I leave the reader with my list of possible research projects.

Do experiments on the differences between the properties of zetas of weighted or quantum graphs and unweighted graphs. See Horton, Stark, and Terras [60], [61]. In particular, consider the connections with random matrix theory. See also Smilansky [115].

(a) Investigate the poles of the Ihara zeta and the Riemann hypothesis for random graphs. How does the distribution of poles depend on the probability of an edge and on the degree sequence?

(b) Can one find a Galois graph covering Y of a base graph X such that the poles of the Ihara zeta of Y behave like those of a random cover of X? One could experiment with various finite non-abelian groups. We considered abelian groups in the preceding chapters; the pole distributions of zetas for abelian covers were seen to be very different from those of random covers. One idea is to imitate the work of Lubotzky, Phillips, and Sarnak [79] using the group SL(2, F), where F is a finite field.

Can you prove an analog of the theorem of Katz and Sarnak [68] for function field zeta functions at least in the case of regular graphs? This theorem says that, for almost all curves over a finite field, as the genus and order of the field go to infinity, the imaginary parts of the zeros of zetas approach the GUE level spacing (meaning that the spacings look like those Odlyzko found for the high zeros of the Riemann zeta, as in Figure 5.5.