Analysis on Graphs and Noncommutative Geometry

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 Lp 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 type. The metric is investigated at some length and compared with other known metrics on graphs.