Let's solve the given question step by step:

Part a: Draw $K_6$

$K_6$ is a complete graph with 6 vertices. In a complete graph, every vertex is connected to every other vertex. Here’s how it is structured:

• There are 6 vertices, $V_1, V_2, \dots, V_6$.
• Each vertex connects to all other 5 vertices.

If you need help visualizing it, imagine a hexagon where every vertex is connected to every other vertex.

Part b: Prove the formula for the number of edges

We need to prove that for all integers $n \geq 1$, the number of edges in $K_n$ is:

$\text{Number of edges} = \frac{n(n-1)}{2}.$

Proof:

1. Total possible pairs of vertices: In $K_n$, every vertex is connected to every other vertex. The total number of edges is equal to the total number of unique pairs of vertices.

   The number of ways to choose 2 vertices out of $n$ vertices is given by the combination formula:

   $\binom{n}{2} = \frac{n(n-1)}{2}.$

2. Conclusion: Since each pair of vertices corresponds to an edge in $K_n$, the number of edges is:

   $\text{Number of edges} = \frac{n(n-1)}{2}.$

Thus, the formula is proved.

Would you like me to provide a detailed diagram for $K_6$, more examples, or clarifications? Let me know!

Follow-up questions:

1. How does the structure of a complete graph change as $n$ increases?
2. Why does the formula involve combinations and not permutations?
3. Can you derive the number of edges for $K_3, K_4,$ and $K_5$ using this formula?
4. How is the complete graph $K_n$ related to real-world network problems?
5. Can you think of a practical scenario where finding the number of edges in $K_n$ is useful?

Tip: When working with graph theory problems, always start by understanding the relationship between vertices and edges!