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Assuming stochastic completeness of the underlying graph, we further derive an explicit pointwise formula for this operator:\n  \\[\n  \\log(-\\Delta)\\:u(x)\n  =\\frac{1}{\\mu(x)}\\sum_{y\\neq x}W_{\\log}(x,y)\\,(u(x)-u(y))\n  -\\frac{1}{\\mu(x)}\\sum_{y}W(x,y)\\,u(y)\n  +\\Gamma'(1)\\,u(x).\n  \\] In the case of weighted lattice graphs with uniformly positive vertex measures, we obtain sharp two-sided bounds for the associated logarithmic kernel. 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