  In this paper we study spectral zeta functions associated to finite and
infinite graphs. First we establish a meromorphic continuation of these
functions under some general conditions. Then we study special values in the
case of standard lattice graphs associated to free abelian groups. In
particular we connect it to Catalan numbers in several ways, and obtain some
non-trivial special values and functional symmetries. Furthermore we relate the
values at the negative integers with the more studied Ihara zeta functions, and
prove a few minor results that seem not to have been recorded before. Finally
we consider the characteristic polynomial of the graph Laplacians, and in
particular completely determine its coefficients for cyclic graphs using new
analytical methods.
