Graph theory

Graph theory is the branch of mathematics that examines the properties of graphs.

A graph with 6 vertices and 7 edges.

Informally, a graph is a set of objects called vertices or nodes connected by links called edges or arcs. Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges).

For more and formal definitions, see Glossary of graph theory and Graph (mathematics).

Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, i.e. numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) we have a directed graph, or digraph.

Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be formulated as questions about certain graphs. For example, the link structure of Wikipedia could be represented by a directed graph: the vertices are the articles in Wikipedia, and there's a directed edge from article A to article B if and only if A contains a link to B. Directed graphs are also used to represent finite state machines. The development of algorithms to handle graphs is therefore of major interest in computer science.

Table of contents

1 History 2 Graph problems 3 Important algorithms 4 Generalizations 5 Related areas of mathematics

History

Leonhard Euler's paper on Seven Bridges of Königsberg is considered to be the first result in graph theory. It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements. This illustrates the deep connection between graph theory and topology.

Graph problems

• Coloring graphs: the four color theorem

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

• Flows:
  • Max flow min cut theorem

• reconstruction conjecture

• Isomorphism problems (Graph matching)
  • Canonical Labeling
  • subgraph isomorphism and monomorphisms
  • Maximal common subgraph

Important algorithms

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

Generalizations

In a hypergraph an edge can connect more than two vertices.

An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.

Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.

In model theory, a graph is just a structure. But in that case, there is no limitations on the number of edges: it can be any cardinal number.

Related areas of mathematics

• Ramsey theory
• Combinatorics