* 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 ).

In particular, the term " graph " was introduced by Sylvester in a paper published in 1878 in Nature, where he draws an analogy between " quantic invariants " and " co-variants " of algebra and molecular diagrams:

Paronomastically, dao is equated with its homonym 蹈 dao < d ' ôg, " to trample ," " tread ," and from that point of view it is nothing more than a " treadway ," " headtread ," or " foretread "; it is also occasionally associated with a near synonym ( and possible cognate ) 迪 ti < d ' iôk, " follow a road ," " go along ," " lead ," " direct "; " pursue the right path "; a term with definite ethical overtones and a graph with an exceedingly interesting phonetic, 由 yu < djôg ," " to proceed from.

In these cases, the term ' waveform ' refers to the shape of a graph of the varying quantity against time or distance.

By extension, the term ' waveform ' also describes the shape of the graph of any varying quantity against time.

The term has been generalized to include measurements in much more general mathematical spaces ; for example, in graph theory, one might consider a geodesic between two vertices / nodes of a graph.

* Lift, another term for a covering graph

This term dominates the low-speed side of the L / D graph, the left side of the U.

extended the proxy temperature record back 1000 years, and the term " Hockey Stick " graph was coined by Jerry Mahlman, director of the Geophysical Fluid Dynamics Laboratory, when he saw this graph.

The term Eulerian graph has two common meanings in graph theory.

The term " Eulerian graph " is also sometimes used in a weaker sense to denote a graph where every vertex has even degree.

This holds for every term except when the process cannot be performed on every Ferrers graph with n dots.

If the edges of a graph are thought of as lines drawn from one vertex to another ( as they are usually depicted in illustrations ), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.

In terms of the implication graph, two terms belong to the same strongly connected component whenever there exist chains of implications from one term to the other and vice versa.

More recently the term has come to mean 1st person RPGs, particularly ones which are aligned to a grid system and can be mapped on graph paper.

But in graph theory, when the term is used without any qualification, it almost always refers to the chromatic number of a graph.

In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph.

More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a coloring must be small and which therefore requires at least colors in any coloring.

The odd girth and even girth of a graph are the lengths of a shortest odd cycle and shortest even cycle respectively.

These are illustrated above in the Cycle graph ( algebra ) | cycle graph format, along with the 180 ° edge ( blue arrows ) and 120 ° vertex ( reddish arrows ) rotation s that permutation | permute the tetrahedron through the positions.

In mathematics, Tait's conjecture states that " Every 3-connected planar cubic graph has a Hamiltonian cycle ( along the edges ) through all its vertices ".

The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side ; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.

In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle, and a third color for all remaining edges.

Alternatively, a 4-coloring of the faces of a Hamiltonian cubic planar graph may be constructed directly, using two colors for the faces inside the cycle and two more colors for the faces outside.

If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex ( and either one of the lower ones ).

* Grinberg's theorem, a necessary condition on the existence of a Hamiltonian cycle that can be used to show that a graph forms a counterexample to Tait's conjecture

Animated graph of a the paths of totality of a solar eclipse cycle.

The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices.

A theorem by Nash-Williams says that every < span class =" texhtml " >< var > k </ var ></ span >‑ regular graph on < span class =" texhtml " > 2 < var > k </ var > + 1 </ span > vertices has a Hamiltonian cycle.

The energy lost per cycle is proportional to the area of the hysteresis loop in the BH graph.

In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph ( whether directed or undirected ).