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## Definition
> [!tldr] Definition
> The **complete graph on $n$ vertices** is denoted $K_n$ and consists of:
> - The [[Graph|vertex]] set is $V = \{1,2,\dots, n\}$
> - The [[Graph|edge]] set is the set of all two-element subsets of $V$.
>
> In other words, $K_n$ is the [[graph]] with $n$ vertices in which there is an edge between every possible pair of distinct vertices.
Notes:
- The number of vertices in $K_n$ is $n$, and the number of edges is $\binom{n}{2}$ where the notation here is the [[Binomial coefficient|binomial coefficient]]. It follows from the [[Closed formula|closed formula]] for the binomial coefficient that there are $\dfrac{n(n-1)}{2}$ edges in $K_n$.
## Examples
Here are visualizations of $K_1$ through $K_6$:
![[Pasted image 20240705121105.png]]