Glossary of graph theory#Basics

As an infinite graph is called in graph theory a graph whose nodes or number of edges is infinite. One speaks of a graph, as is often assumed that nodes and number of edges are finite. A graph is called a wegendlich, if he, in spite of possibly infinite number of nodes does not have infinitely long way.

Statements about infinite graphs can be derived by means of a compactness argument often made ​​corresponding statements about finite graphs. For example, each infinite planar graph is vierfärbbar, because this applies to every finite planar graph. This is based on the lemma of King.

Other statements are not necessarily transferable to infinite graphs.

Examples

In many internal and external mathematical applications are expander graphs of importance.

Locally finite graphs

A graph is called locally finite if each node has only finitely many neighbors.

Fine graphs

An important in geometric group theory class of graphs are graphs Fine, they include locally finite graph, and, for example, the Farey graph.

Application

In the Functional Analysis infinite graphs occur as so-called Bratteli diagrams on in the study of AF C * - algebras.