I know that a graph whose vertices can be divided into two sets $U$ and $V$ such that every edge can only connect a vertex in $U$ to one in $V$ is called a bipartite graph. Is there a name for a type of graph where edges are allowed from $U$ to $V$ and from $V$ to $V$, but not from $U$ to $U$?
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Well, according to your last comment, you have probably some chaos in quantifiers. Because there are three (in the case I have not overlooked some) possible interpretations of your question (no matter whether you allow or disallow loops and/or multiple edges):
If you have meant some other interpretation, please specify your question.