I want to know if you could could help me in establishing a proof for connected graphs?
The question is prove that a finite graph $G= (V,E)$ satisfying the inequality $|E|> \frac{(|V|-1)(|V|-2)}{2}$ is connected.
I have the basic knowledge of what a finite graph is and what a connected graph is but I do not know how to start the proof.
Assume it's not connected. Then it's separated in at least two components, namely, $a_1$ and $a_2$. It satisifies $|a_1|+|a_2|=|V|$. Then $E\le\frac{ a_1\cdot(a_1+1)+a_2\cdot(a_2+1)}2\le\frac{(|V|-1)(|V|-2)}{2}$ with equality occur iff $|a_1|=1$ or $|a_2|=1$. This is because if $n=a+b=c+d$, then $(n-k)(n-k+1)+k(k+1)$ is maximised when $k$ is minimized or maximised. Try prove the last one by yourself.