There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
Explicit construction of linear sized tolerant networks
Discrete Mathematics - First Japan Conference on Graph Theory and Applications
On the second eigenvalue of a graph
Discrete Mathematics
Approximating s-t minimum cuts in Õ(n2) time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Matrix computations (3rd ed.)
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Graph partitioning using single commodity flows
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Expanders via random spanning trees
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Extensions and limits to vertex sparsification
Proceedings of the forty-second ACM symposium on Theory of computing
Subgraph sparsification and nearly optimal ultrasparsifiers
Proceedings of the forty-second ACM symposium on Theory of computing
Rumour spreading and graph conductance
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Sparse reliable graph backbones
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming: Part II
A general framework for graph sparsification
Proceedings of the forty-third annual ACM symposium on Theory of computing
A unified framework for approximating and clustering data
Proceedings of the forty-third annual ACM symposium on Theory of computing
Spectral Sparsification of Graphs
SIAM Journal on Computing
Sparse reliable graph backbones
Information and Computation
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Spectral sparsification via random spanners
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Dimension reduction for finite trees in l1
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On multiplicative λ-approximations and some geometric applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
How user behavior is related to social affinity
Proceedings of the fifth ACM international conference on Web search and data mining
Subsampling mathematical relaxations and average-case complexity
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Graph Sparsification by Effective Resistances
SIAM Journal on Computing
A matrix hyperbolic cosine algorithm and applications
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Algorithms, graph theory, and the solution of laplacian linear equations
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
Constructing all self-adjoint matrices with prescribed spectrum and diagonal
Advances in Computational Mathematics
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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.