Runtime of algorithm to transpose a graph
It all depends on how exactly you store the graph. For example:


If you store the graph as an adjacency matrix, the complexity is $\Theta(|V|^2)$.
If you store the graph as a linked list of edges per vertex (i.e., adjacency lists), then the complexity is $\Theta(|V|+|E|)$.
If you store the graph as a linked list of all edges (and don't store the vertices explicitly), then the complexity is $\Theta(|E|)$.


The representations usually used are the first two. I have never heard of the third being used. In most cases, the number of edges exceeds the number of vertices, and so there would be the third method would have no advantage over the second one, but it would have many disadvantages.