  In this paper, we consider three typical problems on a locally finite
connected graph. The first one is to study the Bochner formula for the
Laplacian operator on a locally finite connected graph. We use the Bochner
formula to derive the Bernstein type estimate of the heat equation. The second
is to derive the Reilly type formula of the Laplacian operator. The last one is
to obtain global positive solution to porous-media equation via the use of
Aronson-Benilan argument. There is not much work in the direction of the study
of nonlinear heat equations on locally finite connected graphs.
