Analysis on graphs and noncommutative geometry

@article{Davies1993AnalysisOG,
  title={Analysis on graphs and noncommutative geometry},
  author={E. B. Davies},
  journal={Journal of Functional Analysis},
  year={1993},
  volume={111},
  pages={398-430}
}
  • E. Davies
  • Published 1 February 1993
  • Mathematics
  • Journal of Functional Analysis
Abstract We study the form of the continuous time heat kernel for a second order discrete Laplacian on a weighted graph. The analysis is shown to be closely related to the theory of symmetric Markov semigroups on noncommutative L p spaces and to the noncommutative geometry of Connes. The paper obtains better pointwise upper bounds on the heat kernels than those previously known, by the use of a novel metric on the graph. In certain cases it is shown that the new estimates are optimal of their… 
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TLDR
The history of the heat kernel Gaussian estimates started with the works of Nash and Aronson and the Aronson’s upper bound for the case of time-independent coefficients which is of interest reads as follows.
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