In

_{ij}'' specifying the number of connections from vertex ''i'' to vertex ''j''. For a simple graph, $A\_\backslash in\backslash $, indicating disconnection or connection respectively, meanwhile $A\_=0$ (that is, an edge can not start and end at the same vertice). Graphs with self-loops will be characterized by some or all $A\_$ being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all $A\_$ being equal to a positive integer. Undirected graphs will have a symmetric adjacency matrix ($A\_=A\_$).

_{''E''} and ''ϕ''_{''A''} defined as above. Directed and undirected graphs are special cases.

_{1}, ''v''_{2}, …, ''v''_{''n''} such that the edges are the where ''i'' = 1, 2, …, ''n'' − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph.

_{1}, ''v''_{2}, …, ''v''_{''n''} such that the edges are the where ''i'' = 1, 2, …, ''n'' − 1, plus the edge . Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.

WTF: The who-to-follow system at Twitter

''Proceedings of the 22nd international conference on World Wide Web''. . *Particularly regular examples of directed graphs are given by the

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, and more specifically in graph theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form
A diagram is a symbolic representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy ...

as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics
Discrete mathematics is the study of mathematical structures
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...

.
The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if any edge from a person ''A'' to a person ''B'' corresponds to ''A'' owes money to ''B'', then this graph is directed, because owing money is not necessarily reciprocated. The former type of graph is called an undirected graph while the latter type of graph is called a directed graph.
Graphs are the basic subject studied by graph theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

. The word "graph" was first used in this sense by in 1878 in a direct relation between mathematics and chemical structure
A chemical structure determination includes a chemist
A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a scientist
A scientist is a person who conducts Scientific method, scientific re ...

(what he called chemico-graphical image).
Definitions

Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and relatedmathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
Graph

A graph (sometimes called ''undirected graph'' for distinguishing from adirected graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a Graph (discrete mathematics), graph that is made up of a set of Vertex (graph theory), vertices connected by directed Edge (graph theory), edges often called ...

, or ''simple graph'' for distinguishing from a multigraph
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

) is a pair
Pair or PAIR or Pairing may refer to:
Government and politics
* Pair (parliamentary convention), matching of members unable to attend, so as not to change the voting margin
* ''Pair'', a member of the Prussian House of Lords
* ''Pair'', the Frenc ...

, where is a set whose elements are called ''vertices'' (singular: vertex), and is a set of paired vertices, whose elements are called ''edges'' (sometimes ''links'' or ''lines'').
The vertices and of an edge are called the ''endpoints'' of the edge. The edge is said to ''join'' and and to be ''incident'' on and . A vertex may belong to no edge, in which case it is not joined to any other vertex.
A multigraph
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a generalization that allows multiple edges to have the same pair of endpoints. In some texts, multigraphs are simply called graphs.
Sometimes, graphs are allowed to contain '' loops'', which are edges that join a vertex to itself. For allowing loops, the above definition must be changed by defining edges as multiset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of two vertices instead of two-sets. Such generalized graphs are called ''graphs with loops'' or simply ''graphs'' when it is clear from the context that loops are allowed.
Generally, the set of vertices ''V'' is supposed to be finite; this implies that the set of edges is also finite. Infinite graph
This is a glossary of graph theory. Graph theory is the study of graph (discrete mathematics), graphs, systems of nodes or vertex (graph theory), vertices connected in pairs by lines or #edge, edges.
Symbols
A
...

s are sometimes considered, but are more often viewed as a special kind of binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...

, as most results on finite graphs do not extend to the infinite case, or need a rather different proof.
An empty graphIn the mathematical field of graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made u ...

is a graph that has an empty set #REDIRECT Empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

of vertices (and thus an empty set of edges). The ''order'' of a graph is its number of vertices . The ''size'' of a graph is its number of edges . However, in some contexts, such as for expressing the computational complexity of algorithms, the size is (otherwise, a non-empty graph could have a size 0). The ''degree'' or ''valency'' of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice.
In a graph of order , the maximum degree of each vertex is (or if loops are allowed), and the maximum number of edges is (or if loops are allowed).
The edges of a graph define a symmetric relation
A symmetric relation is a type of binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and ...

on the vertices, called the ''adjacency relation''. Specifically, two vertices and are ''adjacent'' if is an edge. A graph may be fully specified by its adjacency matrix
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theor ...

''A'', which is an $n\backslash times\; n$ square matrix, with ''ADirected graph

A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, a directed graph is a pair $G=(V,E)$ comprising: * $V$, a set of ''vertices'' (also called ''nodes'' or ''points''); * $E\; \backslash subseteq\; \backslash $, a set of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs'') which areordered pair
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s of vertices (that is, an edge is associated with two distinct vertices).
To avoid ambiguity, this type of object may be called precisely a directed simple graph.
In the edge $(x,y)$ directed from $x$ to $y$, the vertices $x$ and $y$ are called the ''endpoints'' of the edge, $x$ the ''tail'' of the edge and $y$ the ''head'' of the edge. The edge is said to ''join'' $x$ and $y$ and to be ''incident'' on $x$ and on $y$. A vertex may exist in a graph and not belong to an edge. The edge $(y,x)$ is called the ''inverted edge'' of $(x,y)$. ''Multiple edges Multiple edges joining two vertices.
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this conte ...

'', not allowed under the definition above, are two or more edges with both the same tail and the same head.
In one more general sense of the term allowing multiple edges, a directed graph is an ordered triple $G=(V,E,\backslash phi)$ comprising:
* $V$, a set of ''vertices'' (also called ''nodes'' or ''points'');
* $E$, a set of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs'');
* $\backslash phi\; :\; E\; \backslash to\; \backslash $, an ''incidence function'' mapping every edge to an ordered pair
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of vertices (that is, an edge is associated with two distinct vertices).
To avoid ambiguity, this type of object may be called precisely a directed multigraph.
A '' loop'' is an edge that joins a vertex to itself. Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex $x$ to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph) $(x,x)$ which is not in $\backslash $. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of $E$ should be modified to $E\; \backslash subseteq\; \backslash $. For directed multigraphs, the definition of $\backslash phi$ should be modified to $\backslash phi\; :\; E\; \backslash to\; \backslash $. To avoid ambiguity, these types of objects may be called precisely a directed simple graph permitting loops and a directed multigraph permitting loops (or a ''quiver
A quiver is a container for holding arrows, Crossbow bolt, bolts, Dart (missile), darts, or javelins. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers ...

'') respectively.
The edges of a directed simple graph permitting loops $G$ is a homogeneous relation
Homogeneity and heterogeneity are concepts often used in the sciences and statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sc ...

~ on the vertices of $G$ that is called the ''adjacency relation'' of $G$. Specifically, for each edge $(x,y)$, its endpoints $x$ and $y$ are said to be ''adjacent'' to one another, which is denoted $x$ ~ $y$.
Mixed graph

A ''mixed graph'' is a graph in which some edges may be directed and some may be undirected. It is an ordered triple for a ''mixed simple graph'' and for a ''mixed multigraph'' with ''V'', ''E'' (the undirected edges), ''A'' (the directed edges), ''ϕ''Weighted graph

A ''weighted graph'' or a ''network'' is a graph in which a number (the weight) is assigned to each edge. Such weights might represent for example costs, lengths or capacities, depending on the problem at hand. Such graphs arise in many contexts, for example inshortest path problem
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph the ...

s such as the traveling salesman problem
The travelling salesman problem (also called the traveling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city ...

.
Types of graphs

Oriented graph

One definition of an ''oriented graph'' is that it is a directed graph in which at most one of and may be edges of the graph. That is, it is a directed graph that can be formed as anorientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building design ...

of an undirected (simple) graph.
Some authors use "oriented graph" to mean the same as "directed graph". Some authors use "oriented graph" to mean any orientation of a given undirected graph or multigraph.
Regular graph

A ''regular graph'' is a graph in which each vertex has the same number of neighbours, i.e., every vertex has the same degree. A regular graph with vertices of degree ''k'' is called a ''k''‑regular graph or regular graph of degree ''k''.Complete graph

A ''complete graph'' is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.Finite graph

A ''finite graph'' is a graph in which the vertex set and the edge set arefinite set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s. Otherwise, it is called an ''infinite graph''.
Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.
Connected graph

In an undirected graph, an unordered pair of vertices is called ''connected'' if a path leads from ''x'' to ''y''. Otherwise, the unordered pair is called ''disconnected''. A ''connected graph'' is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it is called a ''disconnected graph''. In a directed graph, an ordered pair of vertices is called ''strongly connected'' if a directed path leads from ''x'' to ''y''. Otherwise, the ordered pair is called ''weakly connected'' if an undirected path leads from ''x'' to ''y'' after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called ''disconnected''. A ''strongly connected graph'' is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a ''weakly connected graph'' if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a ''disconnected graph''. A ''k-vertex-connected graph
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theo ...

'' or ''k-edge-connected graph
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theor ...

'' is a graph in which no set of vertices (respectively, edges) exists that, when removed, disconnects the graph. A ''k''-vertex-connected graph is often called simply a ''k-connected graph''.
Bipartite graph

A ''bipartite graph
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

'' is a simple graph in which the vertex set can be into two sets, ''W'' and ''X'', so that no two vertices in ''W'' share a common edge and no two vertices in ''X'' share a common edge. Alternatively, it is a graph with a chromatic number
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph t ...

of 2.
In a complete bipartite graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex (graph theory), vertex of the first set is connected to every vertex of the second set..Electronic edition pa ...

, the vertex set is the union of two disjoint sets, ''W'' and ''X'', so that every vertex in ''W'' is adjacent to every vertex in ''X'' but there are no edges within ''W'' or ''X''.
Path graph

A ''path graph'' or ''linear graph'' of order is a graph in which the vertices can be listed in an order ''v''Planar graph

A ''planar graph'' is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.Cycle graph

A ''cycle graph'' or ''circular graph'' of order is a graph in which the vertices can be listed in an order ''v''Tree

A ''tree'' is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connectedacyclic
Acyclic may refer to:
* In chemistry, a compound which is an open-chain compound, e.g. alkanes and acyclic aliphatic compounds
* In mathematics:
** A graph without a Cycle (graph theory), cycle, especially
*** A directed acyclic graph
** An acyclic ...

undirected graph.
A ''forest'' is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of trees.
Polytree

A ''polytree'' (or ''directed tree'' or ''oriented tree'' or ''singly connected network'') is adirected acyclic graph
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(DAG) whose underlying undirected graph is a tree.
A ''polyforest'' (or ''directed forest'' or ''oriented forest'') is a directed acyclic graph whose underlying undirected graph is a forest.
Advanced classes

More advanced kinds of graphs are: *Petersen graph
In the mathematical field of graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made ...

and its generalizations;
* perfect graph 240px, The Paley graph of order 9, colored with three colors and showing a clique of three vertices. In this graph and each of its induced subgraphs the chromatic number equals the clique number, so it is a perfect graph.
In graph theory, a perfect ...

s;
* cograph
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theor ...

s;
* chordal graph
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s;
* other graphs with large automorphism groups: vertex-transitive
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertex (geometry), vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the sam ...

, arc-transitive, and distance-transitive graph The Biggs-Smith graph, the largest 3-regular distance-transitive graph.
In the mathematics, mathematical field of graph theory, a distance-transitive graph is a Graph (discrete mathematics), graph such that, given any two vertices ''v'' and ''w'' ...

s;
* strongly regular graphImage:Paley13.svg, upright=1.1, The Paley graph of order 13, a strongly regular graph with parameters srg(13,6,2,3).
In graph theory, a strongly regular graph is defined as follows. Let ''G'' = (''V'', ''E'') be a regular graph with ''v'' vertice ...

s and their generalizations distance-regular graph
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
Properties of graphs

Two edges of a graph are called ''adjacent'' if they share a common vertex. Two edges of a directed graph are called ''consecutive'' if the head of the first one is the tail of the second one. Similarly, two vertices are called ''adjacent'' if they share a common edge (''consecutive'' if the first one is the tail and the second one is the head of an edge), in which case the common edge is said to ''join'' the two vertices. An edge and a vertex on that edge are called ''incident''. The graph with only one vertex and no edges is called the ''trivial graph''. A graph with only vertices and no edges is known as an ''edgeless graph''. The graph with no vertices and no edges is sometimes called the '' null graph'' or ''empty graph'', but the terminology is not consistent and not all mathematicians allow this object. Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called ''vertex-labeled''. However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called ''edge-labeled''. Graphs with labels attached to edges or vertices are more generally designated as ''labeled''. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called ''unlabeled''. (In the literature, the term ''labeled'' may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.) Thecategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

of all graphs is the comma category
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

Set ↓ ''D'' where ''D'': Set → Set is the functor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

taking a set ''s'' to ''s'' × ''s''.
Examples

* The diagram is a schematic representation of the graph with vertices $V\; =\; \backslash $ and edges $E\; =\; \backslash .$ * Incomputer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

, directed graphs are used to represent knowledge (e.g., conceptual graph
A conceptual graph (CG) is a formalism for knowledge representation. In the first published paper on CGs, John F. Sowa used them to represent the conceptual schemas used in database systems. The first book on CGs applied them to a wide range of t ...

), finite state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation
A model is an informative representation of an object, person or system. ...

s, and many other discrete structures.
* A binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...

''R'' on a set ''X'' defines a directed graph. An element ''x'' of ''X'' is a direct predecessor of an element ''y'' of ''X'' if and only if ''xRy''.
* A directed graph can model information networks such as Twitter
Twitter is an American microblogging
Microblogging is an online Broadcasting, broadcast medium that exists as a specific form of blogging. A micro-blog differs from a traditional blog in that its content is typically smaller in both actu ...

, with one user following another.Pankaj Gupta, Ashish Goel, Jimmy Lin, Aneesh Sharma, Dong Wang, and Reza Bosagh ZadeWTF: The who-to-follow system at Twitter

''Proceedings of the 22nd international conference on World Wide Web''. . *Particularly regular examples of directed graphs are given by the

Cayley graph on two generators ''a'' and ''b''
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

s of finitely-generated groups, as well as Schreier coset graphs
*In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

, every small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows Associ ...

has an underlying directed multigraph whose vertices are the objects of the category, and whose edges are the arrows of the category. In the language of category theory, one says that there is a forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...

from the category of small categoriesIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

to the category of quivers.
Graph operations

There are several operations that produce new graphs from initial ones, which might be classified into the following categories: * ''unary operations'', which create a new graph from an initial one, such as: **edge contraction
In graph theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ...

,
** line graph
In the mathematical discipline of graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context i ...

,
** dual graph
In the mathematical discipline of graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context i ...

,
** complement graph
In graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a graph, one fills in all the m ...

,
** graph rewriting
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of compu ...

;
* ''binary operations'', which create a new graph from two initial ones, such as:
** disjoint union of graphs
In graph theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ) ...

,
** cartesian product of graphs
In graph theory, the Cartesian product ''G'' \square ''H'' of graphs ''G'' and ''H'' is a graph such that
* the vertex set of ''G'' \square ''H'' is the Cartesian product ''V''(''G'') × ''V''(''H''); and
* two vertices (''u,u' '') and (''v,v' '') ...

,
** tensor product of graphs
In graph theory, the tensor product ''G'' × ''H'' of graphs ''G'' and ''H'' is a graph such that
* the vertex set of ''G'' × ''H'' is the Cartesian product ''V''(''G'') × ''V''(''H''); and
* vertices (''g,h'') and (''g',h) are adjacent in ''G'' ...

,
** strong product of graphs
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theor ...

,
** lexicographic product of graphsImage:Graph-lexicographic-product.svg, 300px, The lexicographic product of graphs.
In graph theory, the lexicographic product or (graph) composition of graphs and is a graph such that
* the vertex set of is the cartesian product ; and
* any two ...

,
** series–parallel graphImage:Series parallel composition.svg, upright=1.25, Series and parallel composition operations for series–parallel graphs
In graph theory, series–parallel graphs are graphs with two distinguished vertices called ''terminals'', formed recursively ...

s.
Generalizations

In ahypergraph
File:PAOH hypergraph representation.png, alt=PAOH visualization of a hypergraph, Alternative representation of the hypergraph reported in the figure above, called PAOH. Edges are vertical lines connecting vertices. V7 is an isolated vertex. Vertice ...

, an edge can join more than two vertices.
An undirected graph can be seen as a simplicial complex
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

consisting of 1- (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
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most si ...

.
In model theory
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...

, a graph is just a structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

. But in that case, there is no limitation on the number of edges: it can be any cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

, see continuous graph.
In computational biology
Computational biology involves the development and application of data-analytical and theoretical methods, mathematical modelling
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelat ...

, power graph analysis introduces power graphs as an alternative representation of undirected graphs.
In geographic information systems
A geographic information system (GIS) is a type of database containing Geographic data and information, geographic data (that is, descriptions of phenomena for which location is relevant), combined with Geographic information system software, sof ...

, geometric networks are closely modeled after graphs, and borrow many concepts from graph theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

to perform spatial analysis on road networks or utility grids.
See also

*Conceptual graph
A conceptual graph (CG) is a formalism for knowledge representation. In the first published paper on CGs, John F. Sowa used them to represent the conceptual schemas used in database systems. The first book on CGs applied them to a wide range of t ...

* Graph (abstract data type)
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of comp ...

* Graph database
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...

* Graph drawing
Graph drawing is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

* List of graph theory topics
This is a list of graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Ver ...

* List of publications in graph theory
* Network theory
Network theory is the study of graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas ...

Notes

References

* * * * * * * * * * * * *Further reading

*External links

* * {{DEFAULTSORT:Graph (Discrete mathematics) Graph theory