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Ramanujan graph : 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 \lambda_0 \geq \lambda_1 \geq \ldots \geq \lambda_ be the eigenvalues of the adjacency matrix of G. Because G is connected and d-regular, its eigenvalues satisfy d = \lambda_0 > \lambda_1 \geq \ldots \geq \lambda_ \geq -d . Whenever there exists \lambda_i with |\lambda_i| < d, define
: \lambda(G) = \max_ |\lambda_i|.\,
A d-regular graph G is a Ramanujan graph if \lambda(G) \leq 2\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)
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