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Is there a natural connection between the Ihara zeta function of a graph, and (for instance) the Riemann zeta function of certain varieties over finite fields ? Thanks.

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RH for the Ihara zeta function will correspond to the graph being Ramanujan (if the graph is (q+1)-regular).

The zeta function for varieties over finite fields is more related to Ruelle's zeta function, but you can see Ihara zeta function as a special instance of it, using symbolic dynamics representation of a walk in your graph as a dynamical system.

A nice reference for this material is Audrey Terras' book - "Zeta Functions of Graphs: A Stroll through the Garden"

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Ah, thanks. Are there other zeta functions for graphs which relate more directly to the Riemann zeta of varieties over finite fields ? –  THC Jan 25 '12 at 8:21

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