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In this monograph, we consider natural operations on directed graphs. And we find the connections between our operations on graphs and the groupoid-perations on "graph" groupoids. Remark that we cannot guarantee the (algebraic or categorial) groupoids generated by the grouopoid-operations; sum, product, quotient or complement; of graph groupoids are again graph groupoids. By defining suitable operations on graphs, we can conclude the groupoids generated by the groupoid-operations of graph groupoids are again graph groupoids; for example, the product groupoid of two graph groupoids is groupoid-isomorphic to the graph groupoid of the product graph, etc. This provides another bridge connceting combinatorics and algebra. Recently, the von Neumann algebras generated by graph groupoids, called graph von Neumann algebras, have been studied. By using the fundamental techniques from graph von Neumann algebra theory, we can characterize the properties of groupoid von Neumann algebras, generated by groupoids obtained from the groupoid-operations, as certain graph von Neumann algebras.