A labeled graph of 6 vertices and 7 edges.

Informally, G=(V,E) consists of vertices, the elements of V, which are connected by edges, the elements of E. Formally, a graph, G, is defined as an ordered pair, G=(V,E), where V is a set (usually finite) and E is a set consisting of two element subsets of V.

Choices of representation

Two main data structures for the representation of graphs are used in practice. The first is called an adjacency list, and is implemented by representing each node as a data structure that contains a list of all adjacent nodes. The second is an adjacency matrix, in which the rows and columns of a two-dimensional array represent source and destination vertices and entries in the array indicate whether an edge exists between the vertices. Adjacency lists are preferred for sparse graphs; otherwise, an adjacency matrix is a good choice. Finally, for very large graphs with some regularity in the placement of edges, a symbolic graph is a possible choice of representation.

Comparison with other data structures

Graph data structures are non-hierarchical and therefore suitable for data sets where the individual elements are interconnected in complex ways. For example, a computer network can be modeled with a graph.

Hierarchical data sets can be represented by a binary or nonbinary tree. It is worth mentioning, however, that trees can be seen as a special form of graph.

Operations

Graph algorithms are a significant field of interest within computer science. Typical operations associated with graphs are: finding a path between two nodes, like depth-first search and breadth-first search and finding the shortest path from one node to another, like Dijkstra's algorithm. A solution to finding the shortest path from each node to every other node also exists in the form of the Floyd-Warshall algorithm.

A directed graph can be seen as a flow network, where each edge has a capacity and each edge receives a flow. The Ford-Fulkerson algorithm is used to find out the maximum flow from a source to a sink in a graph.

The graphs can be represented in two ways. One is adjacency matrix and adjacency list.

For example, let us consider the following graph

             A----------->B
             |            ^
             |            |
             |            |
             V            |
             C ------------

Adjacency Matrix

          A B C
       A  0 1 1
       B  0 0 0 
       C  0 1 0

Adjacency List

       A  ----> | B | ----> | C | ---- NULL
       B  ----> ---- NULL 
       C  ----> | B | ---- NULL

External links

• Interactive visualisations (not working with Mozilla Firefox) of graphs and other data structures.
• Notes (PDF, 280 KiB)
• Boost Graph Library: a powerful C++ graph library
• Perl graph routines
• QuickGraph: Graph Data Structures And Algorithms for .NET
• Algraf Project: Graphical tool to draw graphs, apply several algorithms to them and export to XML
• Graphviz - Graph Visualization Software (Open Source)

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