Studying the heat semigroup, we prove Li-Yau inequality for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE’(n,0), which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities.

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