  By studying the heat semigroup, we prove Li-Yau type estimates for bounded
and positive solutions of the heat equation on graphs, under the assumption of
the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a
notion of curvature for graphs. Furthermore, we derive that if a graph has
non-negative curvature then it has the volume doubling property, from this we
can prove the Gaussian estimate for heat kernel, and then Poincar\'e inequality
and Harnack inequality. As a consequence, we obtain that the dimension of space
of harmonic functions on graphs with polynomial growth is finite, which
original is a conjecture of Yau on Riemannian manifold proved by Colding and
Minicozzi. Under the assumption of positive curvature on graphs, we derive the
Bonnet-Myers type theorem that the diameter of graphs is finite and bounded
above in terms of the positive curvature by proving some Log Sobolev
inequalities.
