The line graph () of a simple graph is defined as follows:
There is exactly one vertex () in () for each edge in .
That being said, how can I prove that the maximum number of edges in L(G) is: $$\frac{n^3 - 3n^2 + 2n}{2}$$
where n is the number of vertices of G.
It would be amazing to have a detailed explanation, since I am not the brightest when it comes to proofs, but tips are also appreciated.
You missed a part of the definition of L(G). An edge in L(G) is generated with endpoints v(e) and v(f) when e and f share a vertex in G.
Thus you can associate each edge in L(G) with a unique vertex in G, and if the vertex has degree $n-1$ there are ${n-1\choose 2}$ such edges (picking any pair of edges).
But as we have $n$ vertices and a simple graph, each vertex has degree $n-1$ at most and thus the number of edges in L(G) is bounded by $n{n-1 \choose 2} = \frac{n^3 - 3n^2 + 2n}{2}$