Graph (mathematics)

A drawing of a labeled graph on 6 vertices and 7 edges.

In mathematics and computer science, a graph is the basic object of study in graph theory. Informally speaking, a graph is a set of objects called points, nodes, or vertices connected by links called lines or edges. In a proper graph, which is by default undirected, a line from point A to point B is considered to be the same thing as a line from point B to point A. In a digraph, short for directed graph, the two directions are counted as being distinct arcs or directed edges. Typically, a graph is depicted in diagrammatic form as a set of dots (for the points, vertices, or nodes), joined by curves (for the lines or edges).

Definitions

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

Graph

A graph or undirected graph Failed to parse (Missing texvc executable; please see math/README to configure.): G

is an ordered pair Failed to parse (Missing texvc executable; please see math/README to configure.): G := (V, E)
that is subject to the following conditions:

• Failed to parse (Missing texvc executable; please see math/README to configure.): V

is a set, whose elements are called vertices or nodes,

• Failed to parse (Missing texvc executable; please see math/README to configure.): E

is a set of pairs (unordered) of distinct vertices, called edges or lines.

The vertices belonging to an edge are called the ends, endpoints, or end vertices of the edge.

Failed to parse (Missing texvc executable; please see math/README to configure.): V

(and hence Failed to parse (Missing texvc executable; please see math/README to configure.): E

) are usually taken to be finite sets, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is Failed to parse (Missing texvc executable; please see math/README to configure.): |V|

(the number of vertices). A graph's size is Failed to parse (Missing texvc executable; please see math/README to configure.): |E|

, the number of edges. The degree of a vertex is the number of other vertices it is connected to by edges.

The edge set Failed to parse (Missing texvc executable; please see math/README to configure.): E

induces a symmetric binary relation ~ on Failed to parse (Missing texvc executable; please see math/README to configure.): V
that is called the adjacency relation of Failed to parse (Missing texvc executable; please see math/README to configure.): G

. Specifically, for each edge {u,v} the vertices u and v are said to be adjacent to one another, which is denoted u ~ v.

For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv.

Types of graphs

Directed graph

A directed graph or digraph Failed to parse (Missing texvc executable; please see math/README to configure.): G

is an ordered pair Failed to parse (Missing texvc executable; please see math/README to configure.): G := (V, A)
with

• Failed to parse (Missing texvc executable; please see math/README to configure.): V

is a set, whose elements are called vertices or nodes,

• Failed to parse (Missing texvc executable; please see math/README to configure.): A

is a set of ordered pairs of vertices, called directed edges, arcs, or arrows.

An arc Failed to parse (Missing texvc executable; please see math/README to configure.): e = (x, y)

is considered to be directed from Failed to parse (Missing texvc executable; please see math/README to configure.): x
to Failed to parse (Missing texvc executable; please see math/README to configure.): y

Failed to parse (Missing texvc executable; please see math/README to configure.): y

is called the head and Failed to parse (Missing texvc executable; please see math/README to configure.): x
is called the tail of the arc; Failed to parse (Missing texvc executable; please see math/README to configure.): y
is said to be a direct successor of Failed to parse (Missing texvc executable; please see math/README to configure.): x

, and Failed to parse (Missing texvc executable; please see math/README to configure.): x

is said to be a direct predecessor of Failed to parse (Missing texvc executable; please see math/README to configure.): y

. If a path leads from Failed to parse (Missing texvc executable; please see math/README to configure.): x

to Failed to parse (Missing texvc executable; please see math/README to configure.): y

, then Failed to parse (Missing texvc executable; please see math/README to configure.): y

is said to be a successor of Failed to parse (Missing texvc executable; please see math/README to configure.): x

, and Failed to parse (Missing texvc executable; please see math/README to configure.): x

is said to be a predecessor of Failed to parse (Missing texvc executable; please see math/README to configure.): y

. The arc Failed to parse (Missing texvc executable; please see math/README to configure.): (y, x)

is called the arc Failed to parse (Missing texvc executable; please see math/README to configure.): (x, y)
inverted.

A directed graph is called symmetric if every arc belongs to it together with the corresponding inverted arc. A symmetric loopless directed graph is equivalent to an undirected graph with the pairs of inverted arcs replaced with edges; thus the number of edges is equal to the number of arcs halved.

A variation on this definition is the oriented graph, which is a graph (or multigraph; see below) with an orientation or direction assigned to each of its edges. A distinction between a directed graph and an oriented simple graph is that if Failed to parse (Missing texvc executable; please see math/README to configure.): x

and Failed to parse (Missing texvc executable; please see math/README to configure.): y
are vertices, a directed graph allows both Failed to parse (Missing texvc executable; please see math/README to configure.): (x, y)
and Failed to parse (Missing texvc executable; please see math/README to configure.): (y, x)
as edges, while only one is permitted in an oriented graph. A more fundamental difference is that, in a directed graph (or multigraph), the directions are fixed, but in an oriented graph (or multigraph), only the underlying graph is fixed, while the orientation may vary.

A directed acyclic graph, occasionally called a dag or DAG, is a directed graph with no directed cycles.

In the theory of Lie groups, a quiver Q is a directed graph serving as the domain of, and thus characterizing the shape of, a representation V defined as a functor, specifically an object of the functor category FinVct_K^F(Q) where F(Q) is the free category on Q consisting of paths in Q and FinVct_K is the category of finite dimensional vector spaces over a field K. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with linear transformations between them, and transform via natural transformations.

Undirected graph

A graph G = {V,E} in which every edge is undirected. This is the same as a digraph (look above) where for an edge (v,u) there is an edge from v to u and u to v.

Finite graph

A finite graph is a graph G = <V,E> such that V(G) and E(G) are finite sets.

Simple graph

A simple graph is an undirected graph that has no self-loops.

Regular graph

A regular graph is a graph where each vertex has the same number of neighbors, i.e. every vertex has the same degree or valency. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.

Weighted graph

A graph is a weighted graph if a positive number (weight) is assigned to each edge. Such weights might represent, for example, costs, lengths or capacities, etc. depending on the problem.

Weight of the graph is sum of the weight given to all edges.

Mixed graph

A mixed graph G is a graph in which some edges may be directed and some may be undirected. It is written as an ordered triple G := (V, E, A) with V, E, and A defined as above. Directed and undirected graphs are special cases.

Complete graph

Main article: Complete graph

Complete graphs have the feature that each pair of vertices has an edge connecting them.

Variations in the definitions

As defined above, edges of undirected graphs have two distinct ends, and E and A are sets (with distinct elements, like all sets). Many applications require more general possibilities, but terminology varies.

Loop

A loop is an edge (directed or undirected) which starts and ends on the same vertex; these may be permitted or not permitted according to the application. In this context, an edge with two different ends is called a link.

Multiset

Sometimes Failed to parse (Missing texvc executable; please see math/README to configure.): E

and Failed to parse (Missing texvc executable; please see math/README to configure.): A
are allowed to be multisets, so that there can be more than one edge (called multiple edges) between the same two vertices. Another way to allow multiple edges is to make Failed to parse (Missing texvc executable; please see math/README to configure.): E
a set, independent of Failed to parse (Missing texvc executable; please see math/README to configure.): V

, and to specify the endpoints of an edge by an incidence relation between Failed to parse (Missing texvc executable; please see math/README to configure.): V

and Failed to parse (Missing texvc executable; please see math/README to configure.): E

. The same applies to a directed edge set Failed to parse (Missing texvc executable; please see math/README to configure.): A , except that there must be two incidence relations, one for the head and one for the tail of each edge.

Multi graph

The term "multigraph" is generally understood to mean that multiple edges (and sometimes loops) are allowed. Where graphs are defined so as to allow loops and multiple edges, a multigraph is often defined to mean a graph without loops,^[1] however, where graphs are defined so as to disallow loops and multiple edges, the term is often defined to mean a "graph" which can have both multiple edges and loops,^[2] although many use the term "pseudograph" for this meaning.^[3]

Half-edges, loose edges

In exceptional situations it is even necessary to have edges with only one end, called half-edges, or no ends (loose edges); see for example signed graphs and biased graphs.

Properties of graphs

For more definitions see Glossary of graph theory.

Two edges of a graph are called adjacent (sometimes coincident) if they share a common vertex. Two arrows of a directed graph are called consecutive if the head of the first one is at the nock (notch end) of the second one. Similarly, two vertices are called adjacent if they share a common edge (consecutive if they are at the notch and at the head of an arrow), 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 not all mathematicians allow this object.

In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal routing problems such as the traveling salesman problem.

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; then the graph may be called unlabeled. (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 that graphs which have labeled edges are called edge-labeled graphs. 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.)

Examples

A graph with six nodes.

The picture is a graphic representation of the following graph

• Failed to parse (Missing texvc executable; please see math/README to configure.): V := \{1, 2, 3, 4, 5, 6\}

• Failed to parse (Missing texvc executable; please see math/README to configure.): E := \{\{1, 2\}, \{1, 5\}, \{2, 3\}, \{2, 5\}, \{3, 4\}, \{4, 5\}, \{4, 6\}\}

The fact that vertex 1 is adjacent to vertex 2 is sometimes denoted by 1 ~ 2.

• In category theory a category can be considered a directed multigraph with the objects as vertices and the morphisms as directed edges. The functors between categories induce then some, but not necessarily all, of the digraph morphisms.
• In computer science directed graphs are used to represent finite state machines and many other discrete structures.
• A binary relation Failed to parse (Missing texvc executable; please see math/README to configure.): R

on a set Failed to parse (Missing texvc executable; please see math/README to configure.): X
is a directed graph. Two edges Failed to parse (Missing texvc executable; please see math/README to configure.): x

, Failed to parse (Missing texvc executable; please see math/README to configure.): y

of Failed to parse (Missing texvc executable; please see math/README to configure.): X
are connected by an arrow if Failed to parse (Missing texvc executable; please see math/README to configure.): xRy

.

Important graphs

Basic examples are:

• In a complete graph each pair of vertices is joined by an edge, that is, the graph contains all possible edges.
• In a complete bipartite graph, the vertex set is the union of two disjoint subsets, 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.
• In a bipartite graph, the vertices can be divided into two sets, W and X, so that every edge has one vertex in each of the two sets.
• In a path of length n, the vertices can be listed in order, v₀, v₁, ..., v_n, so that the edges are v_i−1v_i for each i = 1, 2, ..., n.
• A cycle or circuit of length n is a closed path without self-intersections; equivalently, it is a connected graph with degree 2 at every vertex. Its vertices can be named v₁, ..., v_n so that the edges are v_i−1v_i for each i = 2,...,n and v_nv₁
• A planar graph can be drawn in a plane with no crossing edges (i.e., 'embedded in a plane).
• A forest is a graph with no cycles.
• A tree is a connected graph with no cycles.

More advanced kinds of graphs are:

• The Petersen graph and its generalizations
• Perfect graphs
• Cographs
• Other graphs with large automorphism groups: vertex-transitive, arc-transitive, and distance-transitive graphs.
• Strongly regular graphs and their generalization distance-regular graphs.

Operations on graphs

Main article: Operations on graphs

There are several operations that produce new graphs from old ones. They may be separated into three categories

• Elementary operations, sometimes called "editing operations" on graphs, which create a new graph from the original one by a simple, local change, such as addition or deletion of a vertex or an edge, merging and splitting of vertices, etc.
• Unary operations, which create a significantly new graph from the old one. Examples:
  • Line graph
  • Dual graph
  • Complement graph
• Binary operations, which create new graph from two initial graphs. Examples:
  • Disjoint union of graphs
  • Cartesian product of graphs
  • Tensor product of graphs
  • Strong product of graphs
  • Lexicographic product of graphs

Generalizations

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.

Notes

References

• Balakrishnan, V. K., Graph Theory, McGraw-Hill; 1st edition (February 1, 1997). ISBN 0-07-005489-4.
• Bollobas, Bela, Modern Graph Theory, Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7.
• Bang-Jensen, J. and Gutin, G., "Digraphs: Theory, Algorithms and Applications", Springer, 2000, free at www.cs.rhul.ac.uk/books/dbook/
• Diestel, Reinhard, Graph Theory, Springer; 2nd edition (February 18, 2000). ISBN 0-387-98976-5.
• Gross, Jonathan L., and Yellen, Jay, Graph Theory and Its Applications, CRC Press (December 30, 1998). ISBN 0-8493-3982-0.
• Gross, Jonathan L., and Yellen, Jay (eds.), Handbook of Graph Theory. CRC (December 29, 2003). ISBN 1-58488-090-2.
• Harary, Frank, Graph Theory, Addison Wesley Publishing Company (January 1995). ISBN 0-201-41033-8.
• Zwillinger, Daniel, CRC Standard Mathematical Tables and Formulae, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3.

See also

External links

• Graph theory tutorial
• Some graph theory algorithm animations
  • Step through the algorithm to understand it.
• The compendium of algorithm visualisation sites
• Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring
• Image gallery : Some real-life graphs
• VisualComplexity.com - A visual exploration on mapping complex networks
• Grafos Spanish copyleft software
• Edge Addition Planarity Algorithm — Online version of a paper that describes the Boyer-Myrvold planarity algorithm.
• Edge Addition Planarity Algorithm Source Code — Free C source code for reference implementation of Boyer-Myrvold planarity algorithm, which provides both a combinatorial planar embedder and Kuratowski subgraph isolator.
• Library of Efficient Models and Optimization in Networks — It is an open source C++ template library aimed at combinatorial optimization tasks, especially those working with graphs and networks.
• Eric W. Weisstein, Graph at MathWorld.
• TORSCHE Scheduling Toolbox for Matlab is a freely available toolbox of scheduling and graph algorithms.bg:Граф (математика)

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