Signed graph

In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.

Two fundamental questions about a signed graph are: Is it balanced? What is the largest size of a balanced edge set in it? The first question is not difficult; the second is computationally intractable (technically, it is NP-hard).

Signed graphs were first introduced by Harary to handle a problem in social psychology (Cartwright and Harary, 1956). They have been rediscovered many times because they come up naturally in many unrelated areas. For instance, they enable one to describe and analyze the geometry of subsets of the classical root systems. They appear in topological graph theory and group theory. They are a natural context for questions about odd and even cycles in graphs. They appear in computing the ground state energy in the non-ferromagnetic Ising model; for this one needs to find a largest balanced edge set in Σ. They have been applied to data classification in correlation clustering.