Flexibility of eigenvalues for graph Laplacians arising from genus 3 surfaces

TL;DR

This paper characterizes the possible eigenvalues of weighted graph Laplacians derived from genus 3 surfaces' pair of pants decompositions, linking geometric surface properties to spectral graph theory.

Contribution

It provides a complete description of eigenvalue sets for all four-vertex graphs representing genus 3 surface decompositions.

Findings

Eigenvalues of graph Laplacians can be explicitly characterized for genus 3 surface decompositions.

The results connect geometric surface structures with spectral properties of associated graphs.

Abstract

It is known that the small eigenvalues of the Laplacian of a Riemann surface close to the boundary of the modular space can be well approximated by the eigenvalues of the discrete Laplacian on a certain graph coming from the pair of pants decomposition of the surface. In this paper, we provide a complete description of the sets of eigenvalues of the weighted graph Laplacian for all graphs on four vertices that correspond to a valid pair of pants decomposition of a surface of genus 3.