Randomized Algorithms 2017A – Lecture 11 Graph Laplacians and Spectral Sparsification∗
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چکیده
Alternative definition: Then LG is the matrix with diagonal entries (LG)ii = di, and off-diagonal entries (LG)ij = −wij . Fact 1: The matrix L = LG is symmetric, non-diagonals entries are Lij = −wij , and its diagonal entries are Lii = di, where di = ∑ j:ij∈E wij is the degree of vertex i. It is useful to put these values in a diagonal matrixD = diag(d⃗). If G is unweighted, then L = D−A where A is the adjacency matrix. ∗These notes summarize the material covered in class, usually skipping proofs, details, examples and so forth, and possibly adding some remarks, or pointers. The exercises are for self-practice and need not be handed in. In the interest of brevity, most references and credits were omitted.
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