I have already proven that the complement of a simple, disconnected graph G results in a complementary graph that is always connected.
But what about the case for a simple, connected graph G? Is its complementary graph always disconnected? How would this be proven?
No, the complement of a connected graph can be either connected or non-connected.
There are even graphs that are isomorphic to their complements, such as the cycle graph with 5 vertices, or the path graph with 4 vertices.