  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.
