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I conjecture that, in a Directed Acyclic Graph, $O(|V|) = O(|E|)$. Is this statement correct, can it be refined? This is probably standard material; is there a simple reference about this?

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The maximum number of edges in a DAG with n vertices is $\Theta(n^2)$.

Refer https://stackoverflow.com/questions/11699095/how-many-edges-can-there-be-in-a-dag

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Take the complete bipartite graph $K_{n,n}$ and direct all edges from left to right. This is a DAG with $2n$ vertices and $n^2$ edges.

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