Graph theory

Graph theory isbranchmathematics that examinespropertiesgraphs.

Image:6n-graf.png

A graph6 vertices7 edges.

Informally,graph issetobjects called vertices (or nodes) connected by links called edges (or arcs). Typically,graphdepicted assetdots (i.e., vertices) connected by lines (i.e., edges).

For moreformal definitions, see Glossarygraph theoryGraph (mathematics).

Depending onapplications, edges may or may not havedirection; edges joiningvertexitself may or may not be allowed,vertices and/or edges may be assigned weights, i.e. numbers. Ifedges havedirection associatedthem (indicated by an arrow ingraphical representation) we havedirected graph, or digraph.

A graphonly one vertexno edges istrivial graph or "the dot".

Structures that can be represented as graphsubiquitous,many problemspractical interest can be formulated as questions about certain graphs. For example,link structureWikipedia could be represented bydirected graph:vertices arearticlesWikipedia,there'sdirected edge from article Aarticle B ifonly if A containslinkB. Directed graphsalso usedrepresent finite state machines. The developmentalgorithmshandle graphsthereforemajor interestcomputer science.

Tablecontents

1 History 2 Graph problems 3 Important algorithms 4 Generalizations 5 Related areasmathematics 6 See also

History

Leonhard Euler's paper on Seven BridgesKönigsbergconsideredbefirst resultgraph theory. Italso regarded as one offirst topological resultsgeometry; that is,does not depend on any measurements. This illustratesdeep connection between graph theorytopology.

Graph problems

• Graph coloring:four color theorem

• Route problems:
  • Seven bridgesKönigsberg
  • Minimum spanning tree
  • Shortest path problem
  • Route inspection problem (also called"Chinese Postman Problem")
  • Traveling salesman problem

• Flows:
  • Max flow min cut theorem

• reconstruction conjecture

• Isomorphism problems (Graph matching)
  • Canonical Labeling
  • subgraph isomorphismmonomorphisms
  • Maximal common subgraph

Important algorithms

• Dijkstra's algorithm
• Kruskal's algorithm
• Nearest neighbour algorithm
• Prim's algorithm

Generalizations

Inhypergraph an edge can connect more than two vertices.

An undirected graph can be seen assimplicial complex consisting1-simplices (the edges)0-simplices (the vertices). As such, complexesgeneralizationsgraphs sinceallowhigher-dimensional simplices.

Every graph gives rise tomatroid, butgeneralgraph cannot be recovered from its matroid, so matroidsnot truly generalizationsgraphs.

In model theory,graphjuststructure. Butthat case, thereno limitations onnumberedges:can be any cardinal number.

Related areasmathematics

• Ramsey theory
• Combinatorics

See also

• Ordered tree data structure - DAGs, binary treesother special formsgraph.
• Listgraph theory topics