1. A nonempty but finite set of vertices (or nodes) together with a set of edges that join pairs of distinct vertices. If an edge e joins vertices v1 and v2, then v1 and v2 are said to be incident with e and the vertices are said to be adjacent; e is the unordered pair (v1,v2).
A graph is usually depicted in a pictorial form in which the vertices appear as dots or other shapes, perhaps labeled for identification purposes, and the edges are shown as lines joining the appropriate points. If direction is added to each edge of a graph, a directed graph or digraph is obtained. The edges then form a finite set of ordered pairs of distinct vertices, and are often called arcs. In the pictorial representation, arrows can be placed on each edge. With no direction specified, the graph is said to be undirected.
Although helpful visually these representations are not suitable for manipulation by computer. More useful representations use an incidence matrix or an adjacency matrix.
Graphs are used in a wide variety of ways in computing: the vertices will usually represent objects of some kind and the edges will represent connections of a physical or logical nature between the vertices. So graphs can be used to model in a mathematical fashion such diverse items as a computer and all its attached peripherals, a network of computers, parse trees, logical dependencies between subroutines or nonterminals in a grammar, VLSI diagrams, related items in databases for molecules and reaction networks (for chemoinformatics and bioinformatics). Trees and lists are special kinds of graphs.
Variations exist in the definition of a graph. There is some dispute about whether one edge can join a vertex to itself, whether empty sets are involved, whether an infinite number of vertices and edges are permitted, and so on.
See also connected graph, network, weighted graph.
2. of a function f. The set of all ordered pairs (x,y) with the property that y = f(x). Often such a graph is represented by a curve.
graph1 / graf/ • n. a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles. ∎ Math. a collection of points whose coordinates satisfy a given relation. • v. [tr.] plot or trace on a graph. graph2 • n. Linguistics a visual symbol representing a unit of sound or other feature of speech. Graphs include not only letters of the alphabet but also punctuation marks.
graph, figure that shows relationships between quantities. The graph of a function y=f (x) is the set of points with coordinates [x, f (x)] in the xy-plane, when x and y are numbers. A similar definition can be given for functions involving more general kinds of variables. In mathematics interest is almost exclusively in line graphs and what these reveal about the functions they represent. Statistics makes extensive use of both line graphs and bar graphs, in which the lengths of the various bars show the quantities to be compared. Graph is also a mathematical term used in combinatorics to designate a geometric object consisting of vertices and edges (joining pairs of vertices). Such objects have been studied considerably in recent years because of the applicability to such diverse fields as computer networks, game theory, and social psychology.