In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume growth and nonnegative curvature $CDE'(n,0)$ the Sobolev inequality, Nash inequality, Faber-Krahn inequality, Log-Sobolev inequalities, discrete and continuous-time uniform upper estimate of heat kernel are all true on graph.
Categories: math.DG math.FA