Narrow Results

A lot of attention has been paid recently to the construction of symmetric covers of symmetric graphs. After a new approach given by Conder and the author [Arc-transitive abelian regular covers of cubic graphs, J. Algebra387 (2013) 215–242], the group of covering transformations can be extended to more general abelian groups rather than cyclic or elementary abelian groups. In this paper, by using the Conder–Ma approach, we investigate the symmetric covers of 4-valent symmetric graphs. As an application, all the arc-transitive abelian regular covers of the 4-valent complete graph $K_5$ which can be obtained by lifting the arc-transitive subgroups of automorphisms $A_5$ and $\mathrm{\text{A}}GL(1,5)$ are classified.

Xu, Zhang and Zhou [Chin. Ann. Math.25B(4) (2004) 545–554] classified all arc-transitive cubic graphs of order $4p$ with $p$ a prime. In this paper, we shall generalize this result by classifying all connected arc-transitive cubic graphs of order four times an odd square-free integer. It is shown that, for each of such graphs $\mathrm{\Gamma }$, either the full automorphism group $Aut\mathrm{\Gamma }$ is isomorphic to $PSL(2,p)$, $PGL(2,p)$, $PSL(2,p)\times {\mathbb{Z}}_2$ or $PGL(2,p)\times {\mathbb{Z}}_2$, or $\mathrm{\Gamma }$ is isomorphic to one of 5 graphs.

A Cayley graph $\mathrm{\Gamma }$ is called arc-transitive if its automorphism group $Aut\mathrm{\Gamma }$ is transitive on the set of arcs in $\mathrm{\Gamma }$. In this paper, we give a characterization of cubic arc-transitive Cayley graphs on a class of Frobenius groups.

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

In this paper, a characterization of all pentavalent arc-transitive graphs is given. It is shown that each pentavalent arc-transitive covering graph Γ is a regular simple or elementary abelian covering graph. In particular, the elementary abelian covering groups are ℤ₃, ℤ₅ or a subgroup of ${\mathbb{Z}}_2^5$.

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

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 ${\mathsf{K}}_{11,11}$, the normal Cayley graphs of dihedral groups and the graphs associated with the simple group ${\mathrm{J}}_1$ and $\mathrm{PSL}(2,p)$, where $p$ is a prime.

Let $X$ be a connected $G$-locally primitive arc-transitive graph for some subgroup $G$ of $\mathrm{Aut}(X)$. Suppose that $X$ is a $(G,s)$-transitive graph. In this paper, we give a characterization of the vertex-stabilizer $G_v$ when $X$ has valency 8.

In this paper, we classify connected arc-transitive graphs of valency 11 and of order $4n$, where $n$ is an odd square-free integer. In particular, there are four infinite families of such graphs, which have a minimal arc-transitive automorphism group isomorphic to $\mathrm{PSL}(2,r)$, $\mathrm{PGL}(2,r)$, $Z_2\times \mathrm{PSL}(2,r)$ and $\mathrm{PSL}(2,r)\times Z_{11}$ separately, where $r\equiv \pm 1(\mathrm{mod}11)$ is a prime. Moreover, examples for each class are constructed.