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This article presents the essential definitions. For a more complete account see graph theory. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects For the graph of a function, see graph of a function. In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)
A drawing of a labeled graph on 6 vertices and 7 edges.
A drawing of a labeled graph on 6 vertices and 7 edges. Graph drawing, as a branch of Graph theory, applies Topology and Geometry to derive two-dimensional representations of graphs Graph drawing is

In mathematics and computer science, a graph is the basic object of study in graph theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects 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).

Contents

Definitions

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

Graph

A general example of a graph with three vertices and six edges.
A general example of a graph with three vertices and six edges.

A graph or undirected graph G is an ordered pair G: = (V,E) that is subject to the following conditions:

  • V is a set, whose elements are called vertices or nodes,
  • E is a multiset of unordered pairs of vertices (not necessarily distinct), called edges or lines. In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry In Mathematics, a multiset (or bag) is a generalization of a set.

(Note that this defines the most general type of graph. Some authors call this a multigraph and reserve the term "graph" for simple graphs. A multigraph or pseudograph is a graph which is permitted to have Multiple edges, (also called "parallel edges" that is edges that have the In Mathematics and Computer science, a graph is the basic object of study in Graph theory. )

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

V (and hence E) are usually taken to be finite, 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. Graph theory is a growing area in mathematical research and has a large specialized vocabulary The order of a graph is | V | (the number of vertices). A graph's size is | E | , the number of edges. The degree of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends (a loop) is counted twice. In Graph theory, a loop (also called a self-loop) is an edge that connects a vertex to itself

The edges E induce a symmetric binary relation ~ on V which is called the adjacency relation of 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.
A directed graph.

A directed graph or digraph G is an ordered pair G: = (V,A) with

An arc e = (x,y) is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y is said to be a direct successor of x, and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x, and x is said to be a predecessor of y. In Graph theory, a path in a graph is a Sequence of vertices such that from each of its vertices there is an edge to the next vertex The arc (y,x) is called the arc (x,y) inverted.

A directed graph G is called symmetric if, for every arc that belongs to G, the corresponding inverted arc also belongs to G. 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 x and y are vertices, a directed graph allows both (x,y) and (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 Computer science and Mathematics, a directed acyclic graph, also called a DAG, is a with no; that is for any vertex v, there In Graph theory, a cycle graph is a graph that consists of a single cycle, or in other words some number of vertices connected in a closed chain

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 FinVctKF(Q) where F(Q) is the free category on Q consisting of paths in Q and FinVctK is the category of finite dimensional vector spaces over a field K. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a quiver is a Directed graph where loops and multiple arrows between two vertices are allowed In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, the Functors between two given categories can themselves be turned into a category the morphisms in this functor In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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 with three vertices and three edges. Each vertex has degree two, so this is also a regular graph.
A simple graph with three vertices and three edges. Each vertex has degree two, so this is also a regular graph.

A simple graph is an undirected graph that has no self-loops and no more than one edge between any two different vertices. In Graph theory, a loop (also called a self-loop) is an edge that connects a vertex to itself In a simple graph the edges of the graph form a set (rather than a multiset) and each edge is a pair of distinct vertices. In a simple graph with p vertices every vertex has a degree that is less than p.

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 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 weights 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. In the mathematical field of Graph theory, a complete graph is a Simple graph in which every pair of distinct vertices is connected by an

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 Graph theory, a loop (also called a self-loop) is an edge that connects a vertex to itself In this context, an edge with two different ends is called a link.

Multi graph

The term "multigraph" is generally understood to mean that multiple edges (and sometimes loops) are allowed. A multigraph or pseudograph is a graph which is permitted to have Multiple edges, (also called "parallel edges" that is edges that have the 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. A multigraph or pseudograph is a graph which is permitted to have Multiple edges, (also called "parallel edges" that is edges that have the [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. In the area of Graph theory in Mathematics, a signed graph is a graph in which each edge has a positive or negative sign In Mathematics, a biased graph is a graph with a list of distinguished circles (edge sets of Simple cycles, such that if two circles in the list are contained

Properties of graphs

For more definitions see Glossary of graph theory. Graph theory is a growing area in mathematical research and has a large specialized vocabulary

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. The Travelling salesman problem ( TSP) in Operations research is a problem in discrete or Combinatorial optimization.

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.
A graph with six nodes.

The picture is a graphic representation of the following graph

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

Important graphs

Basic examples are:

More advanced kinds of graphs are:

Operations on graphs

Main article: Operations on graphs

There are several operations that produce new graphs from old ones. Operations on graphs produce new graphs from old ones They may be separated into the following major categories They may be separated into three categories

Generalizations

In a hypergraph, an edge can join more than two vertices. Operations on graphs produce new graphs from old ones They may be separated into the following major categories In Graph theory, the cartesian product G ◻ H of graphs G and H is a graph such that the vertex set of G In Graph theory, the tensor product G × H of graphs G and H is a graph such that the vertex set of G × A graph product is a Binary operation on graphs There are several similarly defined Operations on graphs, called "product" In Graph theory, the lexicographic product or graph composition G ∙ H of graphs G and H is a graph such that In Mathematics, a hypergraph is a generalization of a graph, where edges can connect any number of vertices.

An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle 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 ( or independence structure is a structure that captures the essence of a notion of "independence"

In model theory, a graph is just a structure. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models In Universal algebra and in Model theory, a structure consists of an underlying set along with a collection of Finitary functions and relations But in that case, there is no limitation on the number of edges: it can be any cardinal number. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

Notes

  1. ^ For example, see Balakrishnan, p. 1, Gross (2003), p. 4, and Zwillinger, p. 220.
  2. ^ For example, see. Bollobas, p. 7 and Diestel, p. 25.
  3. ^ Gross (1998), p. 3, Gross (2003), p. 205, Harary, p. 10, and Zwillinger, p. 220.

References

See also

External links

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