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complete_graph [2014/02/09 02:40] nikolaj |
complete_graph [2014/03/21 11:11] |
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| - | ===== Complete graph ===== | ||
| - | ==== Set ==== | ||
| - | | @#88DDEE: $V,E$ ... set | | ||
| - | | @#FFBB00: $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ | | ||
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| - | | @#55EE55: $ \langle V,E,\psi\rangle $ ... simple graph | | ||
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| - | | @#FFFDDD: $ u,v \in V$ | | ||
| - | |||
| - | | @#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 === | ||
| - | 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]] | ||
