TheInfoList

OR:

In
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuo ...
, and more specifically in
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, a graph is a structure amounting to a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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 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 that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuo ...
. 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 an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing money is not necessarily reciprocated. Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by J. J. Sylvester in 1878 due to a direct relation between mathematics and
chemical structure A chemical structure determination includes a chemist's specifying the molecular geometry and, when feasible and necessary, the electronic structure of the target molecule or other solid. Molecular geometry refers to the spatial arrangement of a ...
(what he called a chemico-graphical image).

# Definitions

Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
s.

## Graph

A graph (sometimes called an ''undirected graph'' to distinguish it from a
directed graph 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 directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
, or a ''simple graph'' to distinguish it from a
multigraph In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by more ...
) is a pair , 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, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by more ...
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. To allow loops, the above definition must be changed by defining edges as
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
s of two vertices instead of 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 is supposed to be finite; this implies that the set of edges is also finite. Infinite graphs 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 sets and is a new set of ordered pairs consisting of elements in and i ...
, as most results on finite graphs do not extend to the infinite case, or need a rather different proof. An empty graph is a graph that has an
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
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 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, because a loop contributes 2 to the degree), and the maximum number of edges is (or if loops are allowed). The edges of a graph define a symmetric relation 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 and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite sim ...
, which is an square matrix, with specifying the number of connections from vertex to vertex . For a simple graph, is either 0, indicating disconnection, or 1, indicating connection; moreover because an edge in a simple graph cannot start and end at the same vertex. Graphs with self-loops will be characterized by some or all being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all being equal to a positive integer. Undirected graphs will have a
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

## Directed 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 comprising: * , a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''vertices'' (also called ''nodes'' or ''points''); * , a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'', or ''arcs''), which are
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of distinct vertices: $E \subseteq \$. To avoid ambiguity, this type of object may be called precisely a directed simple graph. In the edge directed from to , the vertices and are called the ''endpoints'' of the edge, the ''tail'' of the edge and the ''head'' of the edge. The edge is said to ''join'' and and to be ''incident'' on and on . A vertex may exist in a graph and not belong to an edge. The edge is called the ''inverted edge'' of . ''
Multiple edges In graph theory, multiple edges (also called parallel edges or a multi-edge), are, in an undirected graph, two or more edges that are incident to the same two vertices, or in a directed graph, two or more edges with both the same tail vertex ...
'', 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 comprising: * , a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''vertices'' (also called ''nodes'' or ''points''); * , a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs''); * , an ''incidence function'' mapping every edge to an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
of vertices (that is, an edge is associated with two distinct vertices): $\phi : E \to \$. 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) $\left(x,x\right)$ which is not in $\$. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of $E$ should be modified to $E \subseteq \$. For directed multigraphs, the definition of $\phi$ should be modified to $\phi : E \to \$. 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, bolts, ammo, projectiles, 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 were tr ...
'') respectively. The edges of a directed simple graph permitting loops is a
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
~ on the vertices of that is called the ''adjacency relation'' of . Specifically, for each edge , its endpoints and are said to be ''adjacent'' to one another, which is denoted .

## 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 , (the undirected edges), (the directed edges), and defined as above. Directed and undirected graphs are special cases.

## 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 in
shortest path problem In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between ...
s such as the
traveling salesman problem The travelling salesman problem (also called the travelling 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 cit ...
.

# 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 an
orientation 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 de ...
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 are
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...
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, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is th ...
'' or '' k-edge-connected graph'' 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 field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V ...
'' is a simple graph in which the vertex set can be partitioned 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, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertice ...
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 of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...
, 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''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.

## 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''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.

## Tree

A ''tree'' is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected 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, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of trees.

## Polytree

A ''polytree'' (or ''directed tree'' or ''oriented tree'' or ''singly connected network'') is a
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one v ...
(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.

More advanced kinds of graphs are: * Petersen graph and its generalizations; * perfect graphs; *
cograph In graph theory, a cograph, or complement-reducible graph, or ''P''4-free graph, is a graph that can be generated from the single-vertex graph ''K''1 by complementation and disjoint union. That is, the family of cographs is the smallest class ...
s; *
chordal graph In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced c ...
s; * other graphs with large automorphism groups:
vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face ...
, arc-transitive, and distance-transitive graphs; * strongly regular graphs and their generalizations distance-regular graphs.

# 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.) The
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
of all graphs is the
comma category In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become obje ...
Set ↓ ''D'' where ''D'': Set → Set is the
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
taking a set ''s'' to ''s'' × ''s''.

# Examples

* The diagram is a schematic representation of the graph with vertices $V = \$ and edges $E = \.$ * In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (inclu ...
, 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 o ...
),
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. It is an abstract machine that can be in exactly one of a finite number ...
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 sets and is a new set of ordered pairs consisting of elements in and i ...
''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 online social media and social networking service owned and operated by American company Twitter, Inc., on which users post and interact with 280-character-long messages known as "tweets". Registered users can post, like, a ...
, with one user following another.Pankaj Gupta, Ashish Goel, Jimmy Lin, Aneesh Sharma, Dong Wang, and Reza Bosagh Zade
''Proceedings of the 22nd international conference on World Wide Web''. .
*Particularly regular examples of directed graphs are given by the Cayley graphs of finitely-generated groups, as well as Schreier coset graphs *In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, 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 ass ...
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 categories In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-c ...
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, ** line graph, ** dual graph, **
complement graph In the mathematical field of 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 ...
, ** graph rewriting; * ''binary operations'', which create a new graph from two initial ones, such as: ** disjoint union of graphs, **
cartesian product of graphs Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to: Mathematics * Cartesian closed category, a closed category in category theory *Cartesian coordinate system, moder ...
, **
tensor product of graphs In graph theory, the tensor product of graphs and is a graph such that * the vertex set of is the Cartesian product ; and * vertices and are adjacent in if and only if ** is adjacent to in , and ** is adjacent to in . The tensor pro ...
, **
strong product of graphs In graph theory, the strong product is a way of combining two graphs to make a larger graph. Two vertices are adjacent in the strong product when they come from pairs of vertices in the factor graphs that are either adjacent or identical. The str ...
, **
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 In mathematics Mathematics is an area of knowledge that includes the topics of number ...
, ** series–parallel graphs.

# Generalizations

In a
hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) w ...
, an edge can join more than two vertices. An undirected graph can be seen as a
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
consisting of 1-
simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
(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 axiomatically, the most significant being ...
. In
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
, a graph is just a
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
. But in that case, there is no limitation on the number of edges: it can be any
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
, see continuous graph. In
computational biology Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has fo ...
, 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 (that is, descriptions of phenomena for which location is relevant), combined with software tools for managing, analyzing, and visualizing those data. In a ...
, geometric networks are closely modeled after graphs, and borrow many concepts from
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
to perform spatial analysis on road networks or utility grids.

*
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 o ...
*
Graph (abstract data type) In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutab ...
*
Graph database A graph database (GDB) is a database that uses graph structures for semantic queries with nodes, edges, and properties to represent and store data. A key concept of the system is the '' graph'' (or ''edge'' or ''relationship''). The graph rel ...
*
Graph drawing Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, c ...
* List of graph theory topics * List of publications in graph theory *
Network theory Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. In computer science and network science, network theory is a part of graph theory: a network can be def ...

# References

* * * * * * * * * * * * *