Simplest way to check edge set for transitivity
There's a very simple algorithm for this:

for each edge (u,v) in the graph:
    for each edge (v,w) in the graph:
        if (u,w) is not in the graph, return "Not transitive"
return "Transitive"


Basically, if the graph is not transitive, then you can always find some path of length two $u\to v \to w$ such that the edge $u \to w$ is not present in the graph.  If the graph is transitive, there won't be any such path of length 2.  So, just check this condition.