They can always be drawn with the vertices arranged so that all edges point downward as in Fig 1. Also all cycles that can be arranged like this are acyclic.

Fig 1.

From the proof of this fact one can deduce an algorithm for finding if a network is acyclic or not:

Furthermore, the adjacency matrix of such a graph can always be made upper-diagonal with zeros on diagonal (as no self-loops). The eigenvalues of an acyclic graph are thus all zero. One can also show the opposite, thus: