  Consider the complete graph on \(n\) vertices where each edge is
independently open with probability \(p,\) or closed otherwise. Phase
transitions for such graphs for \(p = \frac{C}{n}\) have previously been
studied using techniques like branching processes and random walks. In this
paper, we use an alternate component counting argument for establishing phase
transition and obtaining estimates on the sum size of the non-giant components.
As a corollary, we also obtain estimates on the size of the giant component for
\(C\) large: If \(C\) is sufficiently large, there is a positive constant \(M_0
= M_0(C)\) so that with probability at least \(1-e^{-C/100},\) there is a giant
component containing at least \(n - ne^{-C/8}\) vertices and every other
component contains less than \(M_0 \log{n}\) vertices.
