# Graph theory

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In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.wikipedia
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### Directed graph

directed edgedirecteddigraph
A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered.
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.

### Graph (discrete mathematics)

graphundirected graphgraphs
A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

### Vertex (graph theory)

verticesvertexnode
A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines.
In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).

### Degree (graph theory)

degreedegree sequencemaximum degree
The degree or valency of a vertex is the number of edges that connect to it, where an edge that connects a vertex to itself (a loop) is counted twice.
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice.

### Network science

networknetworksdiameter
And the subject that expresses and understands the real-world systems as a network is called network science.
The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology.

### Loop (graph theory)

looploopsself-loop
The degree or valency of a vertex is the number of edges that connect to it, where an edge that connects a vertex to itself (a loop) is counted twice.
In graph theory, a loop (also called a self-loop or a "buckle") is an edge that connects a vertex to itself.

### Multigraph

pseudographdirected multigraphmultidigraph
Many authors call this type of object a multigraph or pseudograph.
In mathematics, and more specifically in graph theory, a multigraph (in contrast to a simple graph) is a graph which is permitted to have multiple edges (also called parallel edges ), that is, edges that have the same end nodes.

### Directed acyclic graph

acyclicDAGacyclic directed graph
More contemporary approaches such as head-driven phrase structure grammar model the syntax of natural language using typed feature structures, which are directed acyclic graphs.
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG ), is a finite directed graph with no directed cycles.

### Graph database

graph databasesgraphgraph data processing
Complementary to graph transformation systems focusing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data.
A graph within graph databases is based on graph theory.

### Discrete mathematics

discretediscrete mathdiscrete structure
Graphs are one of the prime objects of study in discrete mathematics.
It draws heavily on graph theory and mathematical logic.

### Social network analysis

network analysissocial networking potentialsocial network
Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore rumor spreading, notably through the use of social network analysis software.
Social network analysis (SNA) is the process of investigating social structures through the use of networks and graph theory.

### Algebraic graph theory

Algebraic graph theory has close links with group theory.
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs.

### Leonhard Euler

EulerEuler, LeonhardEulerian
The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.
Leonhard Euler (15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

### Tree (graph theory)

treetreesforest
More than one century after Euler's paper on the bridges of Königsberg and while Listing was introducing the concept of topology, Cayley was led by an interest in particular analytical forms arising from differential calculus to study a particular class of graphs, the trees.
In mathematics, and, more specifically, in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path.

### Chemical graph theory

Chemical graph theory uses the molecular graph as a means to model molecules.
Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena.

### Graph (abstract data type)

graphgraphsgraph structure
Complementary to graph transformation systems focusing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data.
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from mathematics; specifically, the field of graph theory.

### Glossary of graph theory terms

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A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines.
This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

### Molecular graph

chemical graph
Chemical graph theory uses the molecular graph as a means to model molecules.
In chemical graph theory and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory.

### Frank Harary

HararyHarary, F.Harary, Frank
Another book by Frank Harary, published in 1969, was "considered the world over to be the definitive textbook on the subject", and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other.
Frank Harary (March 11, 1921 – January 4, 2005) was an American mathematician, who specialized in graph theory.

### Lattice graph

grid graphgridlattice
Still, other methods in phonology (e.g. optimality theory, which uses lattice graphs) and morphology (e.g. finite-state morphology, using finite-state transducers) are common in the analysis of language as a graph.
Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space).

### Dénes Kőnig

KőnigD. KőnigKőnig Dénes
The first textbook on graph theory was written by Dénes Kőnig, and published in 1936.
Dénes Kőnig (September 21, 1884 – October 19, 1944) was a Hungarian mathematician of Jewish heritage who worked in and wrote the first textbook on the field of graph theory.

### Knight's tour

knight tourknight problemknight's tours
This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz.
The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory.

### George Pólya Prize

Pólya PrizeGeorge Polya PrizePolya Prize
Harary donated all of the royalties to fund the Pólya Prize.
Starting in 1969 the prize money was provided by Frank Harary, who donated the profits from his Graph Theory book.

### Algorithm

algorithmscomputer algorithmalgorithm design
The development of algorithms to handle graphs is therefore of major interest in computer science.
Many problems (such as playing chess) can be modeled as problems on graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithms, branch and bound enumeration and backtracking.

### Peter Tait (physicist)

Peter TaitPeter Guthrie TaitTait
The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus.
His name is known in graph theory mainly for Tait's conjecture.