Relating Zeta Functions of Discrete and Quantum Graphs

TL;DR

This paper establishes a relationship between the spectral zeta functions of Laplace operators on equilateral metric graphs and their underlying discrete graphs, revealing how eigenvalue multiplicities depend on graph geometry and applying results to specific graph types.

Contribution

It introduces a method to express the spectral zeta function of quantum graphs in terms of discrete graph spectra, linking geometry and spectral properties.

Findings

Derived a formula relating quantum and discrete graph zeta functions.

Determined eigenvalue multiplicity dependence on graph geometry.

Calculated vacuum energy and spectral determinant for specific graph types.

Abstract

We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation between the spectrum of the Laplacian on a discrete graph and that of the Laplacian on an equilateral metric graph. As a by-product, we determine how the multiplicity of eigenvalues of the quantum graph, that are also in the spectrum of the graph with Dirichlet conditions at the vertices, depends on the graph geometry. Finally we apply the result to calculate the vacuum energy and spectral determinant of a complete bipartite graph and compare our results with those for a star graph, a graph in which all vertices are connected to a central vertex by a single edge.