Coloring Sierpiński graphs and Sierpiński gasket graphs

Dirichlet Forms on the Sierpiński Gasket

We study not necessarily self-similar Dirichlet forms on the Sierpiński gasket that can be described as limits of compatible resistance networks on the sequence of graphs approximating the gasket. We describe the compatibility conditions in detail, and we also present an alternative description, based on just 3 conductance values and the 3-dimensional space of harmonic functions. In addition, w...

Uniform spanning trees on Sierpiński graphs

We study spanning trees on Sierpiński graphs (i.e., finite approximations to the Sierpiński gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpiński graph and show that this construction gives rise to a multi-type Galton-Watson tree. We derive a number of structural results, for instance on the degree distribution....

Generalized Sierpiński graphs

Sierpiński graphs, S(n, k), were defined originally in 1997 by Klavžar and Milutinović. The graph S(1, k) is simply the complete graph Kk and S(n, 3) are the graphs of Tower of Hanoi problem. We generalize the notion of Sierpiński graphs, replacing the complete graph appearing in the case S(1, k) with any graph. The newly introduced notion of generalized Sierpiński graphs can be seen as a crite...

Random Walks on Sierpiński Graphs: Hyperbolicity and Stochastic Homogenization

We introduce two new techniques to the analysis on fractals. One is based on the presentation of the fractal as the boundary of a countable Gromov hyperbolic graph, whereas the other one consists in taking all possible “backward” extensions of the above hyperbolic graph and considering them as the classes of a discrete equivalence relation on an appropriate compact space. Illustrating these tec...

Multiple solutions for a class of nonlinear elliptic equations on the Sierpiński gasket