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Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs). Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges).

For more and formal definitions, see Glossary of graph theory and Graph (mathematics).

Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, that is, numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) then it is a directed graph, or digraph.

The graph with only one vertex and no edges is the trivial graph or "the dot". A graph with only vertices and no edges is known as an empty graph; the graph with no vertices and no edges is the Null graph. See Glossary of graph theory.

Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be formulated as questions about certain graphs. Various networks are conveniently described by means of graphs. For example, the link structure of Wikipedia could be represented by a directed graph: the vertices are the articles in Wikipedia and there's a directed edge from article A to article B if and only if A contains a link to B. Directed graphs are also used to represent finite state machines. The development of algorithms to handle graphs is therefore of major interest in computer science.

1 History

2 Graph representations

In general, there are four ways to represent a graph in a computer system: The incidence list representation, the incidence matrix representation, adjacency list representation, and the adjacency matrixIn mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry a is the number of edges from v to v in. There exists a unique adjacency matrix for each graph, and it is not the adjacency matrix representation.

• Incidence List - This representation uses an arrayIn computer programming, an array also known as a vector or list, is one of the simplest data structures. Arrays hold a fixed number of equally-sized data elements, generally of the same data type. Individual elements are accessed by index using a consecu. Each element in the array corresponds with a single edge. Each edge contains a list of size two which corresponds to the endpoints of the edge. This can also apply to directed graphs by ensuring the first vertex be defined as the source or destination while the second should be defined as the opposite.
• Incidence matrix - This representation uses a matrixAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number of M edges by N vertices. If the vertex is an endpoint to the edge, a value of 1 is assigned to their crossing, otherwise, a value of 0 is assigned. This is a terrible waste of space as every column or row represented by the edge can only have two values of 1 while the rest are labelled 0.

These two representations are most useful when information about edges is more desirable than information about vertices.

• Adjacency list - Much like the incidence list, each node has a list of which nodes it is adjacent to. This can sometimes result in "overkill" in an undirected graph as vertex 3 may be in the list for node 2, then node 2 must be in the list for node 3. Either the programmer may choose to use the unneeded space anyway, or he/she may choose to list the adjacency once. This representation is easier to find all the nodes which are connected to a single node, since these are explicitly listed.
• Adjacency matrix - there is an N by N matrix, where N is the total number of vertices in the graph. If there is an edge from some vertex x to some vertex y, then the element would be 1, otherwise it would be 0. This makes it easier to find subgraphs, and to reverse graphs if needed.

These two representations are most useful when information about the vertices is more desirable than information about the edges. These two representations are also most popular since information about the vertices is often more desirable than edges in most applications.