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

Zeta functions of graphs, their symmetries and extended Catalan numbers