# Signed Complete Graphs on Six Vertices

## TL;DR

This paper classifies all sixteen signed complete graphs on six vertices up to switching isomorphism, extending known classifications for smaller complete graphs.

## Contribution

It provides a complete enumeration of signed K6 graphs up to switching isomorphism, filling a gap in the classification of signed complete graphs.

## Key findings

- Sixteen signed K6 graphs up to switching isomorphism
- Extension of classification from smaller complete graphs
- Provides a foundation for further studies in signed graph theory

## Abstract

A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs. It is also known that, up to switching isomorphism, there are two signed $K_3$'s, three signed $K_4$'s, and seven signed $K_5$'s. In this paper, we prove that there are sixteen signed $K_6$'s upto switching ismomorphism.