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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 product ： ウィキペディア英語版 Graph product In mathematics, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs ''G''1 and ''G''2 and produces a graph ''H'' with the following properties: * The vertex set of ''H'' is the Cartesian product ''V''(''G''1) × ''V''(''G''2), where ''V''(''G''1) and ''V''(''G''2) are the vertex sets of ''G''1 and ''G''2, respectively. * Two vertices (''u''1, ''u''2) and (''v''1, ''v''2) of ''H'' are connected by an edge if and only if the vertices ''u''1, ''u''2, ''v''1, ''v''2 satisfy conditions of a certain type (see below). The following table shows the most common graph products, with $\backslash sim$; denoting “is connected by an edge to”, and $\backslash not\backslash sim$ denoting non-connection. The operator symbols listed here are by no means standard, especially in older papers. \rightarrow J_ | |- | Tensor product (Categorical product) $G\; \backslash times\; H$ | $u\_1$ $\backslash sim$ $v\_1$ and  $u\_2$ $\backslash sim$ $v\_2$ | $G\_\; \backslash times\; H\_\; \backslash rightarrow\; J\_$ | |- | Lexicographical product $G\; \backslash cdot\; H$ or $G()$ | ''u''1 ∼ ''v''1 or ( ''u''1 = ''v''1 and ''u''2 ∼ ''v''2 ) | $G\_\; \backslash cdot\; H\_\; \backslash rightarrow\; J\_$ | |- | Strong product (Normal product, AND product) $G\; \backslash boxtimes\; H$ | ( ''u''1 = ''v''1 and ''u''2 ∼ ''v''2 ) or ( ''u''1 ∼ ''v''1 and ''u''2 = ''v''2 ) or ( ''u''1 ∼ ''v''1 and ''u''2 ∼ ''v''2 ) | $G\_\; \backslash boxtimes\; H\_\; \backslash rightarrow\; J\_$ | |- | Co-normal product (disjunctive product, OR product) $G$ * H | ''u''1 ∼ ''v''1 or ''u''2 ∼ ''v''2 | | |- | Modular product | $(u\_1\; \backslash sim\; v\_1\; \backslash text\; u\_2\; \backslash sim\; v\_2)$ or $(u\_1\; \backslash not\backslash sim\; v\_1\; \backslash text\; u\_2\; \backslash not\backslash sim\; v\_2)$ | | |- | Rooted product | see article | $G\_\; \backslash cdot\; H\_\; \backslash rightarrow\; J\_$ | |- | Kronecker product | see article | see article | see article |- | Zig-zag product | see article | see article | see article |- | Replacement product | | | |- | Homomorphic product $G\; \backslash ltimes\; H$ | $(u\_1\; =\; v\_1)$ or $(u\_1\; \backslash sim\; v\_1\; \backslash text\; u\_2\; \backslash not\backslash sim\; v\_2)$ | | |} In general, a graph product is determined by any condition for (''u''1, ''u''2) ∼ (''v''1, ''v''2) that can be expressed in terms of the statements ''u''1 ∼ ''v''1, ''u''2 ∼ ''v''2, ''u''1 = ''v''1, and ''u''2 = ''v''2. ==Mnemonic== Let $K\_2$ be the complete graph on two vertices (i.e. a single edge). The product graphs $K\_2\; \backslash square\; K\_2$, $K\_2\; \backslash times\; K\_2$, and $K\_2\; \backslash boxtimes\; K\_2$ look exactly like the glyph representing the operator. For example, $K\_2\; \backslash square\; K\_2$ is a four cycle (a square) and $K\_2\; \backslash boxtimes\; K\_2$ is the complete graph on four vertices. The $G()$ notation for lexicographic product serves as a reminder that this product is not commutative. 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「Graph product」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.