Ihara zeta function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Ihara zeta function is a <a href="/facts/Zeta_function/lf43angF">zeta function</a> associated with a finite <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a>. It closely resembles the <a href="/facts/Selberg_zeta_function/DfgbLy7F">Selberg zeta function</a>, and is used to relate closed walks to the <a href="/facts/Spectrum_of_a_matrix/BIBnA4qf">spectrum</a> of the <a href="/facts/Adjacency_matrix/jYXNhYGs">adjacency matrix</a>. The Ihara zeta function was first defined by <a href="/facts/Yasutaka_Ihara/5kWNlclS">Yasutaka Ihara</a> in the 1960s in the context of <a href="/facts/Discrete_group/HIjkpJhS">discrete subgroups</a> of the two-by-two <a href="/facts/P-adic_number/SDgDbkLF">p-adic</a> <a href="/facts/Special_linear_group/rsahah9w">special linear group</a>. <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Jean-Pierre Serre</a> suggested in his book <i>Trees</i> that Ihara's original definition can be reinterpreted graph-theoretically. It was <a href="/facts/Toshikazu_Sunada/8lkWp16o">Toshikazu Sunada</a> who put this suggestion into practice in 1985. As observed by Sunada, a <a href="/facts/Regular_graph/j9dSoLE6">regular graph</a> is a <a href="/facts/Ramanujan_graph/T0lXTlmW">Ramanujan graph</a> if and only if its Ihara zeta function satisfies an analogue of the <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a>.
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Ihara zeta function open-in-new

Ihara zeta function