Ancient caloric functions and parabolic frequency on graphs

We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\text{bb}Z,$ we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency.