Simple Graph of Maximum Size is Complete Graph

Theorem
Let $G$ be a simple graph of order $n$ such that $n \ge 1$.

Let $G$ have the largest size of all simple graphs of order $n$.

Then:
 * $G$ is the complete graph $K_n$
 * its size is $\dfrac {n \paren {n - 1} } 2$.

Proof
By definition, $K_n$ is the simple graph of order $n$ such that every vertex of $K_n$ is adjacent to all other vertices.

So, let $G$ have the largest size of all simple graphs of order $n$.

Then by definition of largest size, it is not possible for another edge to be added to $G$.

The only way that could be is if all the vertices of $G$ are already adjacent to all other vertices.

Hence $G = K_n$ by definition.

The size of $G$ then follows from Size of Complete Graph.

Also see

 * Size of Complete Graph