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Twice-Ramanujan Sparsifiers
[article]
2009
arXiv
pre-print
We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every d>1 and every undirected, weighted graph G=(V,E,w) on n vertices, there exists a weighted graph H=(V,F,w̃) with at most d(n-1) edges such that for every x ∈^V, x^TL_Gx ≤ x^TL_Hx ≤
arXiv:0808.0163v3
fatcat:szswkz4lcrccjmgeoc2lr7vbbi