In graph theory a multigraph a particular type of graph. A multigraph is a set of vertices and for each unordered pair of distinct vertices a set of edges between these.
The terminology is used in distinction to
simple graphs, which are multigraphs with at most one edge between any unordered pair of distinct vertices;
pseudographs, which are like multigraphs but admitted to have “loops” in the sense of edges between a vertex and itself.
Beware that the terminology is not completely consistent across different authors. Some authors may allows loops when they speak of multigraphs.
More formally this means that a multigraph is
a function $E \to \left\{ {\,\atop \,} \{v_1, v_2\} = \{v_2, v_1\} \;\vert\; v_1, v_2 \in V \,,\; v_1 \neq v_2 {\, \atop \,} \right\}$ (sending any edge to the unordered pair of distinct vertices that it goes between).
Specifically a finite multigraph is, after choosing a linear order on its set of vertices, the same as
a natural number $v \in \mathbb{N}$ (the number of vertices);
for each $i \lt j \in \{1, \cdots, v\}$ a natural number $e_{i,j} \in \mathbb{N}$ (the number of edges between the $i$th and the $j$th vertex)
Discussion with an eye towards Feynman diagrams includes
See also
Last revised on August 2, 2018 at 07:11:43. See the history of this page for a list of all contributions to it.