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"Quiver" is the word used for "directed graph" in some parts of representation theory. The main reason to use the term quiver is to indicate an interest in considering representations of the quiver.
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Are the underlying undirected graphs of two mutation-equivalent acylic quivers isomorphic?
It is important in the representation theory of quivers, in the theory of cluster algebras, and in physics. … In the theory of cluster algebras the quivers encode exchange relations.
You might check Keller's java program to see quiver mutation in action. …