To find the maximum number of edges in a simple graph with $n$ vertices, we can analyze the conditions for an edge's existence in a graph. A simple graph is one in which there are no multiple edges between vertices and no loops (edges connecting a vertex to itself). Let's proceed with a detailed proof.

Proof

1. Total Possible Connections:

   • In a simple graph with $n$ vertices, each vertex can potentially connect to every other vertex, but without forming a loop (no self-connections) and without multiple connections between any pair.
   • Each connection between two distinct vertices is a potential edge.

2. Counting Unique Pairs of Vertices:

   • To determine the maximum number of edges, we need to count the number of unique pairs of vertices that can be connected.
   • The number of ways to choose 2 vertices from $n$ vertices is given by the binomial coefficient $\binom{n}{2}$, which is calculated as: $\binom{n}{2} = \frac{n(n - 1)}{2}$

3. Interpreting the Result:

   • This count, $\frac{n(n - 1)}{2}$, represents the maximum number of edges possible because:
     • It ensures each pair of vertices is counted exactly once.
     • It abides by the rules of a simple graph, avoiding loops and multiple edges.

4. Conclusion:

   • Therefore, the maximum number of edges in a simple graph with $n$ vertices is: $\frac{n(n - 1)}{2}$

This result is intuitive because the formula $\frac{n(n - 1)}{2}$ captures all possible unique pairs of vertices without violating the conditions of a simple graph.

Would you like further details on any part of this proof or have additional questions?

Related Questions

1. How many edges are there in a complete graph with 5 vertices?
2. Why does removing loops or multiple edges matter in calculating the maximum edge count?
3. How would the formula change if self-loops were allowed in the graph?
4. Can you derive the same result using induction on the number of vertices?
5. How does this maximum edge count formula relate to the concept of a "dense" graph?

Tip

To generalize, the formula $\frac{n(n - 1)}{2}$ applies to any simple graph regardless of specific vertex labels, as it’s purely based on the number of vertices.