Cubic symmetric graphs having an abelian automorphism group with two orbits

TL;DR

This paper classifies all finite connected cubic symmetric graphs with an abelian automorphism group that has exactly two orbits on vertices, extending previous classifications of girth 6 graphs.

Contribution

It determines all such cubic symmetric graphs with an abelian automorphism group and two orbits, including new graphs beyond those of girth 6.

Findings

Most graphs have girth 6, except for a few known exceptions.

Identified all graphs with the specified symmetry and automorphism properties.

Extended previous classifications to include new cases.

Abstract

Finite connected cubic symmetric graphs of girth 6 have been classified by K. Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an abelian automorphism group with two orbits on the vertex set. In this paper all cubic symmetric graphs with the latter property are determined. In particular, with the exception of the graphs K_4, K_{3,3}, Q_3, GP(5,2), GP(10,2), F40 and GP(24,5), all the obtained graphs are of girth 6.