Oriented 2-cell embeddings of a graph and their symmetry classification: Generating algorithms and case study of the Mobius-Kantor graph graph

We discuss a method to derive all symmetry-distinct oriented 2-cell embeddings of a given graph and classify them based on their symmetry. As an example, we apply the algorithm to the highly symmetrical trivalent Mobius-Kantor graph. Considering the derived 2-cell embeddings as carbon networks leads to some interesting negative curvature carbon allotropes.