A gain graph is a graph whose edges are labelled "invertibly", or "orientably", by elements of a group G. This means that, if an edge e in one direction has label g (a group element), then in the other direction it has label g −1.
The label function φ therefore has the property that it is defined differently, but not independently, on the two different orientations, or directions, of an edge e. The group G is called the gain group, φ is the gain function, and the value φ(e) is the gain of e (in some indicated direction).
Some reasons to be interested in gain graphs are their connections to network flow theory in combinatorial optimization, to geometry, and to physics.
The term "gain graph" is more usual in other contexts, e.g., biased graph theory and matroid theory.
The term group-labelled graph has also been used, but it is ambiguous since "group labels" may be—and have been—treated as weights.