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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Graph diameter ： ウィキペディア英語版 Distance (graph theory) In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance. Notice that there may be more than one shortest path between two vertices.〔 〕 If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. In the case of a directed graph the distance $d(u,v)$ between two vertices $u$ and $v$ is defined as the length of a shortest path from $u$ to $v$ consisting of arcs, provided at least one such path exists.〔F. Harary, Graph Theory, Addison-Wesley, 1969, p.199.〕 Notice that, in contrast with the case of undirected graphs, $d(u,v)$ does not necessarily coincide with $d(v,u)$, and it might be the case that one is defined while the other is not. ==Related concepts== 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. The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected. The eccentricity $\backslash epsilon(v)$ of a vertex $v$ is the greatest geodesic distance between $v$ 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. The radius $r$ of a graph is the minimum eccentricity of any vertex or, in symbols, $r\; =\; \backslash min\_\; \backslash epsilon(v)$. The diameter $d$ of a graph is the maximum eccentricity of any vertex in the graph. That is, $d$ it is the greatest distance between any pair of vertices or, alternatively, $d\; =\; \backslash max\_\backslash epsilon(v)$. To find the diameter of a graph, first find the shortest path between each pair of vertices. The greatest length of any of these paths is the diameter of the graph. A central vertex in a graph of radius $r$ is one whose eccentricity is $r$—that is, a vertex that achieves the radius or, equivalently, a vertex $v$ such that $\backslash epsilon(v)\; =\; r$. A peripheral vertex in a graph of diameter $d$ is one that is distance $d$ from some other vertex—that is, a vertex that achieves the diameter. Formally, $v$ is peripheral if $\backslash epsilon(v)\; =\; d$. A pseudo-peripheral vertex $v$ has the property that for any vertex $u$, if $v$ is as far away from $u$ as possible, then $u$ is as far away from $v$ as possible. Formally, a vertex ''u'' is pseudo-peripheral, if for each vertex ''v'' with $d(u,v)\; =\; \backslash epsilon(u)$ holds $\backslash epsilon(u)=\backslash epsilon(v)$. The partition of a graphs vertices into subsets by their distances from a given starting vertex is called the level structure of the graph. A graph such that for every pair of vertices there is a unique shortest path connecting them is called a geodetic graph. For example, all trees are geodetic.〔Øystein Ore, Theory of graphs (ed., 1967 ), Colloquium Publications, American Mathematical Society, 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Distance (graph theory)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.