# Directed graph

directed edgedirecteddigraphoutdegreeindegreearcin-degreedigraphsout-degreearcs
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.wikipedia
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### Vertex (graph theory)

verticesvertexnode
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.
In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).

### Tournament (graph theory)

tournamentstournamenttransitive tournament
A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph.

### Complete graph

completecomplete connectivitycomplete digraph
A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).

### Directed acyclic graph

acyclicDAGacyclic directed graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG ) is a finite directed graph with no directed cycles.

### Cycle (graph theory)

cyclecyclessimple cycle
A directed cycle in a directed graph is a non-empty directed trail in which the only repeated are the first and last vertices.

### Flow network

network flownetworkaugmenting path
In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow.

### Polytree

oriented tree
In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic.

### Signal-flow graph

signal flow graphsignal-flow analysisElements of signal flow graphs
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes.

### Rooted graph

Accessible pointed graphrootedaccessible pointed directed graph
Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots.

### Category theory

categorycategoricalcategories
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

### Quiver (mathematics)

quiverPath algebraquivers
In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph.

### Multigraph

directed multigraphpseudographlabeled multigraph
More specifically, these entities are addressed as directed multigraphs (or multidigraphs).
In this case the multigraph would be a directed graph with pairs of directed parallel edges connecting cities to show that it is possible to fly both to and from these locations.

### Flow graph (mathematics)

flow graphFlow graphs
A flow graph is a form of digraph associated with a set of linear algebraic or differential equations:

### Finite-state machine

finite state machinestate machinefinite automata

### Control-flow graph

control flow graphcontrol flowCFG
: outdegree(A) > 1 or indegree(B) > 1 (or both).

### Free category

freefree categoriesfreely
In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.

### State diagram

StatechartState machine diagramstate transition diagram
A classic form of state diagram for a finite state machine or finite automaton (FA) is a directed graph with the following elements (Q,Σ,Z,δ,q 0,F):

### Path (graph theory)

pathpathsdirected path
If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y.
A directed path (sometimes called dipath ) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.

### Incidence matrix

incidence relationincidence matricesincidence
Another matrix representation for a directed graph is its incidence matrix.
The incidence matrix of a directed graph is a n × m matrix B where n and m are the number of vertices and edges respectively, such that B i,j = −1 if the edge e j leaves vertex v i, 1 if it enters vertex v i and 0 otherwise (many authors use the opposite sign convention).

### Digraph realization problem

directed graph realization problem
The directed graph realization problem is the problem of finding a directed graph with the degree sequence a given sequence of positive integer pairs.
Given pairs of nonnegative integers, the problem asks whether there is a labeled simple directed graph such that each vertex v_i has indegree a_i and outdegree b_i.

### Network theory

network analysisnetworksnetwork
Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects.

The adjacency matrix of a multidigraph with loops is the integer-valued matrix with rows and columns corresponding to the vertices, where a nondiagonal entry a ij is the number of arrows from vertex i to vertex j, and the diagonal entry a ii is the number of loops at vertex i.
In directed graphs, the in-degree of a vertex can be computed by summing the entries of the corresponding column, and the out-degree can be computed by summing the entries of the corresponding row.

### Connectivity (graph theory)

connectedconnected graphconnectivity
A directed graph is weakly connected (or just connected ) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph.
A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph.

### Orientation (graph theory)

orientationorientationsoriented graph
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph.

### Branching factor

For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex and the number of tail ends adjacent to a vertex is its outdegree (called "branching factor" in trees).
In computing, tree data structures, and game theory, the branching factor is the number of children at each node, the outdegree.