Zeta functions of graphs, their symmetries and extended Catalan numbers

TL;DR

This paper investigates spectral zeta functions of graphs, exploring their meromorphic properties, special values, symmetries, and connections to Catalan numbers, Ihara zeta functions, and graph Laplacian characteristics.

Contribution

It introduces new analytical methods to study spectral zeta functions of graphs, including their meromorphic continuation, special value analysis, and explicit determination of Laplacian polynomial coefficients for cyclic graphs.

Findings

Established meromorphic continuation of spectral zeta functions.

Connected special values to Catalan numbers and symmetries.

Determined coefficients of Laplacian characteristic polynomial for cyclic graphs.

Abstract

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.