# Complete Graph **A simple undirected [[graph]] whose vertices are all connected to each other.** ![[CompleteGraph.svg|300]] *The complete graph of 5 vertices, $K_5$* > [!Note] Complete Graph > A *complete graph* $K_{n}$ is a graph in which every pair of its $n$ vertices are *adjacent*. A complete graph $K_{n}$ has $\frac{n(n-1)}{2}$ total edges, the $n$th [[triangular number]]. It is also a regular graph of degree $n - 1$. Complete graphs $K_{1}$ through $K_{4}$ are also all [[Planar Graph|planar graphs]]. A *complete [[Graph#Directed graph|directed graph]]* is one such that every pair of distinct vertices is connected by a pair of *unique edges*.