 A labeled graph of 6 vertices and 7 edges. In computer science, a graph is a kind of data structure, specifically an abstract data type (ADT), that consists of a set of nodes (also called vertices) and a set of edges that establish relationships (connections) between the nodes. The graph ADT follows directly from the graph concept from mathematics. Image File history File links 6n-graf. ...
Image File history File links 6n-graf. ...
Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
A binary tree, a simple type of branching linked data structure. ...
In computing, an abstract data type (ADT) is a specification of a set of data and the set of operations that can be performed on the data. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
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. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
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. In graph theory, an adjacency list is the representation of all edges or arcs in a graph as a list. ...
In mathematics and computer science, the adjacency matrix of a finite directed or undirected graph G on n vertices is the n × n matrix where the nondiagonal entry is the number of edges from vertex i to vertex j, and the diagonal entry is either twice the number of loops...
In the mathematical subfield of numerical analysis a sparse matrix is a matrix populated primarily with zeros. ...
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. A hierarchy (in Greek: , derived from — hieros, sacred, and — arkho, rule) is a system of ranking and organizing things or people, where each element of the system (except for the top element) is a subordinate to a single other element. ...
A computer network is an interconnection of a group of computers. ...
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. In computer science, a binary tree is a tree data structure in which each node has at most two children. ...
A simple example unordered tree In computer science, a tree is a widely-used data structure that emulates a tree structure with a set of linked nodes. ...
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. Depth-first search (DFS) is an algorithm for traversing or searching a tree, tree structure, or graph. ...
In graph theory, breadth-first search (BFS) is a graph search algorithm that begins at the root node and explores all the neighboring nodes. ...
Dijkstras algorithm, named after its discoverer, Dutch computer scientist Edsger Dijkstra, is a greedy algorithm that solves the single-source shortest path problem for a directed graph with non negative edge weights. ...
In computer science, the Floyd–Warshall algorithm (sometimes known as the Roy–Floyd algorithm, since Bernard Roy described this algorithm in 1959) is a graph analysis algorithm for finding shortest paths in a weighted, directed graph. ...
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. In graph theory, a flow network is a directed graph where each edge has a capacity and each edge receives a flow. ...
The Ford-Fulkerson algorithm (named for L. R. Ford, Jr. ...
The max flow min cut theorem is a statement in optimization theory about optimal flows in networks. ...
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