L^p-L^q 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 L^p-bounds and decay-type L^p-L^q 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.