ON ARC-TRANSITIVE CIRCULANT DIGRAPHS

Denote by C n(S) the circulant digraph with vertex set Z n={0,1,2,...,n-1} and symbol set S(≠-S)Z n }. Let X be the automorphism group of C n(S) and X 0 the stabilizer of 0 in X. Then C n(S) is arc transitive if and only if X 0 acts transitively on S. In this paper, C n(S) with X 0| S being the symmetric group is characterized by its symbol set. By the way all the arc transitive circulant digraphs of degree 2 and 3 are given.

ARC-TRANSITIVE CUBIC GRAPHS OF ORDER 4_p

In this paper, a complete classification of arc-transitive cubic graphs of order 4p is given.

Locally-primitive Arc-transitive lO-Valent Graphs of Square-free Order

In this paper, we present a complete classification for locally-primitive arctransitive graphs which have square-free order and valency 10. The classification involves nine graphs and three infinite families of graphs.

On Pentavalent Arc-Transitive Graphs

In this paper, a characterization of all pentavalent arc-transitive graphs is given. It is shown that each pentavalent arc-transitive covering graph F is a regular simple or elementary abelian covering graph. In particular, the elementary abelian covering groups are Z3,Z5or a subgroup of Z2^5.

On fixity of arc-transitive graphs

The relative xity of a permutation group is the maximum proportion of the points xed by a non-trivial element of the group,and the relative xity of a graph is the relative xity of its automorphism group,viewed as a permutation group on the vertex-set of the graph.We prove in this paper that the relative xity of connected 2-arc-transitive graphs of a xed valence tends to 0 as the number of vertices grows to in nity.We prove the same result for the class of arc-transitive graphs of a xed prime valence,and more generally,for any class of arc-transitive locally-L graphs,where L is a xed quasiprimitive graph-restrictive permutation group.

Arc-Transitive Graphs of Square-free Order with Valency 11

In this paper,we present a complete list of connected arc-transitive graphs of square-free order with valency 11.The list includes the complete bipartite graph K11,11,the normal Cayley graphs of dihedral groups and the graphs associated with the simple group J1 and PSL(2,p),where p is a prime.

Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal;that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.

Two sufficient conditions for non-normal Cayley graphs and their applications

A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G, S). In this paper, two sufficient conditions for non-normal Cayley graphs are given and by using the conditions, five infinite families of connected non-normal Cayley graphs are constructed. As an application, all connected non-normal Cayley graphs of valency 5 on A 5 are determined, which generalizes a result about the normality of Cayley graphs of valency 3 or 4 on A 5 determined by Xu and Xu. Further, we classify all non-CI Cayley graphs of valency 5 on A 5, while Xu et al. have proved that A 5 is a 4-CI group.

Nonorientable regular embeddings of graphs of order pq

In [J. Algeb. Combin. 19(2004), 123-141], Du et al. classified the orientable regular embeddings of connected simple graphs of order pq for any two primes p and q. In this paper, we shall classify the nonorientable regular embeddings of these graphs, where p ≠ q. Our classification depends on the classification of primitive permutation groups of degree p and degree pq but is independent of the classification of the arc-transitive graphs of order pq.

Semisymmetric graphs admitting primitive groups of degree 9p

Let Γ be a connected regular bipartite graph of order 18 p, where p is a prime. Assume that Γ admits a group acting primitively on one of the bipartition subsets of Γ. Then, in this paper, it is shown that eitherΓ is arc-transitive, or Γ is isomorphic to one of 17 semisymmetric graphs which are constructed from primitive groups of degree 9p.

On Locally Primitive Graphs of Order 18p

The authors investigate locally order 18p. It is shown that such a graph the Gray graph and the Tutte 12-cage. primitive bipartite regular connected graphs of is either arc-transitive or isomorphic to one of