3,317 research outputs found
On the multiplicity of Laplacian eigenvalues and Fiedler partitions
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
The Laplacian spectral excess theorem for distance-regular graphs
The spectral excess theorem states that, in a regular graph G, the average
excess, which is the mean of the numbers of vertices at maximum distance from a
vertex, is bounded above by the spectral excess (a number that is computed by
using the adjacency spectrum of G), and G is distance-regular if and only if
equality holds. In this note we prove the corresponding result by using the
Laplacian spectrum without requiring regularity of G
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