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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 Book (graph theory) ： ウィキペディア英語版 Book (graph theory) In graph theory, a book graph (often written $B\_p$ ) may be any of several kinds of graph. One kind, which may be called a quadrilateral book, consists of ''p'' quadrilaterals sharing a common edge (known as the "spine" or "base" of the book).〔(Eric W. Weisstein, "Book Graph." ) From MathWorld–A Wolfram Web Resource.〕 A book of this type is the Cartesian product of a star and ''K''2 . A second type, which might be called a triangular book, is the complete tripartite graph ''K''1,1,''p''. It is a graph consisting of $p$ triangles sharing a common edge.〔Lingsheng Shi and Zhipeng Song, Upper bounds on the spectral radius of book-free and/or ''K''2,l-free graphs. ''Linear Algebra and its Applications'', vol. 420 (2007), pp. 526–529. 〕 A book of this type is a split graph. This graph has also been called a $K\_e(2,p)$. Given a graph $G$, one may write $bk(G)$ for the largest book (of the kind being considered) contained within $G$. The term "book-graph" has been employed for other uses. Barioli〔Francesco Barioli, Completely positive matrices with a book-graph. ''Linear Algebra and its Applications'', vol. 277 (1998), pp. 11–31. 〕 used it to mean a graph composed of a number of arbitrary subgraphs having two vertices in common. (Barioli did not write $B\_p$ for his book-graph.) ==Theorems on books== Denote the Ramsey number of two (triangular) books by $r(B\_p,\backslash \; B\_q).$ * If $1\backslash leq\; p\backslash leq\; q$, then $r(B\_p,\backslash \; B\_q)=2q+3$ (proved by Rousseau and Sheehan). * There exists a constant $c=o(1)$ such that $r(B\_p,\backslash \; B\_q)=2q+3$ whenever $q\backslash geq\; cp$. * If $p\backslash leq\; q/6+o(q)$, and $q$ is large, the Ramsey number is given by $2q+3$. * Let $C$ be a constant, and $k\; =\; Cn$. Then every graph on $n$ vertices and $m$ edges contains a (triangular) $B\_k$.〔P. Erdos, (On a theorem of Rademacher-Turán ). ''Illinois Journal of Mathematics'', vol. 6 (1962), pp. 122–127.〕 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「Book (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.