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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 Ramanujan graphs ： ウィキペディア英語版 Ramanujan graph In spectral graph theory, a Ramanujan graph, named after Srinivasa Ramanujan, is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. Examples of Ramanujan graphs include the clique, the biclique $K\_$, and the Petersen graph. As (Murty's survey paper ) notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". As observed by Toshikazu Sunada, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis.〔Audrey Terras, ''Zeta Functions of Graphs: A Stroll through the Garden'', volume 128, Cambridge Studies in Advanced Mathematics, Cambridge University Press, (2010).〕 ==Definition== Let $G$ be a connected $d$-regular graph with $n$ vertices, and let $\backslash lambda\_0\; \backslash geq\; \backslash lambda\_1\; \backslash geq\; \backslash ldots\; \backslash geq\; \backslash lambda\_$ be the eigenvalues of the adjacency matrix of $G$. Because $G$ is connected and $d$-regular, its eigenvalues satisfy $d\; =\; \backslash lambda\_0\; >\; \backslash lambda\_1$ $\backslash geq\; \backslash ldots\; \backslash geq\; \backslash lambda\_\; \backslash geq\; -d$. Whenever there exists $\backslash lambda\_i$ with , define : $\backslash lambda(G)\; =\; \backslash max\_\; |\backslash lambda\_i|.\backslash ,$ A $d$-regular graph $G$ is a Ramanujan graph if $\backslash lambda(G)\; \backslash leq\; 2\backslash sqrt$. A Ramanujan graph is characterized as a regular graph whose Ihara zeta function satisfies an analogue of the Riemann Hypothesis. 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「Ramanujan graph」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.