Lesson plan
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 the third circle of the theta graph is also in the list. 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 G is a graph and B is a linear class of circles in G. (A "linear class" 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.
Article summary
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 the third circle of the theta graph is also in the list. 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 G is a graph and B is a linear class of circles in G. (A "linear class" 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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.