Sierpi\' nski Gasket Graphs and Some of Their Properties

The {\it Sierpi'nski fractal} or {\it Sierpi'nski gasket} $\mathrm{\Sigma }$ is a familiar object studied by specialists in dynamical systems and probability. In this paper, we consider a graph $S_n$ derived from the first $n$ iterations of the process that leads to $\mathrm{\Sigma }$, and study some of its properties, including its cycle structure, domination number and pebbling number. Various open questions are posed.