Six signed Petersen graphs, and their automorphisms

TL;DR

This paper classifies all six signed Petersen graphs up to switching isomorphism, analyzing their automorphisms and invariants, and introduces new properties of signed graphs and their automorphism groups.

Contribution

It provides a complete classification of signed Petersen graphs and develops new theoretical properties and invariants in signed graph theory.

Findings

Six distinct signed Petersen graphs identified

Computed switching automorphism groups and invariants

Developed new properties of signed graphs and their automorphism groups

Abstract

Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process we develop new properties of signed graphs, especially of their switching automorphism groups.