Extensions and generalizations of graphs | Matroid theory

Gain graph

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 βˆ’1. The label function Ο† 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, Ο† is the gain function, and the value Ο†(e) is the gain of e (in some indicated direction). A gain graph is a generalization of a signed graph, where the gain group G has only two elements. See Zaslavsky (1989, 1991). A gain should not be confused with a weight on an edge, whose value is independent of the orientation of the edge. (Wikipedia).

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