Narrow Results

A Cayley graph $\mathrm{\Gamma }=Cay(G,S)$ is said to be core-free if $G$ is core-free in some $X$ for $G\le X\le Aut\mathrm{\Gamma }$. A graph $\mathrm{\Gamma }$ is called $s$-regular if $Aut\mathrm{\Gamma }$ acts regularly on its $s$-arcs. It is shown in this paper that if $s\le 2$, then there exist no core-free tetravalent $s$-regular Cayley graphs; and for $s\ge 3$, every tetravalent $s$-regular Cayley graph is a normal cover of one of the three known core-free graphs. In particular, a characterization of tetravalent $s$-regular Cayley graphs is given.

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.

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.