  We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves
the geometric structure of the original continual object. This means that,
speaking of a discrete model, we do not mean just the direct replacement of
differential operators by difference ones but also a discrete analog of the
Riemannian structure. We consider this structure on the appropriate
combinatorial analog of differential forms. Self-adjointness and boundness for
a discrete Laplacian are proved. We define the Green function for this operator
and also derive an explicit formula of the one.
