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Encyclopedia > Loop (graph theory)
A graph with 6 vertices (nodes) and 7 edges.
A graph with 6 vertices (nodes) and 7 edges.

In mathematics and computer science a graph is the basic object of study in graph theory. Informally, a graph is a set of objects called vertices connected by links called edges. Typically, a graph is depicted as a set of dots (vertices, nodes) connected by lines (the edges). Depending on the application some edges can be directed. simple:Image:6n-graf. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Computer science (academically, CS, CSC or compsci) encompasses a variety of topics that relates to computation, like abstract analysis of algorithms, formal grammars, and subjects such as programming languages, program design, software and computer hardware. ... In mathematics and computer science, graph theory studies the properties of graphs. ...

Contents

Definitions

Definitions in graph theory vary in the literature. Here are the conventions used in this encyclopedia.


Undirected graph

An undirected graph or graph G is an ordered pair G:=(V, E) with An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

  • V a set of vertices or nodes,
  • E a set of unordered pairs of distinct vertices, called edges or lines.

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. This article is about sets in mathematics. ...


Directed graph

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

  • V a set of vertices or nodes,
  • A a set of ordered pairs of vertices called directed edges, arcs or arrows.

A variation on this definition is the oriented graph, which is a simple graph with an orientation assigned to all of its edges. The distinction between a directed graph and an oriented graph is that if x and y are verticies, a directed graph allows both (x, y) and (y, x) as edges, while only one is permitted in an oriented graph. This article is about sets in mathematics. ...


A directed graph may or may not be allowed to have loops, that is, edges where the start and end vertices are the same. By definition, this is forbidden in an oriented graph.


Mixed graph

A mixed graph G is a 3-tuple G:=(V,E,A) with V, E and A defined as above.


Variations in the definitions

As defined above, edges of undirected graphs have distinct ends, and E and A are sets (with distinct elements as sets always do). Many applications require more general possibilities, but terminology varies. A loop is an edge (directed or undirected) with both ends the same; these may be permitted or not permitted according to the application. Sometimes E or A are allowed to be multisets, so that there can be more than one edge between the same two vertices. The unqualified word "graph" might allow or disallow multiple edges in the literature, according to the preferences of the author. If it is intended to exclude multiple edges (and, in the undirected case, to exclude loops), the graph can be called simple. On the other hand, if it is intended to allow multiple edges, the graph can be called a multigraph. Sometimes the word pseudograph is used to indicate that both multiple edges and loops are allowed. This article is about sets in mathematics. ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a cardinal number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ...


Further definitions

For more definitions see Glossary of graph theory.

Two edges of a graph are called adjacent (sometimes coincident) if they share a common vertex. Similarly, two vertices are called adjacent if they share a common edge, that is they are connected by an edge. An edge and a vertex on that edge are called incident. One major problem that has plagued graph theory since its inception is the consistent lack of consistency in terminology. ...


The graph with only one vertex and no edges is the trivial graph. A graph with only vertices and no edges is known as an empty graph; the graph with no vertices and no edges is the null graph, but not all mathematicians allow this concept.


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 route problems such as the traveling salesman problem. The traveling salesman problem (TSP), also known as the traveling salesperson problem, is a problem in discrete or combinatorial optimization. ...


Normally, the vertices of a graph by their nature are undistinguishable. (Of course, they may be distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges). Some branches of graph theory require to uniquely identify vertices. If each vertex is given a label, then the graph is said to be a vertex-labeled graph, whereas 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 without labels are called unlabelled.


Examples

Image:6n-graf.png
The picture is a graphic representation of the following graph simple:Image:6n-graf. ...

  • V:={1,2,3,4,5,6}
  • E:={{1,2},{1,5},{2,3},{2,5},{3,4},{4,5},{4,6}}

Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In category theory, a functor is a special type of mapping between categories. ... Computer science (academically, CS, CSC or compsci) encompasses a variety of topics that relates to computation, like abstract analysis of algorithms, formal grammars, and subjects such as programming languages, program design, software and computer hardware. ... Fig. ...

Important graphs

In graph theory, a cograph, or P4-free graph, is a graph that fulfills the following equivalent properties: Can be constructed from isolated vertices by complement, joint union and disjoint union. ... In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. ... A tree with 6 vertices and 5 edges In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. ... In the mathematical field of graph theory, a bipartite graph is a special graph where the set of vertices can be divided into two disjunct sets with two vertices of the same set never sharing an edge. ... In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the clique number of that subgraph. ... In graph theory, a planar graph is a graph that can be embedded in a plane so that no edges intersect. ... In graph theory, the line graph L(G) of a graph G is a graph such that each vertex of L(G) represents an edge of G; and any two vertices of L(G) are adjacent if and only if their corresponding edges are incident, meaning they share a common...

Generalizations

In a hypergraph, an edge can connect more than two vertices. In mathematics, the concept of hypergraph generalizes the notion of a graph. ...


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. In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... simplex refers to a one-way communications channel. ...


Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs. In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence that generalizes linear independence in vector spaces. ...


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 mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...


  Results from FactBites:
 
Quantum Field Theory Resources (5737 words)
Quantum field theory is the basis of the Standard Model of particle physics and is the best tested of all physical theories, more general in application and better tested within its range of application than the existing formulation of general relativity (which needs modification to include quantum field effects), describing all electromagnetic and nuclear phenomena.
This approach, 'loop quantum gravity', is entirely different from that in string theory, which is based on building extra-dimensional speculation upon other speculations, e.g., the speculation that gravity is due to spin-2 gravitons (this is speculative is no experimental evidence for it).
In loop quantum gravity, by contrast to string theory, the aim is merely to use quantum field theory to derive the framework of general relativity as simply as possible.
Glossary of graph theory - Wikipedia, the free encyclopedia (5364 words)
Likewise, a graph G is said to be homomorphic to a graph H if there is a mapping, called homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then their corresponding vertices are adjacent in H.
A vertex of degree 0 is an isolated vertex.
In computers, a finite, directed or undirected graph (with n vertices, say) is often represented by its adjacency matrix : an n -by- n matrix whose entry in row i and column j gives the number of edges from the i -th to the j -th vertex.
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