Relating Zeta Functions of Discrete and Quantum Graphs

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.

arXiv comment12 pages; added reference after Corollary 1 (on page 4)

References in corpus (8)

• Spectra of Schroedinger operators on equilateral quantum graphs
• Vacuum Energy and Repulsive Casimir Forces in Quantum Star Graphs
• Repulsive Casimir Pistons
• Index theorems for quantum graphs
• Zeta functions of quantum graphs
• Spectral determinants and zeta functions of Schrödinger operators on metric graphs
• Zeta Functions of the Dirac Operator on Quantum Graphs
• Mathematical Aspects of Vacuum Energy on Quantum Graphs

Cited by in corpus (2)

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• Eigenspaces of symmetric graphs are not typically irreducible