On the multiplicity of Laplacian eigenvalues and Fiedler partitions

TL;DR

This paper explores how specific graph structures influence Laplacian eigenvalues and their multiplicities, providing methods for eigenvalue reduction and insights into spectral partitioning.

Contribution

It introduces conditions on (m,k)-star and l-dependent graphs that determine eigenvalue multiplicities and offers a vertex reduction method preserving spectral properties.

Findings

Eigenvalue multiplicities depend on graph topology and weights.

Vertex reduction preserves eigenvalues and multiplicities.

Results inform Fiedler spectral partitioning techniques.

Abstract

In this paper we study two classes of graphs, the (m,k)-stars and l-dependent graphs, investigating the relation between spectrum characteristics and graph structure: conditions on the topology and edge weights are given in order to get values and multiplicities of Laplacian matrix eigenvalues. We prove that a vertex set reduction on graphs with (m,k)-star subgraphs is feasible, keeping the same eigenvalues with reduced multiplicity. Moreover, some useful eigenvectors properties are derived up to a product with a suitable matrix. Finally, we relate these results with Fiedler spectral partitioning of the graph. The physical relevance of the results is shortly discussed.