We need a disconnected graph, that too with the maximum number of edges possible. To satisfy both these conditions, we can say that we must have a graph with exactly two components, each of which is a complete graph.
To maximize the number of edges, we should make a complete graph with $9$ vertices, and isolate one vertex.
Hence, we get $36$ edges. (In a complete graph on n vertices, we have ${}^nC_2$ edges. So, for a complete graph on 9 vertices, we have ${}^9C_2 = 36$ Edges)
In general, we can say that, for n vertices disconnected graph, maximum number of edges possible is ${}^{(n-1)}C_2$.