The word ‘graph’ usually refers to some kind of picture or diagram meant to visually represent relationships among certain objects or quantities. Most commonly, we say graph to mean a representation in the Cartesian plane of a relation or function. In statistics graphical representations of data are generally called charts
When speaking of functions, ‘graph’ is informally used to refer to the picture drawn in the Cartesian plane, but formally we identify the graph of a function with the set of ordered pairs that actually make up the function.
In graph theory a graph is a collection of nodes (also called vertices) together with a set of edges joining some of the nodes. Formally, a graph \(G\) is a set of vertices \(V\), a set of edges \(E\), and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being lines drawn between them.