===== Complete graph =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $V,E$ ... set |

| @#FFBB00: definiendum | @#FFBB00: $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ |

| @#55EE55: postulate   | @#55EE55: $ \langle V,E,\psi\rangle $ ... simple graph |

| @#FFFDDD: for all     | @#FFFDDD: $ u,v \in V$ |

| @#55EE55: postulate   | @#55EE55: $ u\neq v\implies \exists !(e\in E).\ \psi(e)=\{u,v\} $ |

==== Discussion ====
In a complete (undirected) graph, every two distinct vertices are connected.

The axiom $\{u\}\notin\mathrm{im}(\psi)$ says that there are no loops on a single vertex.
=== Theorems ===
When dealing with simple graphs, if you add another vertex to a graph with $n$ vertices, you can add $n$ new edges at best. So a complete graph with $n$ vertices has maximally $\sum_{k=1}^{n-1}k=\frac{1}{2}n(n-1)$ edges.
=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Complete_graph|Complete graph]]
==== Parents ====
=== Subset of ===
[[Simple graph]]