Being a https://en.wikipedia.org/wiki/Perfect_graph is not monotonic under adding edges, and it's certainly an interesting property for a graph to have! (But yes, most interesting properties are monotonic under adding edges)

> It seems to me most graph problems that people care about are monotonic under adding edges. Actually, I can't think of a single problem in graph theory I've done where that wouldn't hold.

https://en.wikipedia.org/wiki/Braess%27s_paradox is very famous.