  This paper introduces certain elliptic Harnack inequalities for harmonic
functions in the setting of the product space $M \times X$, where $M$ is a
(weighted) Riemannian Manifold and $X$ is a countable graph. Since some
standard arguments for the elliptic case fail in this "mixed" setting, we adapt
ideas introduced by Thierry Delmotte for the discrete parabolic case. We then
present some useful applications of this inequality, namely, a kernel estimate
for functions of the Laplacian on a graph. This application in turn provides
sharp estimates for certain Markov kernels on graphs. We then close this paper
with an application to convolution power estimates on finitely generated groups
of polynomial growth.
