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| [[File:Directed.svg|125px|thumb|A directed graph.]]
| | CMS provides the best platform to create websites that fulfill all the specifications of SEO. The next step is to visit your Word - Press blog dashboard. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. When you have virtually any concerns concerning where as well as the best way to work with [http://www.spellenaanbod.nl/profile/lofryar wordpress backup plugin], you'll be able to call us on the page. As of now, Pin2Press is getting ready to hit the market. Use this section to change many formatting elements. The first thing you need to do is to choose the right web hosting plan. After that the developer adds the unordered list for navigations. <br><br>A built-in widget which allows you to embed quickly video from popular websites. * Robust CRM to control and connect with your subscribers. Thus it is difficult to outrank any one of these because of their different usages. If you are looking for Hire Wordpress Developer then just get in touch with him. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level. <br><br>Many developers design websites and give them to the clients, but still the client faces problems to handle the website. s ability to use different themes and skins known as Word - Press Templates or Themes. The days of spending a lot of time and money to have a website built are long gone. You should stay away from plugins that are full of flaws and bugs. You can check out the statistics of page of views for your web pages using free tools that are available on the internet. |
| In [[mathematics]], and more specifically in [[graph theory]], a '''directed graph''' (or '''digraph''') is a [[graph (mathematics)|graph]], or set of nodes connected by edges, where the edges have a direction associated with them. In formal terms, a digraph is a pair <math>G=(V,A)</math> (sometimes <math>G=(V,E)</math>) of:<ref>{{harvtxt|Bang-Jensen|Gutin|2000}}. {{harvtxt|Diestel|2005}}, Section 1.10. {{harvtxt|Bondy|Murty|1976}}, Section 10.</ref>
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| * a [[Set (mathematics)|set]] ''V'', whose [[element (mathematics)|elements]] are called ''vertices'' or ''nodes'',
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| * a set ''A'' of [[ordered pair]]s of vertices, called ''arcs'', ''directed edges'', or ''arrows'' (and sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A'').
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| It differs from an ordinary or [[undirected graph]], in that the latter is defined in terms of [[unordered pair]]s of vertices, which are usually called [[edge (graph theory)|edges]].
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| Sometimes a digraph is called a ''simple digraph'' to distinguish it from a ''[[multigraph|directed multigraph]]'', in which the arcs constitute a [[multiset]], rather than a set, of ordered pairs of vertices. Also, in a simple digraph loops are disallowed. (A loop is an arc that pairs a vertex to itself.) On the other hand, some texts allow loops, multiple arcs, or both in a digraph.
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| ==Basic terminology==
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| An arc <math>e = (x, y)</math> is considered to be directed ''from'' <math>x</math> ''to'' <math>y</math>; <math>y</math> is called the ''head'' and <math>x</math> is called the ''tail'' of the arc; <math>y</math> is said to be a ''direct successor'' of <math>x</math>, and <math>x</math> is said to be a ''direct predecessor'' of <math>y</math>. If a [[path (graph theory)|path]] made up of one or more successive arcs leads from <math>x</math> to <math>y</math>, then <math>y</math> is said to be a ''successor'' of <math>x</math>, and <math>x</math> is said to be a ''predecessor'' of <math>y</math>. The arc <math>(y, x)</math> is called the arc <math>(x, y)</math> ''inverted''.
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| An [[Orientation (graph theory)|orientation]] of a [[simple graph|simple undirected graph]] is obtained by assigning a direction to each edge. Any directed graph constructed this way is called an "oriented graph". A directed graph is an oriented simple graph if and only if it has neither self-loops nor 2-cycles.<ref>{{harvtxt|Diestel|2005}}, Section 1.10.</ref>
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| A ''weighted digraph'' is a digraph with weights assigned to its arcs, similar to a [[weighted graph]].
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| In the context of graph theory a digraph with weighted edges is called a ''network''.
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| The [[adjacency matrix]] of a digraph (with loops and multiple arcs) is the integer-valued [[Matrix (mathematics)|matrix]] with rows and columns corresponding to the nodes, where a nondiagonal entry <math>a_{ij}</math> is the number of arcs from node ''i'' to node ''j'', and the diagonal entry <math>a_{ii}</math> is the number of loops at node ''i''. The adjacency matrix of a digraph is unique up to identical permutation of rows and columns.
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| Another matrix representation for a digraph is its [[incidence matrix]].
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| See [[Glossary of graph theory#Direction]] for more definitions.
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| ==Indegree and outdegree==
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| [[File:DirectedDegrees.svg|thumb|A digraph with vertices labeled (indegree, outdegree)]]
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| For a node, the number of head endpoints adjacent to a node is called the ''indegree'' of the node and the number of tail endpoints adjacent to a node is its ''outdegree'' (called "[[branching factor]]" in trees).
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| The indegree is denoted <math>\deg^-(v)</math> and the outdegree as <math>\deg^+(v).</math> A vertex with <math>\deg^-(v)=0</math> is called a ''source'', as it is the origin of each of its incident edges. Similarly, a vertex with <math>\deg^+(v)=0</math> is called a ''sink''.
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| The ''degree sum formula'' states that, for a directed graph,
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| :<math>\sum_{v \in V} \deg^+(v) = \sum_{v \in V} \deg^-(v) = |A|\, .</math> | |
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| If for every node {{math|''v'' ∈ ''V''}}, <math>\deg^+(v) = \deg^-(v)</math>, the graph is called a ''balanced digraph''.<ref>{{citation|page=460|title=Discrete Mathematics and Graph Theory|first1=Bhavanari|last1=Satyanarayana|first2=Kuncham Syam|last2=Prasad|publisher=PHI Learning Pvt. Ltd.|isbn=978-81-203-3842-5}}; {{citation|page=51|title=Combinatorial matrix classes|volume=108|series=Encyclopedia of mathematics and its applications|first=Richard A.|last=Brualdi|publisher=Cambridge University Press|year=2006|isbn=978-0-521-86565-4}}.</ref>
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| ==Digraph connectivity==
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| {{main|Connectivity (graph theory)}}
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| A digraph G is called ''weakly connected'' (or just ''connected''<ref>{{harvtxt|Bang-Jensen|Gutin|2000}} p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).</ref>) if the undirected ''underlying graph'' obtained by replacing all directed edges of G with undirected edges is a [[connected graph]]. A digraph 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''. The ''strong components'' are the maximal strongly connected subgraphs.
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| ==Classes of digraphs==
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| A directed graph ''G'' is called '''symmetric''' if, for every arc that belongs to ''G'', the corresponding reversed arc also belongs to ''G''. A symmetric, loopless directed graph is equivalent to an undirected graph with the edges replaced by pairs of inverse arcs; thus the number of edges is equal to the number of arcs halved.
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| [[File:Directed acyclic graph 2.svg|right|100px|thumb|A simple acyclic directed graph]]
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| An '''acyclic''' directed graph, acyclic digraph, or [[directed acyclic graph]] is a directed graph with no [[directed cycle]]s. Special cases of acyclic directed graphs include [[multitree]]s (graphs in which no two directed paths from a single starting node meet back at the same ending node), [[oriented tree]]s or polytrees (digraphs formed by orienting the edges of undirected acyclic graphs), and [[rooted tree]]s (oriented trees in which all edges of the underlying undirected tree are directed away from the roots).
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| [[File:4-tournament.svg|thumb|right|100px|A tournament on 4 vertices]]
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| A '''[[tournament (mathematics)|tournament]]''' is an oriented graph obtained by choosing a direction for each edge in an undirected [[complete graph]].
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| In the theory of [[Lie group]]s, a '''[[quiver (mathematics)|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]] '''FinVct'''<sub>''K''</sub><sup>F(''Q'')</sup> where ''F''(''Q'') is the [[free category]] on ''Q'' consisting of paths in ''Q'' and '''FinVct'''<sub>''K''</sub> is the category of finite dimensional [[vector space]]s over a [[Field (mathematics)|field]] ''K''. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with [[Linear map|linear transformations]] between them, and transform via [[natural transformation]]s.
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| ==See also==
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| * [[Flow chart]]
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| * [[Preorder]]
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| * [[Quiver (mathematics)|Quiver]]
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| * [[Transpose graph]]
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| * [[Vertical constraint graph]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation
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| | last1=Bang-Jensen | first1=Jørgen
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| | last2=Gutin | first2=Gregory
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| | title=Digraphs: Theory, Algorithms and Applications
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| | publisher=[[Springer Science+Business Media|Springer]]
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| | year=2000
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| | isbn=1-85233-268-9
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| | url=http://www.cs.rhul.ac.uk/books/dbook/
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| }}<br>(the corrected 1st edition of 2007 is now freely available on the authors' site; the 2nd edition appeared in 2009 ISBN 1-84800-997-6).
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| * {{citation
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| | last1=Bondy | first1=John Adrian | authorlink1=John Adrian Bondy
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| | last2=Murty | first2=U. S. R. | authorlink2=U. S. R. Murty
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| | title=Graph Theory with Applications
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| | year=1976
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| | publisher=North-Holland
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| | isbn=0-444-19451-7
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| | url=http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html
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| }}.
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| *{{Citation
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| | last=Diestel | first=Reinhard
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| | title=Graph Theory
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| | publisher=[[Springer Science+Business Media|Springer]]
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| | year=2005
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| | edition=3rd
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| | isbn=3-540-26182-6
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| | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
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| }} (the electronic 3rd edition is freely available on author's site).
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| *{{citation|last1=Harary|first1=Frank|last2=Norman|first2=Robert Z.|last3=Cartwright|first3=Dorwin|title=Structural Models: An Introduction to the Theory of Directed Graphs|place=New York|publisher=Wiley|year=1965}}.
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| *{{citation | title=Number of directed graphs (or digraphs) with n nodes. | url=http://oeis.org/A000273}}
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| {{DEFAULTSORT:Directed Graph}}
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| [[Category:Directed graphs]]
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As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister.
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