In this paper a classification of tetravalent edge-transitive metacirculants is given. It is shown that a tetravalent edge-transitive metacirculant $Γ$ is a normal graph except for four known graphs. If further, $Γ$ is a Cayley graph of a non-abelian metacyclic group, then $Γ$ is half-transitive.
亚循环图是一类基本的点传递图，弱亚循环图是它的推广。一个图称为弱亚循环图，如果它有一个点传递亚循环自同构群。本文研究了奇素数幂阶的弱亚循环。首先证明了奇素幂次弱亚循环是亚循环的当且仅当它有一个点传递分裂亚循环自同构群。然后我们证明了对于任何奇素数$p$和整数$\ell\geq 4$，存在$p^\ell$阶的弱亚循环，它们是任何亚循环群的Cayley图，但不是Cayley图;这回答了Li et al.（2013）中的一个问题。我们构造这样的图明确引入一个建设，这是广义彼得森图的推广。最后，我们确定了任何亚循环群的所有可能的奇素数幂阶的最小亚循环图是Cayley图而不是Cayley图.
A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order $12p$.
Let $s$ be a positive integer. A graph is $s$-transitive if its automorphism group is transitive on s-arcs but not on $(s + 1)$-arcs. In this paper, we study all tetravalent s-transitive graphs of order $6p^2$.
In this paper, we introduce a new family of graphs, $Γ(n,a)$. We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of $Γ(n,a)$.
Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order $2^s$ for every $s\geqslant3$ has been wide ope…
We prove that, given a finite graph $Σ$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $Σ$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.
A graph $Γ$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(Γ)$ is transitive on the set of vertices of $Γ$ and the set of $s$-arcs of $Γ$, where for an integer $s \ge 1$ an $s$-arc of $Γ$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $Γ$ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$. $Γ$ is called 2-transitive if it is $(\mathrm{Aut}(Γ), 2)$-arc-transitive but not $(\mathrm{Aut}(Γ), 3)$-arc-transitive. A Cayley graph $Γ$ of a group $G$ is called normal if $G$ is normal in $\mathrm{Aut}(Γ)$ and non-normal othe…
