Enumeration method for the hole line expansion diagrams

We point out that to any order of the hole line expansion simple graph theoretical arguments yield an upper bound on the number of hole lines contained in the diagrams. The derivation of this upper bound suggests a procedure which could be useful in the actual enumeration of the hole line expansion diagrams. The cases of the binding energy and of the mass operator are treated in sufficient detail to make obvious the extension of our results to other quantities. {copyright} {ital 1997} {ital The American Physical Society}