  It is shown explicitly how self-similar graphs can be obtained as `blow-up'
constructions of finite cell graphs $\hat C$. This yields a larger family of
graphs than the graphs obtained by discretising continuous self-similar
fractals.
  For a class of symmetrically self-similar graphs we study the simple random
walk on a cell graph $\hat C$, starting in a vertex $v$ of the boundary of
$\hat C$. It is proved that the expected number of returns to $v$ before
hitting another vertex in the boundary coincides with the resistance scaling
factor.
  Using techniques from complex rational iteration and singularity analysis for
Green functions we compute the asymptotic behaviour of the $n$-step transition
probabilities of the simple random walk on the whole graph. The results of
Grabner and Woess for the Sierpi\'nski graph are generalised to the class of
symmetrically self-similar graphs and at the same time the error term of the
asymptotic expression is improved. Finally we present a criterion for the
occurrence of oscillating phenomena of the $n$-step transition probabilities.
