On the multiplicity of Laplacian eigenvalues and Fiedler partitions

In this paper we investigate the relation between eigenvalue distribution and graph structure of two classes of graphs: the (m,k)-stars and l-dependent graphs. We give conditions on the topology and edge weights 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 and the physical relevance of the results is shortly discussed.