# Spectral asymptotics of the Laplacian on Platonic solids graphs

## TL;DR

This paper studies the high-energy eigenvalue behavior of quantum graphs based on Platonic solids, revealing how different vertex couplings influence spectral asymptotics and highlighting the role of vertex parity.

## Contribution

It provides a comparative analysis of spectral asymptotics for two types of vertex couplings on Platonic solid graphs, emphasizing the impact of vertex parity.

## Key findings

- Octahedron graph differs in spectral asymptotics due to vertex parity.
- Other Platonic solid graphs' spectra approach Dirichlet Laplacian behavior.
- Vertex coupling type significantly influences high-energy eigenvalue distribution.

## Abstract

We investigate the high-energy eigenvalue asymptotics quantum graphs consisting of the vertices and edges of the five Platonic solids considering two different types of the vertex coupling. One is the standard $\delta$-condition, the other is the preferred-orientation one introduced in [ET18]. The aim is to provide another illustration of the fact that the asymptotic properties of the latter coupling are determined by the vertex parity by showing that the octahedron graph differs in this respect from the other four for which the edges at high energies effectively disconnect and the spectrum approaches the one of the Dirichlet Laplacian on an interval.