Question

How many edges are there in a complete graph with \(n\) vertices?

• A. \(n(n-1)\)
• B. \( \frac{n(n-1)}{2} \)
• C. \(n^2\)
• D. \( \frac{n(n+1)}{2} \)

Hint:

In a complete graph \(K_n\), every vertex connects to all other vertices. Hence, the total number of edges is always \( \frac{n(n-1)}{2} \).

Answer 1

The Correct Option is B

Concept: A complete graph, denoted by \(K_n\), is a simple undirected graph in which every pair of distinct vertices is connected by exactly one edge. Thus, each vertex is connected to every other vertex in the graph.
Step 1: Count connections from each vertex.
In a complete graph with \(n\) vertices, each vertex is connected to: \[ n - 1 \] other vertices.
Step 2: Compute total connections.
If we count edges from all vertices, we initially get: \[ n(n-1) \] However, this counts each edge twice (once from each endpoint).
Step 3: Correct the double counting.
To avoid counting edges twice, we divide by \(2\): \[ \text{Number of edges} = \frac{n(n-1)}{2} \]
Step 4: Conclusion.
Therefore, the number of edges in a complete graph with \(n\) vertices is: \[ \frac{n(n-1)}{2} \]