GRAPH (MATHEMATICS)

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).

Contents

Definitions

Graph

Directed graph

Mixed graph

Variations in the definitions

Loop

Multiset

Multigraph

Halfedge

Properties of graphs

Examples

Important graphs

Operations on graphs

Generalizations

Notes

References

See also

External links

Definitions

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

Graph

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 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.
$V$ (and hence $E$) 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 $|V|$ (the number of vertices). A graph's 'size' is $|E|$, the number of edges. The 'degree' of a vertex is the number of other vertices it is connected to by edges.
The edge set $E$ induces a symmetric binary relation ~ on $V$ that 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''.

Directed graph

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

★ $V$ is a set, whose elements are called 'vertices' or 'nodes',

★ $A$ is a set of ordered pairs of vertices, called 'directed edges', 'arcs', or 'arrows'.
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 $y$. The arc $(y,\; x)$ is called the arc $(x,\; y)$ '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 $x$ 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.
A quiver is simply a directed graph, but the context is different. When discussing quivers emphasis is placed on representations of the graph where vector spaces are attached to the vertices and linear transformations are attached to the arcs.

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.

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) with both ends the same; 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 $E$ and $A$ 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 $E$ a set, independent of $V$, and to specify the endpoints of an edge by an incidence relation between $V$ and $E$. The same applies to a directed edge set $A$, except that there must be two incidence relations, one for the head and one for the tail of each edge.

Multigraph

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]

Halfedge

In exceptional situations it is even necessary to have edges with only one end, called 'halfedges', or no ends ('loose edges'); see for example signed 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 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 ''unlabelled''. (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

The picture is a graphic representation of the following graph

★ $V\; :=\; \{1,\; 2,\; 3,\; 4,\; 5,\; 6\}$

★ $E\; :=\; \{\{1,\; 2\},\; \{1,\; 5\},\; \{2,\; 3\},\; \{2,\; 5\},\; \{3,\; 4\},\; \{4,\; 5\},\; \{4,\; 6\}\}$
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 $R$ on a set $X$ is a directed graph. Two edges $x$, $y$ of $X$ are connected by an arrow if $xRy$..............................

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, ''v''₀, ''v''₁, ..., ''v''_n, so that the edges are ''v''_i−1''v''_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 ''v''₁, ..., ''v''_n so that the edges are ''v''_i−1''v''_''i'' for each ''i'' = 2,...,''n'' and ''v''_n''v''₁

★ 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

★ Pultasek graphs pultasek

★ 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 articles: 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:

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★ Line graph

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★ Dual graph

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★ Complement graph

★ Binary operations, which create new graph from two initial graphs. Examples:

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★ Disjoint union of graphs

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★ Cartesian product of graphs

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★ Tensor product of graphs

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★ Strong product of graphs

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★ 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

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

★ 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.

★ 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

★ Dual graph

★ Glossary of graph theory

★ Graph (data structure)

★ Graph drawing

★ Linked list

★ Important publications in graph theory

★ List of graph theory topics

★ Network theory

★ Polygon

★ Quantum graph

★ Tessellation

External links

★ Graph Theory online textbook

★ Graph theory tutorial

★ Some graph theory algorithm animations

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★ ''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.

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