  We study a class of solutions of IIB Supergravity which are asymptotically
AdS_5 x Y^{p,q}. They have an R x SO(4) x SU(2) x U(1) isometry and preserve
half of the 8 supercharges of the background geometry. They are described by a
set of second order differential equations that we have found and analysed in a
previous paper, where we studied 1/8 BPS states in the maximally supersymmetric
AdS_5 x S^5 background. These geometries correspond to certain chiral primary
operators of the N=1 superconformal quiver theories, dual to IIB theory on
AdS_5 x Y^{p,q}.
  We also show how to recover the AdS_5 x Y^{p,q} backgrounds by suitably
doubling the number of preserved supersymmetries. We then solve the
differential equations perturbatively in a large AdS_5 radius expansion,
imposing asymptotic AdS_5 x Y^{p,q} boundary conditions. We compute the global
baryonic and mesonic charges, including the R-charge. As for the computation of
the mass, i.e. the conformal dimension D of the dual field theory operators,
which is notoriously subtle in asymptotically AdS backgrounds, we adopt the
general formalism due to Wald and collaborators, which gives a finite result,
and verify the relation D=3 R / 2, demanded by the N=1 superconformal algebra.
