* In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges.

There are also connections to string theory, game theory, graph matchings, solitons and integer programming.

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

Some mathematicians, especially in set theory, do not consider the sets and to be part of the relation, and therefore define a binary relation as being a subset of x, that is, just the graph.

One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas.

Five binary tree s on three Vertex ( graph theory ) | vertices, an example of Catalan number s.

It should be noted that while there are very strong connections between graph theory and combinatorics, these two are sometimes thought of as separate subjects.

* Chord ( graph theory ), an edge joining two not-adjacent nodes in a cycle

* Conjugate ( graph theory ), an alternative term for a line graph

* Digraph or directed graph, in graph theory

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be graph coloring | colored with four color theorem | only four colors.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory.

Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.

It draws heavily on graph theory and logic.

* Eccentricity ( graph theory ) of a vertex in a graph

In graph theory, the term fullerene refers to any 3-regular, planar graph with all faces of size 5 or 6 ( including the external face ).

** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

where s < sub > max </ sub > is the maximum value of s ( H ) for H in the set of all graphs with degree distribution identical to G. This gives a metric between 0 and 1, where a graph G with small S ( G ) is " scale-rich ", and a graph G with S ( G ) close to 1 is " scale-free ".

If the graph is a metric space then there is an efficient approximation algorithm that finds a Hamiltonian cycle with maximum edge weight being no more than twice the optimum.

As an example, the stress majorization approach to metric MDS can be applied to graph drawing as described above.

This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric.

A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric.

The vertex set ( of an undirected graph ) and the distance function form a metric space, if and only if the graph is connected.

During the route creation and maintenance phases, nodes use a height metric to establish a directed acyclic graph ( DAG ) rooted at destination.

The word metric on G is very closely related to the Cayley graph of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G.

The Fry readability formula ( or Fry readability graph ) is a readability metric for English texts, developed by Edward Fry.

Concretely, this can be achieved for instance by defining a topological similarity, by using ontologies to define a distance between words ( a naive metric for terms arranged as nodes in a directed acyclic graph like a hierarchy would be the minimal distance — in separating edges — between the two term nodes ), or using statistical means such as a vector space model to correlate words and textual contexts from a suitable text corpus ( co-occurrence ).

* Graph metric, a metric defined in terms of distances in a certain graph.

* Bipartite dimension, the minimum number of complete bipartite graphs whose union is the given graph

The dimension of the cycle space of a connected graph is thus related to the number of vertices and edges of the graph.

If the graph has n vertices and m edges then the dimension is m − n + 1.

The most common format is a graph with two geometric dimensions: the horizontal axis represents time, the vertical axis is frequency ; a third dimension indicating the amplitude of a particular frequency at a particular time is represented by the intensity or colour of each point in the image.

where is the adjacency matrix of the line graph of G, B ( G ) is the incidence matrix, and is the identity matrix of dimension q.

The bipartite dimension of a graph is the minimum number of bicliques needed to cover all the edges of the graph.

Let G be a connected finite graph embedded in Euclidean space of dimension n. Let V be a closed regular neighborhood of G. Then V is an n-dimensional handlebody.

The isoperimetric dimension of any finite graph is 0.

In rough terms, this means that a graph " mimicking " a given manifold ( as the grid mimics the Euclidean space ) would have the same isoperimetric dimension as the manifold.

It turns out that a group with polynomial growth of order d has isoperimetric dimension d. This holds both for the case of Lie groups and for the Cayley graph of a finitely generated group.

A theorem of Varopoulos connects the isoperimetric dimension of a graph to the rate of escape of random walk on the graph.

: This paper contains a precise definition of the isoperimetric dimension of a graph, and establishes many of its properties.

The cyclomatic number of the graph is defined as the dimension of this space.

For three dimensions it is possible to find the minimum spanning tree in time O (( n log n )< sup > 4 / 3 </ sup >), and in any dimension greater than three it is possible to solve it in a time that is faster than the quadratic time bound for the complete graph and Delaunay triangulation algorithms.

Cayley's formula follows from Kirchhoff's theorem as a special case, since every vector with 1 in one place, − 1 in another place, and 0 elsewhere is an eigenvector of the Laplacian matrix of the complete graph, with the corresponding eigenvalue being n. These vectors together span a space of dimension n − 1, so there are no other non-zero eigenvalues.

With probability 1, the graph of B < sub > H </ sub >( t ) has both Hausdorff dimension and box dimension of 2 − H.

The edge space is the-vector space freely generated by the edge set E. The dimension of the vertex space is thus the number of vertices of the graph, while the dimension of the edge space is the number of edges.