A graph with $n$ vertices and $p$ components has at least $n-p$ edges. Can I give an upper bound for the number of edges of such a graph? If so, how to prove it?
My attempt:
For each component, we have that the number of edges $m_i$ is given by $m_i \le \binom{n_i}{2}$, with $n_i$ the number of vertices within a component. So $m \le \sum_{i=1}^p \binom{n_i}{2} \le p\binom{n}{2}$.
Thanks.