  The {\it Sierpi\'nski fractal} or {\it Sierpi\'nski gasket} $\Sigma$ 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 $\Sigma$, and study some of its properties, including
its cycle structure, domination number and pebbling number. Various open
questions are posed.
