Signed Complete Graphs on Six Vertices

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.