Penrose tiling fractality in coordination Cayley’s tree graphs representation

We offer the mathematical apparatus for mapping lattice and cellular systems into the generalized coordination Cayley’s tree graphs. These Cayley’s trees have a random branchiness property and an 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 three types of fractal characteristics are introduced, which characterize quasi-crystalline pentagonal Penrose tiling. The quantitative estimate for the frontal-radial fractal percolation on a Cayley’s tree graph of a Penrose tiling leading to the overdimensioned effect is calculated.