# Edge (graph theory)

In graph theory , an edge denotes a part of a graph . An edge indicates whether two nodes are related to one another or whether they are connected in the graphical representation of the graph. In a directed graph an edge is an ordered pair of nodes, in an undirected graph an edge is a set of two nodes. Two nodes that are connected by an edge are called adjacent or adjacent.

## Edge types and their notation

### Unoriented edges

Edges in an undirected graph are called "undirected edges". An undirected edge is therefore a set of two nodes. Sometimes the term is also extended to directed graphs, to express that two nodes “a” and “b” are connected by both the edge and the edge . ${\ displaystyle \ left (a, b \ right)}$ ${\ displaystyle \ left (b, a \ right)}$ ### Directed edges

Edges in a directed graph are called "directed edges". In contrast to an undirected edge, it has an orientation. For an edge , the node is called the start node and the node is called the end node of the edge. A directed edge is also called an "arc" or an "arrow". Two edges , with and called "counter" or "antiparallel". ${\ displaystyle e = \ left (a, b \ right)}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle e_ {1}}$ ${\ displaystyle e_ {2}}$ ${\ displaystyle e_ {1} = \ left (a, b \ right)}$ ${\ displaystyle e_ {2} = \ left (b, a \ right)}$ ## Special edges

• Loop : Connects a knot to itself.
• Multiple edge / multi edge: Several edges of the same type run between two nodes in a multigraph . The individual edges are referred to as “parallel edges”.
• Multiple loop: A directed multiple edge in a multigraph that is also a loop.

## Generalization: hyperedge

In hypergraphs , an edge can also connect more than two nodes as a so-called hyperedge .

## literature

• Dénes Kőnig : Theory of finite and infinite graphs . Academic Publishing Company, Leipzig 1936.