  In this work, an exact renormalization group treatment of honeycomb lattice
leading to an exact relation between the coupling strengths of the honeycomb
and the triangular lattices is presented. Using the honeycomb and the
triangular duality relation, the critical coupling values of honeycomb and
triangular lattices are calculated exactly by the simultaneous solution of the
renormalized relation and the duality relation, without using the so-called
star-triangular transformation. Apparently, the obtained coupling relation is
unique. It, not only takes place the role of the star triangular relation, but
also it is the only exact relation obtained from renormalization group theory
other than the 1D Ising chain. An exact pair correlation functions expression
relating the nearest neighbors and the next nearest neighbor correlation
functions are also obtained for the honeycomb lattice. Utilizing this
correlation relation, an exact expression of the correlation length of the
honeycomb lattice is calculated analytically for the coupling constant values
less than the critical value in the realm of the scaling theory. The critical
exponents $\nu$ and $\alpha$ are also calculated as $\nu=1$ and $\alpha=0$.
