# alternative definition of a multigraph

Many authors tried to formalize the notation of a graph. This problem is relatively simple if we allow at most $1$ edge between vertices. But for multigraphs^{}, i.e. graphs with many edges (possibly infinitely many) between vertices this tends to be problematic formally. We wish to give an alternative definition, which uses so called symmetric power (http://planetmath.org/SymmetricPower).

Definition. A multigraph or non-oriented graph is a triple

$$G=(V,E,\tau )$$ |

where $V$ is a nonempty set whose elements are called vertices, $E$ is a set whose elements are called edges and

$$\tau :E\to {V}_{sym}^{2}$$ |

is a function which takes every edge to a pair of vertices called ends of this edge. On the right side we have a symmetric power (http://planetmath.org/SymmetricPower) of $V$ to ensure that the order of ends is not important.

This definition allows loops and even infinite^{} number of edges between two vertices and is one of the most general and formal.

Title | alternative definition of a multigraph |
---|---|

Canonical name | AlternativeDefinitionOfAMultigraph |

Date of creation | 2013-03-22 19:16:54 |

Last modified on | 2013-03-22 19:16:54 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 05C75 |