Blue Balanced Vs Red Unbalanced
A cycle, or more generally a closed-walk of (G, 𝛴 ) is said to be unbalanced if the product of signs of its edges, repetition considered, is negative. Otherwise it is said to balanced.
Attention As in (G, E(G)) a cycle is unbalanced if and only if it is an odd-cycle, the term "odd-cycle" is frequently used to refer to an "unbalanced-cycle". We strongly recommend against such confusing terminology.
Historical note Harary was first to use the term "unbalanced", however he defined it to be any signed graph with an unbalanced-cycle. That is because in any such graph the friendship rules (friend of a friend is a friend, enemy of a friend is enemy, friend of an enemy is an enemy, enemy of enemy is a friend) leads to an awkward relationship. He then characterized the set of balanced graphs as those which can be resigned to have only positive edges. A generalization of this theorem is provided Zaslavsky in "T. Zaslavsky, Signed graphs. Discrete Appl Math 4(1) (1982), 47–74.":
Theorem Two signatures 𝛴1 and 𝛴2 on a same graph G are equivalent if and only if they induce the same set of unbalanced cycles.