Lᵖ-Lq estimates of the heat kernels on graphs with applications to a parabolic system

Let $G=(V, E)$ be a locally finite connected graph satisfying curvature-dimension conditions ($CDE(n, 0)$ or its strengthened version $CDE'(n, 0))$) and polynomial volume growth conditions of degree m. We systematically establish sharp Lᵖ-bounds and decay-type Lᵖ-Lq estimates for heat operators on G, accommodating both bounded and unbounded Laplacians. The analysis utilizes Li-Yau-type Harnack inequalities and geometric completeness arguments to handle degenerate cases. As a key application, we prove the existence of global solutions to a semilinear parabolic system on G under critical exponents governed by volume growth dimension m.