Graph (discrete mathematics)

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". 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).[1] 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.

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 admiring B, then this graph is directed, because admiration 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. The word "graph" was first used in this sense by James Joseph Sylvester in 1878.[2][3]

A graph with six vertices and seven edges.


Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.


A graph with three vertices and three edges.

A graph (sometimes called undirected graph for distinguishing to from a directed graph, or simple graph for distinguishing from a multigraph)[4][5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of two-sets (set with two distinct elements) of vertices, whose elements are called edges (sometimes links or lines).

The vertices x and y of an edge {x, y} are called the endpoints of the edge. The edge is said to join x and y and to be incident on x and y. A vertex may not belong to any edge.

A multigraph is a generalization that allows multiple edges adjacent to the same pair of vertices. In some texts, multigraphs are simply called graphs.[6][7]

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 multisets 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 graphs are sometimes considered, but are more often viewed as a special kind of binary relation, 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 of vertices (and thus an empty set of edges). The order of a graph is its number of vertices |V|. The size of a graph is its number of edges |E|. However, in some contexts, such that for expressing the computational complexity of algorithms, the size is |V| + |E| (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 n, the maximum degree of each vertex is n − 1 (or n + 1 if loops are allowed), and the maximum number of edges is n(n − 1)/2 (or n(n + 1)/2 if loops are allowed).

The edges of a graph define a symmetric relation on the vertices, called the adjacency relation. Specifically, two vertices x and y are adjacent if {x, y} is an edge.

Directed graph

A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction).

A directed graph or digraph is a graph in which edges have orientations.

In one restricted but very common sense of the term,[8] a directed graph is an ordered pair G = (V, E) comprising:

  • V a set of vertices (also called nodes or points);
  • E ⊆ {(x, y) | (x, y) ∈ V2xy} a set of edges (also called directed edges, directed links, directed lines, arrows or arcs) which are ordered pairs of distinct vertices (i.e., 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 (y, x) is called the inverted edge of (x, y). 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. A loop is an edge that joins a vertex to itself. Multiple edges are two or more edges that join the same two vertices.

In one more general sense of the term allowing multiple edges,[8] a directed graph is an ordered triple G = (V, E, ϕ) 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);
  • ϕ: E → {(x, y) | (x, y) ∈ V2 ∧ x ≠ y} an incidence function mapping every edge to an ordered pair of distinct vertices (i.e., an edge is associated with two distinct vertices).

To avoid ambiguity, this type of object may be called precisely a directed multigraph.

Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex x is the edge (for a directed simple graph) or is incident on (for a directed multigraph) (x, x) which is not in {(x, y) | (x, y) ∈ V2xy}. So to allow loops the definitions must be expanded. For directed simple graphs, E ⊆ {(x, y) | (x, y) ∈ V2 ∧ x ≠ y} should become EV2. For directed multigraphs, ϕ: E → {(x, y) | (x, y) ∈ V2 ∧ x ≠ y} should become ϕ: EV2. 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) respectively.

The edges of a directed simple graph permitting loops G is a homogeneous relation ~ 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 G = (V, E, A) for a mixed simple graph and G = (V, E, A, ϕE, ϕA) for a mixed multigraph with V, E (the undirected edges), A (the directed edges), ϕE and ϕA defined as above. Directed and undirected graphs are special cases.

Weighted graph

Weighted network
A weighted graph with ten vertices and twelve edges.

A weighted graph or a network[9][10] is a graph in which a number (the weight) is assigned to each edge.[11] 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 problems such as the traveling salesman problem.

Types of graphs

Oriented graph

An oriented graph is a directed graph in which at most one of (x, y) and (y, x) may be edges of the graph. That is, it is a directed graph that can be formed as an orientation of an undirected graph. However, some authors use "oriented graph" to mean the same as "directed graph".

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

Complete graph K5
A complete graph with five vertices and ten edges. Each vertex has an edge to every other vertex.

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 sets. 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 {x, y} 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 (x, y) 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 or k-edge-connected graph is a graph in which no set of k − 1 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 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 of 2.

In a complete bipartite graph, 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 n ≥ 2 is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the {vi, vi+1} 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 n ≥ 3 is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the {vi, vi+1} where i = 1, 2, …, n − 1, plus the edge {vn, v1}. 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.


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 of trees.


A polytree (or directed tree or oriented tree or singly connected network) is a directed acyclic graph (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:

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. (Note that 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 of all graphs is the slice category Set ↓ D where D: Set → Set is the functor taking a set s to s × s.


A graph with six vertices and seven edges.
  • The diagram is a schematic representation of the graph with vertices and edges
  • In computer science, directed graphs are used to represent knowledge (e.g., conceptual graph), finite state machines, and many other discrete structures.
  • A binary relation 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, with one user following another.[12][13]
  • 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, every small category 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 from the category of small categories 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:


In a hypergraph, an edge can join more than two vertices.

An undirected graph can be seen as a simplicial complex consisting of 1-simplices (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 model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can be any cardinal number, see continuous graph.

In computational biology, power graph analysis introduces power graphs as an alternative representation of undirected graphs.

In geographic information systems, geometric networks are closely modeled after graphs, and borrow many concepts from graph theory to perform spatial analysis on road networks or utility grids.

See also


  1. ^ Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 19. ISBN 978-0-486-67870-2. Retrieved 8 August 2012. A graph is an object consisting of two sets called its vertex set and its edge set.
  2. ^ See:
  3. ^ Gross, Jonathan L.; Yellen, Jay (2004). Handbook of graph theory. CRC Press. p. 35. ISBN 978-1-58488-090-5.
  4. ^ Bender & Williamson 2010, p. 148.
  5. ^ See, for instance, Iyanaga and Kawada, 69 J, p. 234 or Biggs, p. 4.
  6. ^ Bender & Williamson 2010, p. 149.
  7. ^ Graham et al., p. 5.
  8. ^ a b Bender & Williamson 2010, p. 161.
  9. ^ Strang, Gilbert (2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 978-0-03-010567-8
  10. ^ Lewis, John (2013), Java Software Structures (4th ed.), Pearson, p. 405, ISBN 978-0133250121
  11. ^ Fletcher, Peter; Hoyle, Hughes; Patty, C. Wayne (1991). Foundations of Discrete Mathematics (International student ed.). Boston: PWS-KENT Pub. Co. p. 463. ISBN 978-0-53492-373-0. A weighted graph is a graph in which a number w(e), called its weight, is assigned to each edge e.
  12. ^ Grandjean, Martin (2016). "A social network analysis of Twitter: Mapping the digital humanities community". Cogent Arts & Humanities. 3 (1): 1171458. doi:10.1080/23311983.2016.1171458.
  13. ^ Pankaj Gupta, Ashish Goel, Jimmy Lin, Aneesh Sharma, Dong Wang, and Reza Bosagh Zadeh WTF: The who-to-follow system at Twitter, Proceedings of the 22nd international conference on World Wide Web. doi:10.1145/2488388.2488433.


  • Balakrishnan, V. K. (1997). Graph Theory (1st ed.). McGraw-Hill. ISBN 978-0-07-005489-9.
  • Bang-Jensen, J.; Gutin, G. (2000). Digraphs: Theory, Algorithms and Applications. Springer.
  • Bender, Edward A.; Williamson, S. Gill (2010). Lists, Decisions and Graphs. With an Introduction to Probability.
  • Berge, Claude (1958). Théorie des graphes et ses applications (in French). Paris: Dunod.
  • Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge University Press. ISBN 978-0-521-45897-9.
  • Bollobás, Béla (2002). Modern Graph Theory (1st ed.). Springer. ISBN 978-0-387-98488-9.
  • Diestel, Reinhard (2005). Graph Theory (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-26183-4.
  • Graham, R.L.; Grötschel, M.; Lovász, L. (1995). Handbook of Combinatorics. MIT Press. ISBN 978-0-262-07169-7.
  • Gross, Jonathan L.; Yellen, Jay (1998). Graph Theory and Its Applications. CRC Press. ISBN 978-0-8493-3982-0.
  • Gross, Jonathan L.; Yellen, Jay (2003). Handbook of Graph Theory. CRC. ISBN 978-1-58488-090-5.
  • Harary, Frank (1995). Graph Theory. Addison Wesley Publishing Company. ISBN 978-0-201-41033-4.
  • Iyanaga, Shôkichi; Kawada, Yukiyosi (1977). Encyclopedic Dictionary of Mathematics. MIT Press. ISBN 978-0-262-09016-2.
  • Zwillinger, Daniel (2002). CRC Standard Mathematical Tables and Formulae (31st ed.). Chapman & Hall/CRC. ISBN 978-1-58488-291-6.

Further reading

External links

Bondage number

In mathematics, the bondage number of a nonempty graph is the cardinality of the smallest set E of edges such that the domination number of the graph with the edges E removed is strictly greater than the domination number of the original graph.

The concept was introduced by Fink et. al.

Cluster graph

In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs.

Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers.

Connected dominating set

In graph theory, a connected dominating set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph.

Core (graph theory)

In the mathematical field of graph theory, a core is a notion that describes behavior of a graph with respect to graph homomorphisms.


Gephi is an open-source network analysis and visualization software package written in Java on the NetBeans platform.

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 connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

Refer to the glossary of graph theory for basic definitions in graph theory.

Incidence poset

In mathematics, an incidence poset or incidence order is a type of partially ordered set that represents the incidence relation between vertices and edges of an undirected graph. The incidence poset of a graph G has an element for each vertex or edge in G; in this poset, there is an order relation x ≤ y if and only if either x = y or x is a vertex, y is an edge, and x is an endpoint of y.


Incident may refer to:

A property of a graph (discrete mathematics) (see also glossary of graph theory)

Incident (film), a 1948 film noir

Incident (festival), a cultural festival of The National Institute of Technology in Surathkal, Karnataka, India

Incident (Scientology), a concept in Scientology

Incident Ray, a ray of light that strikes a surface

Nuclear and radiation accidents and incidents, an irregularity with a nuclear installation not classified as a nuclear accident

Klam value

In the parameterized complexity of algorithms, the klam value of a parameterized algorithm is a number that bounds the parameter values for which the algorithm might reasonably be expected to be practical. An algorithm with a higher klam value can be used for a wider range of parameter values than another algorithm with a lower klam value. The klam value was first defined by Downey and Fellows (1999), and has since been used by other researchers in parameterized complexity both as a way of comparing different algorithms to each other and in order to set goals for future algorithmic improvements.

Logical matrix

A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets.

Meurs Challenger

Meurs Challenger is an online graph visualization program, with data analysis and browsing.The software supports several graph layout algorithms, and allows the user to interact with the nodes. The displayed data can be filtered using textual search, node and edge type, or based on the graph distance between nodes. Written in ActionScript, the program runs on Windows, Linux, macOS and other platforms that support the Adobe Flash Player.

Meurs Challenger was the winner at the 2011 edition of the International Symposium on Graph Drawing, in the large graph category.It is publicly available as a Facebook application, which displays the network graph of the user's friends.

Mind map

A mind map is a diagram used to visually organize information. A mind map is hierarchical and shows relationships among pieces of the whole. It is often created around a single concept, drawn as an image in the center of a blank page, to which associated representations of ideas such as images, words and parts of words are added. Major ideas are connected directly to the central concept, and other ideas branch out from those major ideas.

Mind maps can also be drawn by hand, either as "rough notes" during a lecture, meeting or planning session, for example, or as higher quality pictures when more time is available. Mind maps are considered to be a type of spider diagram. A similar concept in the 1970s was "idea sun bursting".

Order dimension

In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.

This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order.

Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992).

Paul A. Catlin

Paul Allen Catlin ((1948-06-25)June 25, 1948 – (1995-04-20)April 20, 1995) was a mathematician, professor of mathematics and Doctor of Mathematics, known for his valuable contributions to graph theory and number theory. He wrote one of the most cited papers in the series of chromatic numbers and Brooks' theorem, titled Hajós graph coloring conjecture: variations and counterexamples.


Pursuit-evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit-evasion, and the graph formulation discrete pursuit-evasion (also called graph searching). Current research is typically limited to one of these two formulations.


In graph theory, a subcoloring is an assignment of colors to a graph's vertices such that each color class induces a vertex disjoint union of cliques. That is, each color class should form a cluster graph.

The subchromatic number χS(G) of a graph G is the least number of colors needed in any subcoloring of G.

Subcoloring and subchromatic number were introduced by Albertson et al. (1989).

Every proper coloring and cocoloring of a graph are also subcolorings, so the subchromatic number of any graph is at most equal to the cochromatic number, which is at most equal to the chromatic number.

Subcoloring is as difficult to solve exactly as coloring, in the sense that (like coloring) it is NP-complete. More specifically,

the problem of determining whether a planar graph has subchromatic number at most 2 is NP-complete, even if it is a

triangle-free graph with maximum degree 4 (Gimbel & Hartman 2003) (Fiala et al. 2003),

comparability graph with maximum degree 4 (Ochem 2017),

line graph of a bipartite graph with maximum degree 4 (Gonçalves & Ochem 2009),

graph with girth 5 (Montassier & Ochem 2015).The subchromatic number of a cograph can be computed in polynomial time (Fiala et al. 2003). For every fixed integer r, it is possible to decide in polynomial time whether the subchromatic number of interval and permutation graphs is at most r (Broersma et al. 2002).

Three-dimensional graph

A three-dimensional graph may refer to

A graph (discrete mathematics), embedded into a three-dimensional space

The graph of a function of two variables, embedded into a three-dimensional space

Well-colored graph

In graph theory, a subfield of mathematics, a well-colored graph is an undirected graph for which greedy coloring uses the same number of colors regardless of the order in which colors are chosen for its vertices. That is, for these graphs, the chromatic number (minimum number of colors) and Grundy number (maximum number of greedily-chosen colors) are equal.


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