Fractality of the penrose tiling in coordination Cayley’s tree graphs representation method

We offer the mathematical apparatus for mapping lattice and cellular systems into the generalized coordination Cayley’s tree graphs. These Cayley’s trees have random branchiness property and intralayer interbush local intersection. Classical Bethe-Cayley tree graphs don’t have these properties. Bush type simplicial decomposition on Cayley’s tree graphs is introduced, on which the enumerating polynomials or enumerating distributions are built. Within the entropy methodology a fractal characteristic in terms of Radon-Nikodym derivative is introduced, which characterizes quasi-crystalline pentagonal Penrose tiling.