Directed edge

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them.

In formal terms, a directed graph is an ordered pair G = (V, A) where

It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, arcs, or lines.

The aforementioned definition does not allow a directed graph to have multiple arrows with same source and target nodes, but some authors consider a broader definition that allow directed graphs to have such multiple arrows (namely, they allow the arrows set to be a multiset). More specifically, these entities are addressed as directed multigraphs (or multidigraphs).
On the other hand, the aforementioned definition allows a directed graph to have loops (that is, arrows that connect nodes with themselves), but some authors consider a narrower definition that doesn't allow directed graphs to have loops. More specifically, directed graphs without loops are addressed as simple directed graphs, while directed graphs with loops are addressed as loop-digraphs (see section Types of directed graphs).

An arrow (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the arrow; y is said to be a direct successor of x and x is said to be a direct predecessor of y. 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. The arrow (y, x) is called the inverted arrow of (x, y).

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. The adjacency matrix of a directed graph is unique up to identical permutation of rows and columns.

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