  This paper deals with some of the algebraic properties of Sierpi\'nski graphs
and a family of regular generalized Sierpi\'nski graphs. For the family of
regular generalized Sierpi\'nski graphs, we obtain their spectrum and
characterize those graphs that are Cayley graphs. As a by-product, a new family
of non-Cayley vertex-transitive graphs, and consequently, a new set of
non-Cayley numbers are introduced. We also obtain the Laplacian spectrum of
Sierpi\'nski graphs in some particular cases, and make a conjecture on the
general case.
