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 $H=(V,F,\tilde{w})$ with at most $\lceil d(n-1)\rceil$ edges such that for every $x\in\mathbb{R}^{V}$, $x^{T}L_{G}x\leq x^{T}L_{H}x\leq\bigl(\frac{d+1+2\sqrt{d}}{d+1-2\sqrt{d}}\bigr)\cdot x^{T}L_{G}x$, 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 $H$.

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 ⌈d(n-1)⌉ edges such that for every x ∈ RV, [xT LG x ≤ xT LH x ≤ ((d+1+2√d)/(d+1-2√d)) • xT LG x] where LG and LH 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.