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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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Why are coherent sheaves on $\Bbb P^1$ derived equivalent to representations of the Kronecke...
Your question about what happens when you have two vertices and $n$ arrows between them was answered independently by Minamoto (https://arxiv.org/abs/math/0702861) and Piontkovski (https://arxiv.org/a …