The number of edges in a complete graph 𝐾𝑛K n​ is:A. 𝑛nB. 𝑛−1n−1C. 𝑛(𝑛−1)22n(n−1)​ D. 2𝑛2n

Break Down the Problem

1. Identify the definition of a complete graph $K_n$.
2. Determine how edges in a complete graph are calculated depending on the number of vertices $n$.

Relevant Concepts

1. A complete graph $K_n$ has every possible edge between its vertices.
2. The formula for the number of edges in $K_n$ is derived from the combination of $n$ vertices taken 2 at a time.

Analysis and Detail

1. The number of ways to choose 2 vertices from $n$ vertices is given by the combination formula: $\binom{n}{2} = \frac{n(n-1)}{2}$ This represents the number of edges since each pair of vertices is connected by exactly one edge.

Verify and Summarize

• The options provided are:

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

• From our analysis, the correct number of edges in a complete graph $K_n$ is represented by option C, which is $\frac{n(n-1)}{2}$.

Final Answer

The number of edges in a complete graph $K_n$ is $\frac{n(n-1)}{2}$, so the answer is C. $\frac{n(n-1)}{2}$.