# 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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### Graph (discrete mathematics)

graphundirected graphgraphs
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 (also called nodes or points) which are connected by edges (also called links 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).

### Loop (graph theory)

looploopsself-loop
A loop is an edge that joins a vertex to itself.
In graph theory, a loop (also called a self-loop or a "buckle") is an edge that connects a vertex to itself.

### Network science

networksnetworkdiameter
Emphasizing their application to real-world systems, the term network is sometimes defined to mean a graph in which attributes (e.g. names) are associated with the vertices and edges, 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.

### Multiple edges

multiple edgeparallel edgesmultiple arcs
Multiple edges are two or more edges that join the same two vertices.
In graph theory, multiple edges (also called parallel edges or a multi-edge), are two or more edges that are incident to the same two vertices.

### Binary relation

relationrelationsidentity relation
The edges of an undirected simple graph permitting loops G induce a symmetric homogeneous relation ~ on the vertices of G that is called the adjacency relation of G.
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.

### 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.

### 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.

### 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.

### 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

EulerLeonard EulerEuler, Leonhard
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, geographer, 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.

### Graph database

graph databasesgraphGraphDB
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.

### Tree (graph theory)

treetreesrooted tree
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 graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph.

### 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 (data 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 (also called nodes or points) which are connected by edges (also called links or lines).
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).

### Topology

topologicaltopologicallytopologist
Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy and L'Huilier, and represents the beginning of the branch of mathematics known as topology.
This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory.

### 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.

### George Pólya Prize

Pólya PrizePólya Prize (SIAM)George Polya 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.

### Peter Tait (physicist)

Peter Guthrie TaitPeter TaitP. G. Tait
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.

### Algorithm

algorithmsalgorithm designcomputer algorithm
The development of algorithms to handle graphs is therefore of major interest in computer science.
Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and parsing techniques.

### Paul Seymour (mathematician)

Paul SeymourSeymourPaul D. Seymour
A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas.
His research interest is in discrete mathematics, especially graph theory.