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.

Read more about Biased GraphTechnical Notes, Examples, Minors, Matroids, Multiary Quasigroups

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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