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**Graph theory** is the branch of mathematics that examines the properties of **mathematical graphs**. See glossary of graph theory for common terms and their definition.

Informally, this type of graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs), which can also have associated directions. Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges), with an arrowhead on a line representing a directed arc.

Such graphs can be used to represent and analyze a variety of systems and problems, including colorability problems, shortest path algorithms and spanning trees.

For information on other types of graphs see graph (disambiguation).