On enumerating certain design problems in terms of bicoloured graphs with no isolates

The enumeration of zero - one matrices with nonzero row and column sums has recently been solved in the context of certain problems arising in architectural design. However, this solution took no account of the symmetries of these configurations, which would be required for any realistic application in an architectural context. In graph-theoretic terms, this new problem calls for counting unlabelled bicoloured graphs with no isolated points. We determine these numbers by deriving their generating function. We also present an equivalent set of recurrence relations, from which the numbers may be easily computed.