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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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Invariant Subvarieties of Variety of Quiver Representations

I need to think about this more carefully, but my hypothesis (very tentative, as I'm just beginning to study quivers) is that for quivers of finite type, these are all the G-invariant varieties. … Ultimately, I'd like to find an answer for certain classes of wild quivers, but understanding this example would be a good start. with thanks, Mike …