On cubic symmetric non-Cayley graphs with solvable automorphism groups

TL;DR

This paper constructs an infinite family of non-Cayley cubic symmetric graphs with solvable automorphism groups, expanding the understanding of their structure beyond previously known classifications.

Contribution

It introduces an infinite family of non-Cayley cubic 2-regular graphs of type 2^2 with solvable automorphism groups, including the smallest graph of order 6174.

Findings

Constructed an infinite family of such graphs.

Identified the smallest graph with order 6174.

Extended classification of cubic symmetric graphs with solvable automorphism groups.

Abstract

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.