Log-Sobolev inequalities on graphs with positive curvature

Based on a global estimate of the heat kernel, some important inequalities such as Poincare´ inequality and log-Sobolev inequality, furthermore a tight logarithm Sobolev inequality are presented on graphs, just under a positive curvature condition C DE′ (n, K ) with some K > 0. As consequences, we obtainexponential integrability of integrable Lipschitz functions and moment bounds at the same assumption on graphs.