Ming Yao Xu

Automorphism groups and isomorphisms of Cayley digraphs

Abstract We call a Cayley digraph X = Cay(G, S) normal for G if the right regular representation R(G) of G is normal in the full automorphism group Aut(X) of X. In this paper we give some examples of normal and nonnormal Cayley digraphs and survey some results about the normality of Cayley digraphs. We also propose some conjectures and problems about them. In the last section of this paper we discuss a problem about isomorphisms of Cayley digraphs.

Vertex-primitive graphs of order a product of two distinct primes

Abstract Let k and p be odd primes with k p . All vertex-primitive graphs of order kp are classified, and those which are symmetric, or are edge-transitive but not symmetric. or are not Cayley graphs are identified. In addition, a classification is given of all vertex-primitive, edge-transitive lantisymmetric) directed graphs or order kp . The classification of the vertex-primitive symmetric graphs of order kp is used by the authors and others to complete the classification of symmetric graphs or order kp . The classification of the vertex-primitive non-Cayley graphs of order kp is used by B.D. McKay and the first author in an investigation of vertex-transitive non-Cayley graphs.

A Classification of Symmetric Graphs of Order 3p

Let ? be a simple undirected graph and G a subgroup of Aut ?. ? is said to be G-symmetric, if G acts transitively on the set of ordered adjacent pairs of vertices of ?. ? is said to be symmetric if it is Aut ?-symmetric. In this paper we give a complete classification for symmetric graphs of order 3p where p is a prime and p > 3. (See the Theorem in the end of Section 4). In the proof of this theorem several consequences of the finite simple group classification, including the classifications of doubly transitive permutation groups and primitive groups of degree mp with p being a prime and m < p, are used.

Symmetric Graphs of Order a Product of Two Distinct Primes

A simple undirected graph ? is said to be symmetric if its automorphism group Aut ? is transitive on the set of ordered pairs of adjacent vertices of ?, and ? is said to be imprimitive if Aut ? acts imprimitively on the vertices of ?. Let k and p be distinct primes with k < p. This paper gives a classification of all imprimitive symmetric graphs on kp vertices for k ? 5. The cases k < 5 have been treated previously by Cheng and Oxley (k = 2) and the second and third authors (k = 3), and the classification of primitive symmetric graphs on kp vertices with k ? 5 was done by the first and third authors.

On edge-transitive Cayley graphs of valency four

A characterization is given of a class of edge-transitive Cayley graphs, providing methods for constructing Cayley graphs with certain symmetry properties. Various new half-arc transitive graphs are constructed.

On cubic Cayley graphs of finite simple groups

For a finite group G, a Cayley graph Cay(G,S) is said to be normal if the group GR of right translations on G is a normal subgroup of the full automorphism group of Cay(G,S). In this paper, we prove that, for most finite simple groups G, connected cubic Cayley graphs of G are all normal. Then we apply this result to study a problem related to isomorphisms of Cayley graphs, and a problem regarding graphical regular representations of finite simple groups. The proof of the main result depends on the classification of finite simple groups.

2-Arc-transitive regular covers of complete graphs Having the covering transformation group Z p 3

A family of 2-arc-transitive regular covers of a complete graph is investigated. In this paper, we classify all such covering graphs satisfying the following two properties: (1) the covering transformation group is isomorphic to the elementary abelian p-group Zp3, and (2) the group of fiber-preserving automorphisms acts 2-arc-transitively. As a result, new infinite families of 2-arc-transitive graphs are constructed.

On cubic s -arc transitive Cayley graphs of finite simple groups

For a positive integer s, a graph Γ is called s-arc transitive if its full automorphism group AutΓ acts transitively on the set of s-arcs of Γ. Given a group G and a subset S of G with S = S-1 and 1 ∉ S, let Γ = Cay(G, S) be the Cayley graph of G with respect to S and GR the set of right translations of G on G. Then GR forms a regular subgroup of AutΓ. A Cayley graph Γ = Cay(G, S) is called normal if GR is normal in AutΓ. In this paper we investigate connected cubic s-arc transitive Cayley graphs Γ of finite non-Abelian simple groups. Based on Lis work (Ph.D. Thesis (1996)), we prove that either Γ is normal with s ≤ 2 or G = A47 with s = 5 and AutΓ ≃ A48. Further, a connected 5-arc transitive cubic Cayley graph of A47 is constructed.

A characterization of a class of symmetric graphs of twice prime valency

This paper characterizes the class of connected symmetric graphs of valency 2pp a prime, whose automorphism groups have abelian normal p-subgroups which are not semiregular on vertices. The examples all belong to a three-parameter family of graphs C(m, r, s), m ≥ 2, r ≥ 3, s ≥ 1; these graphs are discussed in the paper and, in particular, their automorphism groups are determined.

Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D 2 n

A Cayley graph X = Cay(G, S) of group G is said to be normal if R(G) is normal in Aut(X). Let G = «a, b | an = b2 = 1 », S be a generating set of G, |S| = 4. In this paper we show that any one-regular and 4-valent Cayley graph X = Cay(G, S) of dihedral groups G is normal except that n = 4s, and X ≃ Cay(G, {a, a-1, aib, a-ib}), where i2 ≡ ± 1 (mod 2s), 2 ≤ i ≤ 2s - 2.