Upper Bounds for Eigenvalues of the Discrete and Continuous Laplace Operators

@article{Chung1996UpperBF,
  title={Upper Bounds for Eigenvalues of the Discrete and Continuous Laplace Operators},
  author={Fan R. K. Chung and Alexander Grigor’yan and Shing-Tung Yau},
  journal={Advances in Mathematics},
  year={1996},
  volume={117},
  pages={165-178}
}
In this paper, we are concerned with upper bounds of eigenvalues of Laplace operator on compact Riemannian manifolds and finite graphs. While on the former the Laplace operator is generated by the Riemannian metric, on the latter it reflects combinatorial structure of a graph. Respectively, eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics. We develop a universal approach to upper bounds on both continuous and discrete structures based… 
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