A graph is a set V of vertices (or nodes) together with a set E of edges (pairs of vertices from V). If the pairs are unordered, the graph is undirected; such a graph is usually called a graph. If the pairs are ordered, the graph is called a directed graph or digraph.

The notion of edge may be generalized from pairs (connecting two vertices at once) to hyperedges (connecting n vertices at once). The generalization of the graph with hyperedges is a hypergraph.

If f is a function from X to Y, then the set G := { (x,y) | y=f(x) for all f(x) } is said to be the graph of f.
If f is a function from some subset D of R to R, the set G, consisting of ordered pairs of real numbers, can be depicted visually on the coordinate plane.

Take note that the mathematical definition of the graph of a function does not rely on a "graphical" representation of the function, whereas one common definition does.

Graph (?), n. [See -graph.] (Math.)

1.

A curve or surface, the locus of a point whose coördinates are the variables in the equation of the locus.

2.

A diagram symbolizing a system of interrelations by spots, all distinguishable from one another and some connected by lines of the same kind.

© Webster 1913