  Spectral sparsification is a technique that is used to reduce the number of
non-zero entries in a positive semidefinite matrix with little changes to its
spectrum. In particular, the main application of spectral sparsification is to
construct sparse graphs whose spectra are close to a given dense graph. We
study spectral sparsification under the assumption that the edges of a graph
are allocated among sites which can communicate among each other. In this work
we show that if a graph is allocated among several sites, the union of the
spectral sparsifiers of each induced subgraph give us an spectral sparsifier of
the original graph. In contrast to other works in the literature, we present
precise computations of the approximation factor of the union of spectral
sparsifiers and give an explicit calculation of the edge weights. Then we
present an application of this result to data clustering in the
Number-On-Forehead model of multiparty communication complexity when input data
is allocated as a sunflower among sites in the party.
