Induction proof that the number of edges of a complete graph with $x$ vertices is always $\frac{x^2-x}{2}$

Question

I am trying to prove that the number of edges of a complete graph with $x$ vertices is always $\frac{x^2-x}{2}$.

I'm having trouble figuring out this proof by induction. I've only just started doing induction proofs and I'm not entirely sure where to begin. Thank you very much!

Answer 1

Guide:

• Consider base case: a singleton.

• Assume the result hold true for $x$ vertices.

• Think of how to construct a complete graph with $x+1$ vectices by adding a single node. Think of how many edges are needed to be added to the complte graph of $x$ vertices to form a complete graph of $x+1$ nodes. Use the induction hypothesis (number of edges of $x$ vertices plus the number of additional edges that are needed to include the additional node).