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In mathematics, the Ihara zeta-function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta-function, and is used to relate closed paths to the spectrum of the adjacency matrix. The Ihara zeta-function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear group. Jean-Pierre Serre suggested in his book ''Trees'' that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice (1985). As observed by Sunada, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.〔Terras (1999) p.678〕 ==Definition== The Ihara zeta-function can be defined by a formula analogous to the Euler product for the Riemann zeta function: : such that : and is the length of cycle ''p'', as used in the formulae above.〔Terras (2010) p.12〕 This formulation in graph-theoretic setting is due to Sunada. 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「Ihara zeta function」の詳細全文を読む スポンサード リンク
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