In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense “related”. The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line).[1] Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person A can shake hands with a person B only if B also shakes hands with A. In contrast, if an edge from a person A to a person B means that A owes money to B, then this graph is directed, because owing money is not necessarily reciprocated.

Graphs are the basic subject studied by graph theory. The word “graph” was first used in this sense by J. J. Sylvester in 1878 due to a direct relation between mathematics and chemical structure (what he called a chemico-graphical image).[2][3]

A graph with six vertices and seven edges

1 Definitions

1.1 Graph

1.2 Directed graph

1.3 Mixed graph

1.4 Weighted graph

2 Types of graphs

2.1 Oriented graph

2.2 Regular graph

2.3 Symmetric graph

2.4 Complete graph

2.5 Finite graph

2.6 Connected graph

2.7 Bipartite graph

2.8 Path graph

2.9 Planar graph

2.10 Cycle graph

2.11 Tree

2.12 Polytree

2.13 Advanced classes

3 Properties of graphs

4 Examples

5 Graph operations

6 Generalizations

7 See also