[Math] Maximum number of components in a Graph Containing $n$ vertices and $k$ edges

Question

I know that in a Graph $G$ Containing $n$ vertices and $k$ edges,$G$ will contain atleast $n-k$ components.

Explanation-:

Graph containing $n$ vertices and $0$ edges will contain $n$ components.Each time adding an edge will reduce the component by 1.Thus $k$ edge will contain $n-k$ component.
This is the minimum number.

I am interested in finding the maximum number of component.please help me out !!!

Answer 1

In the worst case the edges are used to form a clique. Let $i$ be the smallest integer $\ge 1$ s.t. $\frac{i(i-1)}{2} \ge k$. There are $n-i+1$ connected components in that case, which is maximal posible.