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,˜w) with at most dn edges such that for every

x ∈ R^V,
x^T L_G x ≤ x^T L_H x ≤ ( d+1+2√d / d+1-2√d ) x^T L_G x

where L_G and L_H are the Laplacian matrices of G and H, respectively. Thus, H approximates G spectrally at least as well as a Ramanujan expander with dn/2 edges approximates the complete graph.

We give an elementary deterministic polynomial time algorithm for constructing H.

Joint work with Josh Batson and Dan Spielman.