  Adapting a method developed for the study of quantum chaos on {\it quantum
(metric)} graphs \cite {KS}, spectral $\zeta$ functions and trace formulae for
{\it discrete} Laplacians on graphs are derived. This is achieved by expressing
the spectral secular equation in terms of the periodic orbits of the graph, and
obtaining functions which belongs to the class of $\zeta$ functions proposed
originally by Ihara \cite {Ihara}, and expanded by subsequent authors \cite
{Stark,Sunada}. Finally, a model of "classical dynamics" on the discrete graph
is proposed. It is analogous to the corresponding classical dynamics derived
for quantum graphs \cite {KS}.
