体的点传递图还需要根据图本身的特性来确定它的单特征值.给出一类正则二部图,它们是二面体群的凯莱图,这类图的单特征值中除了它的正、负度数之外还有0或者±1,而它们恰好是Petersdorf和Sachs所给出的单特征值范围内的中间取值.