# Classification of cubic vertex-transitive tricirculants

## TL;DR

This paper classifies all finite connected cubic vertex-transitive tricirculant graphs, revealing their structure and categorizing them into specific families with a few small exceptions.

## Contribution

It provides a complete classification of finite connected cubic vertex-transitive tricirculants, identifying their structure and infinite families, extending previous understanding of symmetric graphs.

## Key findings

- Most such graphs are prisms or Möbius ladders.
- Two infinite families of graphs are characterized for orders 6k with k odd.
- Small exceptions of order less than 54 are explicitly listed.

## Abstract

A finite graph is called a tricirculant if admits a cyclic group of automorphism which has precisely three orbits on the vertex-set of the graph, all of equal size. We classify all finite connected cubic vertex-transitive tricirculants. We show that except for some small exceptions of order less than 54, each of these graphs is either a prism of order 6k with k odd, a M\"obius ladder, or it falls into one of two infinite families, each family containing one graph for every order of the form 6k with k odd.