  Conjecturally, almost all graphs are determined by their spectra. This
problem has also been studied for variants such as the spectra of the Laplacian
and signless Laplacian. Here we consider the problem of determining graphs with
Ihara and Bartholdi zeta functions, which are also computable in polynomial
time. These zeta functions are geometrically motivated, but can be viewed as
certain generalizations of characteristic polynomials. After discussing some
graph properties determined by zeta functions, we show that large classes of
cospectral graphs can be distinguished with zeta functions and enumerate graphs
distinguished by zeta functions on $\le 11$ vertices. This leads us to
conjecture that almost all graphs which are not determined by their spectrum
are determined by zeta functions.
  Along the way, we make some observations about the usual types of spectra and
disprove a conjecture of Setyadi and Storm about Ihara zeta functions
determining degree sequences.
