  We study ancient solutions of polynomial growth to both continuous-time and
discrete-time heat equations on graphs with unbounded Laplacians. We generalize
Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on
graphs with normalized Laplacians to the setting of graphs with unbounded
Laplacians: For a graph admitting an intrinsic metric, which has polynomial
volume growth, the dimension of the space of ancient solutions of polynomial
growth is bounded by the dimension of harmonic functions with the same growth
up to some factor.
