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| {{Redirect|Geodesic distance|distances on the surface of the Earth|Great-circle distance}}
| | The title of the writer is Figures but it's not the most masucline title out there. Managing people is his profession. South Dakota is exactly where me and my spouse live. Playing baseball is the pastime he will by no means stop performing.<br><br>Here is my blog - [http://www.hotporn123.com/user/D71W std testing at home] |
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| In the [[mathematics|mathematical]] field of [[graph theory]], the '''distance''' between two [[vertex (graph theory)|vertices]] in a [[graph (mathematics)|graph]] is the number of edges in a [[shortest path problem|shortest path]] (also called a '''graph geodesic''') connecting them. This is also known as the '''geodesic distance'''.<ref>{{cite journal |last=Bouttier |first=Jérémie|coauthors=Di Francesco,P. ,Guitter, E. |date=July 2003
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| |title=Geodesic distance in planar graphs |journal= Nuclear Physics B|volume=663 |issue=3 |pages=535–567 |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVC-48KW72R-1&_user=3742306&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000061256&_version=1&_urlVersion=0&_userid=3742306&md5=86dd4de63373a7e72d23d16840947661
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| |accessdate= 2008-04-23 |quote=By distance we mean here geodesic distance along the graph, namely the length of any shortest path between say two given faces
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| |doi=10.1016/S0550-3213(03)00355-9}}</ref> Notice that there may be more than one shortest path between two vertices.<ref>
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| {{cite web |url=http://mathworld.wolfram.com/GraphGeodesic.html |title=Graph Geodesic |accessdate= 2008-04-23
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| |last=Weisstein |first=Eric W. |authorlink=Eric W. Weisstein |work=MathWorld--A Wolfram Web Resource
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| |publisher= Wolfram Research
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| |quote=The length of the graph geodesic between these points d(u,v) is called the graph distance between u and v }}
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| </ref> If there is no path connecting the two vertices, i.e., if they belong to different [[connected component (graph theory)|connected component]]s, then conventionally the distance is defined as infinite.
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| In the case of a [[directed graph]] the distance <math>d(u,v)</math> between two vertices <math>u</math> and <math>v</math> is defined as the length of a shortest path from <math>u</math> to <math>v</math> consisting of arcs, provided at least one such path exists.<ref>F. Harary, Graph Theory, Addison-Wesley, 1969, p.199.</ref> Notice that, in contrast with the case of undirected graphs, <math>d(u,v)</math> does not necessarily coincide with <math>d(v,u)</math>, and it might be the case that one is defined while the other is not.
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| ==Related concepts==
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| A [[metric space]] defined over a set of points in terms of distances in a graph defined over the set is called a '''graph metric'''.
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| The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is [[connected (graph theory)|connected]].
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| The '''eccentricity''' <math>\epsilon(v)</math> of a vertex <math>v</math> is the greatest geodesic distance between <math>v</math> and any other vertex. It can be thought of as how far a node is from the node most distant from it in the graph.
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| The '''radius''' <math>r</math> of a graph is the minimum eccentricity of any vertex or, in symbols, <math>r = \min_{v \in V} \epsilon(v)</math>.
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| The '''diameter''' <math>d</math> of a graph is the maximum eccentricity of any vertex in the graph. That is, <math>d</math> it is the greatest distance between any pair of vertices or, alternatively, <math>d = \max_{v \in V}\epsilon(v)</math>. To find the diameter of a graph, first find the [[Shortest path problem|shortest path]] between each pair of [[vertex (graph theory)|vertices]]. The greatest length of any of these paths is the diameter of the graph.
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| A '''central vertex''' in a graph of radius <math>r</math> is one whose eccentricity is <math>r</math>—that is, a vertex that achieves the radius or, equivalently, a vertex <math>v</math> such that <math>\epsilon(v) = r</math>.
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| A '''peripheral vertex''' in a graph of diameter <math>d</math> is one that is distance <math>d</math> from some other vertex—that is, a vertex that achieves the diameter. Formally, <math>v</math> is peripheral if <math>\epsilon(v) = d</math>.
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| A '''pseudo-peripheral vertex''' <math>v</math> has the property that for any vertex <math>u</math>, if <math>v</math> is as far away from <math>u</math> as possible, then <math>u</math> is as far away from <math>v</math> as possible. Formally, a vertex ''u'' is pseudo-peripheral,
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| if for each vertex ''v'' with <math>d(u,v) = \epsilon(u)</math> holds <math>\epsilon(u)=\epsilon(v)</math>.
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| The [[partition of a set|partition]] of a graphs vertices into subsets by their distances from a given starting vertex is called the [[level structure]] of the graph.
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| ==Algorithm for finding pseudo-peripheral vertices==
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| Often peripheral [[sparse matrix]] algorithms need a starting vertex with a high eccentricity. A peripheral vertex would be perfect, but is often hard to calculate. In most circumstances a pseudo-peripheral vertex can be used. A pseudo-peripheral vertex can easily be found with the following algorithm:
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| # Choose a vertex <math>u</math>.
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| # Among all the vertices that are as far from <math>u</math> as possible, let <math>v</math> be one with minimal [[degree (graph theory)|degree]].
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| # If <math>\epsilon(v) > \epsilon(u)</math> then set <math>u=v</math> and repeat with step 2, else <math>v</math> is a pseudo-peripheral vertex.
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| ==See also==
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| * [[Distance matrix]]
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| * [[Resistance distance]]
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| * [[Betweenness]]
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| * [[Centrality]]
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| * [[Closeness (graph theory)|Closeness]]
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| *[[degree diameter|Degree diameter problem]] for [[graph (mathematics)|graph]]s and [[digraph (mathematics)|digraph]]s
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| ==Notes==
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| {{reflist}}
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| [[Category:Graph theory]]
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