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.

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" 

