In mathematics, a **biased graph** is a graph with a list of distinguished circles (edge sets of simple cycles), such that if two circles in the list are contained in a theta graph, then so is the third circle of the theta graph. A biased graph is a generalization of the combinatorial essentials of a gain graph and in particular of a signed graph.

Formally, a **biased graph** Ω is a pair (*G*, * B*) where

*is a*

**B****linear class**of circles; this by definition is a class of circles that satisfies the theta-graph property mentioned above.

A subgraph or edge set whose circles are all in * B* (and which contains no half-edges) is called

**balanced**. For instance, a circle belonging to

*is*

**B***balanced*and one that does not belong to

*is*

**B***unbalanced*.

Biased graphs are interesting mostly because of their matroids, but also because of their connection with multiary quasigroups. See below.

Read more about Biased Graph: Technical Notes, Examples, Minors, Matroids, Multiary Quasigroups

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