Graph (mathematics)
{{dablink|This article presents the essential definitions. For a more complete account see
graph theory. For an alphabetical reference of graph theory terms see
List of graph theory terms. For the graph of a function, see
graph of a function.}}Image:6n-graf.svg|thumb|250px|A
drawing of a labeled graph on 6 vertices and 7 edges.]]In
mathematics, a
graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called
vertices, and the links that connect some pairs of vertices are called
edges. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. For example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge
between two vertices if and only if those two integers have at least one decimal digit in common.In other cases the relationship between vertices is not symmetric: for example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge
from i to j if
i is a divisor of
j. This type of graph is called a
directed graph and the edges are called
directed edges or
arcs; in contrast, a graph where the edges are not directed is called
undirected.Vertices are also called
nodes or
points, and edges are also called
lines. Graphs are the basic subject studied by
graph theory.
Definitions
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.
Graph
Image:Multigraph.svg|thumb|125px|A general example of a graph (actually, a
pseudographpseudographIn the most common sense of the term,
(1)a
graph is an
ordered pair G := (V E)
comprising a
set V
of
vertices or
nodes together with a set
E
of
edges or
lines, which are 2-element subsets of
V
. To avoid ambiguity, this type of graph may be described precisely as
undirected and
simple'''.Other senses of
graph stem from different conceptions of the edge set. In one more generalized notion,
(2) E
is a set together with a relation of
incidence that associates with each edge two vertices. In another generalized notion,
E
is a
multiset of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a
multigraph or
pseudograph.All of these variants and others are described more fully below.The vertices belonging to an edge are called the
ends,
endpoints, or
end vertices of the edge. A vertex may exist in a graph and not belong to an edge.
V
and
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. 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.The edges
E
of an undirected graph
G
induce 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.{{-}}
Types of graphs
Distinction in terms of the main definition
As stated above, in different contexts it may be useful to define the term
graph with different degrees of generality. Whenever it is necessary to draw a strict distinction, the following terms are used. Most commonly, in modern texts in graph theory, unless stated otherwise,
graph means "undirected simple finite graph" (see the definitions below).
Undirected graph
A graph in which edges have no orientation, i.e., they are not ordered pairs, but sets {
u,
v} (or 2-multisets) of vertices.
Directed graph
125px|thumb|A directed graph.A
directed graph or
digraph is an ordered pair
D := (V A)
with
-
V
a set whose elements are called vertices or nodes, and
-
A
a set of ordered pairs of vertices, called arcs, directed edges, or arrows.
An arc
a = (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
reachable from
x
, and
x
is said to be a
predecessor of
y
. The arc
is called the arc
(x y) ∈||ed.A directed gra(h D is cal≤d symmetric if for every arc ∈ D the corres(ond∈g ∈||ed arc also belongs → D. A symmetric loo(≤ss directed gra(h D = (V A) is ≡a≤nt → a si&(lusmn;≤ undirected gra(h G = (V E) where the (airs of ∈verse arcs ∈ A corres(ond 1-→-1 with the ed≥s ∈ E; thus the ed≥s ∈ G νmber ||E|| = ||A||/2 or half the νmber of arcs ∈ D.A variation on this def∈ition is the oriented gra(h ∈ which not more than o≠ of
and
(y x)
may be 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.
Multigraph
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.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,
(3) 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,
(4) although many use the term "
pseudograph" for this meaning.
(5)Simple graph
thumb|125px|A simple graph with three vertices and three edges. Each vertex has degree two, so this is also a regular graph.As opposed to a multigraph, a simple graph is an undirected graph that has no
loops and no more than one edge between any two different vertices. 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
n vertices every vertex has a degree that is less than
n (the converse, however, is not true - there exist non-simple graphs with
n vertices in which every vertex has a degree smaller than
n).
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.The weight of the graph is sum of the weights given to all edges.
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.
Important graph classes
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.
Complete graph
Complete graphs have the feature that each pair of vertices has an edge connecting them.
Finite and infinite graphs
A finite graph is a graph
G = <
V,
E> such that
V(
G) and
E(
G) are
finite sets. An infinite graph is the one with sets of vertices or edges or both
infinite.Most commonly in graph theory it is implied that the discussed graphs are finite, i.e., finite graphs are called simply "graphs", while the infinite graphs are called so in full.
Graph classes in terms of connectivity
In an
undirected graph G, two
vertices u and
v are called
connected if
G contains a
path from
u to
v. Otherwise, they are called
disconnected. A graph is called
connected if every pair of distinct vertices in the graph is connected and
disconnected otherwise. A graph is called
k-vertex-connected or k-edge-connected if removal of
k or more vertices (respectively, edges) makes the graph disconnected. A
k-vertex-connected graph is often called simply
k-connected.A
directed graph is called
weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is
strongly connected or
strong if it contains a directed path from
u to
v and a directed path from
v to
u for every pair of vertices
u,
v.
Properties of graphs
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
thumb|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:
- In a complete graph each pair of vertices is joined by an edge, that is, the graph contains all possible edges.
- 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 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 path of length n, the vertices can be listed in order, v0, v1, ..., vn, so that the edges are vi−1vi 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, ..., vn so that the edges are vi−1vi for each i = 2,...,n and vnv1
- A planar graph can be drawn in a plane with no crossing edges (i.e., embedded in a plane).
- A tree is a connected graph with no cycles.
- A forest is a graph with no cycles (i.e. one or more trees).
More advanced kinds of graphs are:
Operations on graphs
There are several operations that produce new graphs from old ones, which might be classified into the following 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.
- Graph rewrite operations replacing the occurrence of some pattern graph within the host graph by an instance of the corresponding replacement graph.
- Unary operations, which create a significantly new graph from the old one. Examples:
- Binary operations, which create new graph from two initial graphs. Examples:
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.In
computational biology,
power graph analysis introduces power graphs as an alternative representation of undirected graphs.
See also
Notes
-
[See, for instance, Iyanaga and Kawada, 69 J, p. 234.]
-
[See, for instance, Graham et al., p. 5.]
-
[ For example, see Balakrishnan, p. 1, Gross (2003), p. 4, and Zwillinger, p. 220.]
-
[For example, see. Bollobas, p. 7 and Diestel, p. 25.]
-
[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.
- Berge, C., Théorie des graphes et ses applications. Collection Universitaire de Mathématiques, II Dunod, Paris 1958, viii+277 pp. (English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first English edition. Dover, New York 2001)
- 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/
- {{Citation | last1=Diestel | first1=Reinhard | title=Graph Theory | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/GraphTheoryIII.pdf | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 }}.
- Graham, R.L., Grötschel, M., and Lovász, L (eds.), Handbood of Combinatorics, MIT Press, 1995. ISBN 0-262-07169-X.
- 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.
- Iyanaga, Shôkichi and Kawada, Yukiyosi, Encyclopedic Dictionary of Mathematics, MIT Press, 1977. ISBN 0-262-09016-3.
- Zwillinger, Daniel, CRC Standard Mathematical Tables and Formulae, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3.
External links
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