End (graph theory)
In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit-evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.
Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
Definition and characterization
Ends of graphs were defined by Rudolf Halin (1964) in terms of equivalence classes of infinite paths. A ray in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices v0, v1, v2, ... in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays r0 and r1 are equivalent if there is another ray r2 (not necessarily different from either of the first two rays) that contains infinitely many of the vertices in each of r0 and r1. This is an equivalence relation: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be transitive. Therefore, it partitions the set of all rays into equivalence classes, and Halin defined an end as one of these equivalence classes.
Graph
Graph may refer to:
In information science:
In mathematics:
- Graph theory, the study of such graphs
In computer science:
Other uses:
See also
Facebook Graph Search
Facebook Graph Search was a semantic search engine that was introduced by Facebook in March 2013. It was designed to give answers to user natural language queries rather than a list of links. The Graph Search feature combined the big data acquired from its over one billion users and external data into a search engine providing user-specific search results. In a presentation headed by Facebook CEO Mark Zuckerberg, it was announced that the Graph Search algorithm finds information from within a user's network of friends. Additional results were provided by Microsoft's Bing search engine. In July it was made available to all users using the U.S. English version of Facebook. In December 2014, Facebook changed its search features, dropping partnership with Bing, and eliminating most of the search patterns.
Development
The feature was developed under former Google employees Lars Rasmussen and Tom Stocky.
The Graph Search Features was launched in Beta January 2013 as a limited preview for some English users in the United States. Company reports indicate that the service launched to between tens and hundreds of thousands of users. As of August 2013, Graph Search is available to all English users in the US.
Graph (discrete mathematics)
In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices (also called nodes or points), and the links that connect some pairs of vertices are called edges (also called arcs or lines). Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.
The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this is an undirected graph, because if person A shook hands with person B, then person B also shook hands with person A. In contrast, if there is an edge from person A to person B when person A knows of person B, then this graph is directed, because knowledge of someone is not necessarily a symmetric relation (that is, one person knowing another person does not necessarily imply the reverse; for example, many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans). The former type of graph is called an undirected graph and the edges are called undirected edges while the latter type of graph is called a directed graph and the edges are called directed edges.
End
End or Ending may refer to:
End
END
Ending
See also
Conclusion (music)
In music, the conclusion is the ending of a composition and may take the form of a coda or outro.
Pieces using sonata form typically use the recapitulation to conclude a piece, providing closure through the repetition of thematic material from the exposition in the tonic key. In all musical forms other techniques include "altogether unexpected digressions just as a work is drawing to its close, followed by a return...to a consequently more emphatic confirmation of the structural relations implied in the body of the work."
For example:
End (film)
End is a 1984 fiction film by Mahmoud Shoolizadeh; is a story about the children who live near the railways and their lives are full of ups and downs. This film displays the story of a small child who lives in south of Tehran during times of social problems. The film analyses an unjust and unfair society.
Director's View
Inflation, unemployment, divorce and social poverty are the major problems in modern Iran. If the officials, social and cultural experts and planners do not find solutions to these problems, the future society would face crisis due to the consequences.
