Spectral techniques applied to sparse random graphs

@article{Feige2005SpectralTA,
  title={Spectral techniques applied to sparse random graphs},
  author={Uriel Feige and Eran Ofek},
  journal={Random Struct. Algorithms},
  year={2005},
  volume={27},
  pages={251-275}
}
We analyze the eigenvalue gap for the adjacency matrices of sparse random graphs. Let λ1 ≥ . . . ≥ λn be the eigenvalues of an n-vertex graph, and let λ = max[λ2, |λn|]. Let c be a large enough constant. For graphs of average degree d = c log n it is well known that λ1 ≥ d, and we show that λ = O( √ d). For d = c it is no longer true that λ = O( √ d), but we show that by removing a small number of vertices of highest degree in G, one gets a graph G′ for which λ = O( √ d). Our proofs are based… CONTINUE READING

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