# Encyclopedia of Machine Learning

2010 Edition
| Editors: Claude Sammut, Geoffrey I. Webb

# Graphs

• Tommy R. Jensen
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30164-8_352

## Definition

Graph Theory is (dyadic) relations on collections specified objects. In its most common, a graph is a pair G = (V, E) of a (finite) set of verticesV and a set of edgesE (or links). Each edge e is a 2-element subset {u, v} of V , usually abbreviated as e = uv; u and v are called the endvertices of e, they are mutually adjacent and each is incident to e in G. This explains the typical model of a simple graph.

A directed graph or is a more general structure, in which the edges are replaced by ordered pairs of distinct elements of the vertex set V , each such pair being referred to as an arc. Another generalization of a graph is a hypergraph or “set-system” on V , in which the hyperedges may have any size. Various concepts in graph theory extend naturally to multigraphs, in which each pair of (possibly identical) vertices may be adjacent via several edges (respectively loops). Also studied are infinite graphs, for which the vertex and edge sets are not restricted to be...

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## References

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