# Biased Graph

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 B is a 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 B is balanced and one that does not belong to B is unbalanced.

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

### Other articles related to "graph, graphs, biased graph, biased graphs":

Examples - Matroids From Graph Theory
... A second original source for the theory of matroids is graph theory ... Every finite graph (or multigraph) G gives rise to a matroid as follows take as E the set of all edges in G and consider a set of edges independent if and only ... Other matroids on graphs were discovered subsequently The bicircular matroid of a graph is defined by calling a set of edges independent if every connected subset ...
Biased Graph - Multiary Quasigroups
... Just as a group expansion of a complete graph Kn encodes the group (see Dowling geometry), its combinatorial analog expanding a simple cycle of length n + 1 encodes an n-ary (multiary ... to prove theorems about multiary quasigroups by means of biased graphs (Zaslavsky, t.a.) ...

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