  In this paper we propose the precise Wigner-Weyl calculus for the lattice
models defined on the honeycomb lattice. We construct two symbols of operators:
the $\mathscr{B}$-symbol, which is similar to the symbol introduced by F. Buot,
and the $W$ (or, Weyl) symbol. The latter possesses the set of useful
properties. These identities allow us to use it in physical applications. In
particular, we derive topological expression for the Hall conductivity through
the Wigner transformed Green function. This expression may be used for the
description of quantum Hall effect in the systems with artificial honeycomb
lattice, when magnetic flux through the lattice cell is of the order of
elementary quantum of magnetic flux.
