A comma represents the end of a breath group, with associated intonation and pause.
If you don't end a breath group after "edges", it is perfectly grammatical, but parses as
These constraints give rise to [a graph with three edges as shown in Figure 1].
which implies that it is the three edges that are shown in Figure 1, rather than the (more usual) meaning that the graph is shown in Figure 1. In practice, there might not be much difference between the meanings in this case, but in other instances they could be very different.