$L^p$-$L^q$ 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 $CD{E}^{\mathrm{\prime }}(n,0))$) 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 $m$.