On the Spectrum of Laplace Operator Defined on the Metric Graphs of Platonic Solids
Hendri Maulana, Yudi Soeharyadi*, Oki Neswan

Faculty of Mathematics and Natural Science, Bandung Institute of Technology Jalan Ganesha 10, Bandung 40132, Indonesia
*ysoeharyadi[at]itb.ac.id

Abstract

A metric graph is a graph, for which each of the edge is identified with a finite interval. The length of the interval is interpreted as the weight of the edge. In order for a set of functions on the edges of a metric graph is considered to define a function on the whole graph, conditions on each vertex must be imposed. Some of the most common vertex conditions are continuity condition and Kirchhoff condition.
In this study we report on the eigenvalues of the Laplace operator defined on the metric graphs of the Platonic Solids. Laplacian eigenvalue problem on metric graphs is described in a second order ordinary differential equation with Kirchhoff condition of vertices. Specifically, we discuss the algorithm for determining the eigenvalues of the Laplace operator by implementing calculations for two simplest platonic solids, namely the tetrahedron graph and the cube graph. We assume that the length the edges is uniform.

Keywords: Spectrum Laplace operator, Metric platonic solid graph, Continuity conditions, Kirchhoff condition

Topic: MATHEMATICS AND STATISTICS