# Differences

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 complete_graph [2014/02/09 02:40]nikolaj complete_graph [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2014/02/09 02:42 nikolaj 2014/02/09 02:40 nikolaj 2014/02/08 14:14 nikolaj 2014/02/08 13:13 nikolaj 2014/02/08 12:55 nikolaj 2014/02/08 12:55 nikolaj 2014/02/08 02:15 nikolaj 2014/02/08 02:13 nikolaj 2014/02/08 00:33 nikolaj 2014/02/07 23:48 nikolaj created Next revision Previous revision 2014/02/09 02:42 nikolaj 2014/02/09 02:40 nikolaj 2014/02/08 14:14 nikolaj 2014/02/08 13:13 nikolaj 2014/02/08 12:55 nikolaj 2014/02/08 12:55 nikolaj 2014/02/08 02:15 nikolaj 2014/02/08 02:13 nikolaj 2014/02/08 00:33 nikolaj 2014/02/07 23:48 nikolaj created Line 1: Line 1: ===== Complete graph ===== ===== Complete graph ===== ==== Set ==== ==== Set ==== - | @#88DDEE: $V,E$ ... set | + | @#55CCEE: context ​    | @#55CCEE: $V,E$ ... set | - | @#FFBB00: $\langle V,​E,​\psi\rangle \in \mathrm{it}(E,​V)$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\langle V,​E,​\psi\rangle \in \mathrm{it}(E,​V)$ | - | @#55EE55: $\langle V,​E,​\psi\rangle$ ... simple graph | + | @#55EE55: postulate ​  | @#55EE55: $\langle V,​E,​\psi\rangle$ ... simple graph | - | @#FFFDDD: $u,v \in V$ | + | @#FFFDDD: for all     | @#FFFDDD: $u,v \in V$ | - | @#55EE55: $u\neq v\implies \exists !(e\in E).\ \psi(e)=\{u,​v\}$ | + | @#55EE55: postulate ​  | @#55EE55: $u\neq v\implies \exists !(e\in E).\ \psi(e)=\{u,​v\}$ | ==== Discussion ==== ==== Discussion ==== Line 16: Line 16: The axiom $\{u\}\notin\mathrm{im}(\psi)$ says that there are no loops on a single vertex. The axiom $\{u\}\notin\mathrm{im}(\psi)$ says that there are no loops on a single vertex. === Theorems === === Theorems === - A complete graph with $n$ vertices has maximally $\sum_{k=1}^{n-1}k=\frac{1}{2}n(n-1)$ edges. + 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 === === Reference === Wikipedia: [[http://​en.wikipedia.org/​wiki/​Complete_graph|Complete graph]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Complete_graph|Complete graph]] 