We enumerate all connected graphs with minimal vertex degree 2 on at most 11 vertices and determine their Ihara zeta functions. We also count the number of such graphs for which there is another graph with the same zeta function. We then use these graphs to conjecture properties of the graphs determined by the zeta function. In addition, we study switching constructions, proposed by Godsil and McKay, to determine whether they preserve the zeta function. We show that GM switching is not strong enough to preserve the zeta function, but GM* switching, following the notation of Haemers and Spence, does preserve the zeta function.