--- ## 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]]