Fractals are a fascinating branch of mathematics. Apart from its visual appeal, self-similar and recursive structures such as Hilbert curves, Cantor sets, and the Menger sponge serve as important counterexamples in analysis and topology. In this talk, we shall investigate counting problems and spectral properties of a graph called the “Sierpinski gasket”. We will be studying how the number of closed walks converges 3-adically on iterates converging to the Sierpinski gasket and show that this argument generalises to other fractal graphs such as n-flakes.

Among many things, we will present both theoretical and empirical results on growth of the dimension of eigenspaces on the Sierpinski gasket.

In particular, we provide an explicit recursive construction of eigenvectors which obeys the empirical patterns given that eigenvectors satisfy certain symmetry and boundary conditions.