Why matters
Many graph problems reduce to Laplacian solves: max flow, spanning trees, effective resistance, PageRank.
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Sparsifiers
Approximate L by L' with fewer edges but same spectrum (approximately). Solve on sparsifier instead.
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Preconditioners
Approximate inverse using low-stretch spanning trees. Recursive combination via multigrid.
Complexity
Õ(m log^(1/2) n log(1/ε)) modern. Practical implementations exist (Combinatorial BLAS, Julia's Laplacians.jl).