  The connected sum construction, which takes as input Gorenstein rings and
produces new Gorenstein rings, can be considered as an algebraic analogue for
the topological construction having the same name. We determine the graded
Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along
the way, we find the graded Betti numbers for fiber products of graded rings;
an analogous result was obtained in the local case by Geller. We relate the
connected sum construction to the doubling construction, which also produces
Gorenstein rings. Specifically, we show that a connected sum of doublings is
the doubling of a fiber product ring.
