It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.

中文翻译：

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关于具有可解自同构群的三次对称非凯莱图

在[Y.-Q. 冯，CH Li 和 J.-X。Zhou，具有可解自同构群的对称三次图，{\em European J. Combin.} {\bf 45} (2015), 1-11] 具有可解自同构群的三次对称图是 Cayley 图或 $2 $-$2^2$ 类型的正则图，即没有 $2$ 阶自同构交换相邻顶点的图。在本文中，构造了具有可解自同构群的 $2^2$ 类型的非凯莱三次 $2$-正则图的无限族。该系列中最小的图的阶数为 6174。