# 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.

## Keywords

Quantum graph Zeta function Spectral graph theory## Mathematics Subject Classification

05C99 81Q10 81Q35## Notes

### Acknowledgements

The authors would like to thank Gregory Berkolaiko and the anonymous referees, whose suggestions substantially simplified the presentation of the main result. JH would like to thank the University of Warwick for their hospitality during his sabbatical where some of the work was carried out. JH was supported by the Baylor University research leave program. This work was partially supported by a grant from the Simons Foundation (354583 to Jonathan Harrison).

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