A multigraph is a graph with multiple edges, i.e. edges that have the same end nodes. Formally, a multigraph G is an ordered pair G:=(V, E) with An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

• V a set of vertices or nodes,
• E a multiset of unordered pairs of distinct vertices, called edges or lines.

A multidigraph is a directed graph with multiple arcs, i.e., arcs with the same source and target nodes. A multidigraph G is an ordered pair G:=(V,A) with In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ... A labeled graph with 6 vertices (nodes) and 7 edges. ...

• V a set of vertices or nodes,
• A a multiset of ordered pairs of vertices called directed edges, arcs or arrows.

A mixed multigraph G:=(V,E, A) may be defined in the same way as a mixed graph. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A labeled graph with 6 vertices (nodes) and 7 edges. ...

Labeling

Multigraphs and multidigraphs also support the notion of graph labeling, in a similar way. However there is no unity in terminology in this case. In the mathematical discipline of graph theory, a graph labeling is the assignment of unique identifiers to the edges and vertices of a graph. ...

The definitions of labeled multigraphs and multidigraphs are similar, and we define only the latter ones here.

Definition 1: A labeled multidigraph is a labeled graph with labeled arcs. In graph theory (which is an area in mathematics and computer science) a labeled graph is a graph with labels assigned to its nodes and edges. ...

Formally: A labeled multidigraph G is a multigraph with labeled nodes and arcs. Formally it is an 8-tuple where

• V is a set of nodes and A is a multiset of arcs.
• Σ_V and Σ_A are finite alphabets of the available node and arc labels,
• and are two maps indicating the source and target node of an arc,
• and are two maps describing the labeling of the nodes and edges.

Definition 2: A labeled multidigraph is a labeled graph with multiple labeled edges, i.e. edges with the same end nodes and the same edge label (note that this notion of a labeled graph is different to the notion given by the article graph labeling). In graph theory (which is an area in mathematics and computer science) a labeled graph is a graph with labels assigned to its nodes and edges. ... In the mathematical discipline of graph theory, a graph labeling is the assignment of unique identifiers to the edges and vertices of a graph. ...