Analysis on graphs and noncommutative geometry

  title={Analysis on graphs and noncommutative geometry},
  author={E. B. Davies},
  journal={Journal of Functional Analysis},
  • 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… 
Dirichlet Forms on Noncommutative Spaces
We show how Dirichlet forms provide an approach to potential theory of noncommutative spaces based on the notion of energy. The correspondence with KMS-symmetric Markovian semigroups is explained in
Coverings and the heat equation on graphs: Stochastic incompleteness, the Feller property, and uniform transience
We study regular coverings of graphs and manifolds with a focus on properties of the heat equation. In particular, we look at stochastic incompleteness, the Feller property and uniform transience;
Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
As an outgrowth of our investigation of non-regular spaces within the context of quantum gravity and non-commutative geometry, we develop a graph Hilbert space framework on arbitrary (infinite)
Topological Poincar\'e type inequalities and lower bounds on the infimum of the spectrum for graphs
We study topological Poincar\'e type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable
Universal Lower Bounds for Laplacians on Weighted Graphs
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In
Gaussian upper bounds for the heat kernel on arbitrary manifolds
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.