  Stark and Terras introduced the edge zeta function of a finite graph in 1996.
The edge zeta function is the reciprocal of a polynomial in twice as many
variables as edges in the graph and can be computed in polynomial time. We look
at graph properties which we can determine using the edge zeta function. In
particular, the edge zeta function is enough to deduce the clique number, the
number of Hamiltonian cycles, and whether a graph is perfect or chordal.
Actually computing these properties takes exponential time. Finally, we present
a new example illustrating that the Ihara zeta function cannot necessarily do
the same.
