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An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification [chapter]

Nikhil Srivastava, Luca Trevisan
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if G is an n-node weighted undirected graph of average combinatorial degree d (that is, G has dn/  ...  2 edges) and girth g > 2d 1/8 + 1, and if λ 1 ≤ λ 2 ≤ · · · λ n are the eigenvalues of the (non-normalized) Laplacian of G, then  ...  Acknowledgments We would like to thank Alexandra Kolla for helpful conversations, as well as the Simons Institute for the Theory of Computing, where this work was carried out.  ... 
doi:10.1137/1.9781611975031.85 dblp:conf/soda/SrivastavaT18 fatcat:ddqcdx76dbcmlnsfithadfrrti

Graph powering and spectral robustness [article]

Emmanuel Abbe, Enric Boix, Peter Ralli, Colin Sandon
2018 arXiv   pre-print
it is shown that graph powering produces a 'maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph  ...  This paper proposes a simple and generic construction for sparse graphs, ψ(A) = ((I+A)^r >1), where A denotes the adjacency matrix and r is an integer (less than the graph diameter).  ...  We then make more precise the notion of "achieving a maximal spectral gap on Erdős-Rényi graphs", by establishing an Alon-Boppana result for graph powering.  ... 
arXiv:1809.04818v1 fatcat:ty3zirwj4rcfppo4alo6xps73u

Proportional Volume Sampling and Approximation Algorithms for A-Optimal Design [article]

Aleksandar Nikolov and Mohit Singh and Uthaipon Tao Tantipongpipat
2018 arXiv   pre-print
The problem also finds applications in sensor placement in wireless networks, sparse least squares regression, feature selection for k-means clustering, and matrix approximation.  ...  The proportional volume-sampling algorithm also gives approximation algorithms for other optimal design objectives such as D-optimal design and generalized ratio objective matching or improving previous  ...  Nikhil Srivastava and Luca Trevisan. An alon-boppana type bound for weighted graphs and lowerbounds for spectral sparsification. CoRR, abs/1707.06364, 2017. URL 06364.  ... 
arXiv:1802.08318v5 fatcat:dirb5tbukzavho356x7s23nftm