We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates give rise to a very explicit probabilistic interpretation, and can be formulated in terms of a weighted metric. Interestingly, this metric is not equivalent to the intrinsic metric. 1991 Mathematics Subject Classification 39A12.
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Metzger et al. (2000) studied this question.
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