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{{Short description|Directed graph in mathematics and computer science}}
{{About|directed graphs in mathematics and computer science|other uses|Digraph (disambiguation)}}
A '''digraph''' or '''directed graph''' is a type of [[graph (discrete mathematics)|graph]] in which the [[edges]] have a direction. This means that each edge is an ordered pair of vertices, representing a one-way relationship between the vertices. Digraphs are used extensively in [[computer science]], [[mathematics]], and related fields to model relationships and processes.
== Definition ==
A digraph is defined as a pair \( G = (V, E) \), where:
* \( V \) is a set of [[vertices]] (also called nodes or points).
* \( E \) is a set of ordered pairs of vertices, called [[directed edges]] (or arcs).
In a digraph, an edge \( (u, v) \) is directed from vertex \( u \) to vertex \( v \). This directionality distinguishes digraphs from [[undirected graphs]], where edges have no orientation.
== Types of Digraphs ==
There are several types of digraphs, including:
* '''Simple digraphs''': Digraphs with no multiple edges or loops.
* '''Multidigraphs''': Digraphs that allow multiple edges between the same pair of vertices.
* '''Weighted digraphs''': Digraphs where edges have associated weights or costs.
== Properties ==
Digraphs have several important properties:
* '''In-degree and out-degree''': The in-degree of a vertex is the number of edges directed towards it, while the out-degree is the number of edges directed away from it.
* '''Paths and cycles''': A path in a digraph is a sequence of vertices connected by directed edges. A cycle is a path that starts and ends at the same vertex.
* '''Strong connectivity''': A digraph is strongly connected if there is a directed path from any vertex to every other vertex.
== Applications ==
Digraphs are used in various applications, including:
* [[Computer networks]]: Modeling the flow of data between devices.
* [[Social networks]]: Representing relationships and interactions between individuals.
* [[Project management]]: Using [[PERT]] and [[CPM]] charts to model task dependencies.
* [[Linguistics]]: Representing syntactic structures in [[natural language processing]].
== Related Concepts ==
* [[Graph theory]]
* [[Adjacency matrix]]
* [[Adjacency list]]
* [[Topological sorting]]
* [[Directed acyclic graph]]
== See also ==
* [[Graph (discrete mathematics)]]
* [[Undirected graph]]
* [[Weighted graph]]
* [[Multigraph]]
* [[Network theory]]
== References ==
{{Reflist}}
[[Category:Graph theory]]
[[Category:Discrete mathematics]]
[[Category:Computer science]]
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Digraph - Revision history

Prab at 18:29, 18 March 2025

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