# Graph

A graph is a type of visual display that arises in several places in mathematics and statistics.

In statistics, graphs are a way to display data. Pie charts and bar charts are two common types of graphs used to help readers visually understand the content of a data set.

In mathematics, and especially in topology, a graph is an abstract object consisting of a number of points ("vertices") and lines ("edges") connecting those points, and is used to study networks and similar objects. For example, one could create a graph to represent human friendships: this graph has one point for each person, and an edge between every two people who are friends. According to the theory of six degrees of separation, any two points on this graph are connected by a path that goes along at most six edges!

In mathematical terms, a graph is a pair of two sets V (vertices) and E (edges), where E is a subset of Cartesian product $V \times V$. A graph might be directed or undirected. A labeled graph has some data assigned to edges.

Example 1: Vertices are American cities, edges are roads between them, and labels are lengths of the roads. A computer given such graph will find the shortest path between any two cities.

Example 2: A graph might show correlations and relations between concepts.

   atheism → homosexuality
↓    ↙
suicide


Example 3: A bipartite graph is a graph whose vertices may be divided into two types, presented on left and right. It might show membership.

      Pope Benedict XVI ↘
Rick Santorum → Catholic
Mother Teresa ↗

Mitt Romney → Mormon
Joseph Smith ↗

Barack Obama → Muslim

Christopher Hitchens → Atheist
Joseph Stalin ↗


On the left side there are people, and on the right hand side there are religions.

This bipartite graph models a function.

A planar graph is a graph which can be drawn without crossings. The graph in example 1 is not planar, because roads can intersect outside cities. The graphs drawn in examples 2 and 3 are currently planar, but after adding more edges they might become nonplanar.