Enumeration of graphs with signed points and lines

Our object is to enumerate graphs in which the points or lines or both are assigned positive or negative signs. We also treat several associated problems for which these configurations are self-dual with respect to sign change. We find that the solutions to all of these counting problems can be expressed as special cases of one general formula involving the concatenation of the cycle index of the symmetric group with that of its pair group. This counting technique is based on Polya's Enumeration Theorem and the Power Group Enumeration Theorem. Using a suitable computer program, we list the number of graphs of each type considered up to twelve points. Sharp asymptotic estimates are also obtained.