  The combinatorial heat and wave equations on all finite Cayley and coset
graphs with discrete time variable was solved by Lal {\it et al}. In this
paper, the results of the above paper are extended for infinite Cayley and
coset graphs, whenever the associated groups are discrete, abelian and finitely
generated. Furthermore, we study the solution of the combinatorial heat and
wave equations on a $k$-regular tree whose associated group is a non-abelian
free group on $k$ generators, each of order~$2$. It turns out that in case of
Cayley graphs the solutions to combinatorial heat and wave equations are
weighted sum of the initial functions over balls of certain radius which are
dependent on the discrete time variable.
