In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d, $d\le 4$, is at most $c_{d}n$ where $c_3=1$ and $c_4=9$. Whether such a constant $c_d$ exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, $\mathrm{\Gamma }\ncong K_{3,3}$, has a regular orbit, that is, an orbit of $〈g〉$ of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C.

在本文中，我们研究有限顶点传递图自同构的阶数、最长循环和循环数。特别是，我们证明了具有n 个顶点且价为d的连通顶点传递图的每个自同构的阶，$d\le 4$, 至多是$C_{d}n$在哪里$C_3=1$和$C_4=9$。是否有这样的常数$C_d$大于 4 的化合价是否存在仍然是一个悬而未决的问题。此外，我们证明有限连通三价顶点传递图 Г 的每个自同构g ，$\mathrm{\gamma }\ncong K_{3,3}$，有一个规则轨道，即轨道$〈G〉$长度等于g的量级。此外，我们证明在这种情况下，要么 Г 属于一个很好理解的异常图族，要么 Г 的至少 5/12 的顶点属于g的规则轨道。最后，我们根据 Γ 的顶点数和 C 的最长轨道的长度给出连通的三价顶点传递图 Г 的自同构循环群C的轨道数的上限。