Find exact or asymptotic formulae for the number of labelled graphs of order n having a certain property. The property we are interested in in this note is that of being k-coloured and having connectivity at least l. Special cases of this problem have been tackled by many authors; in particular Gilbert [6], Read [9] and Robinson [11] found exact formulae, and Read and Wright [10], and Wright [12], [13] found asymptotic expressions (for many other examples see [7]). Recently Harary and Robinson [8] counted labelled bipartite blocks, that is 2-connected bipartite graphs. (For terms not defined here and general background in graph theory see [1].) Our present investigations have been prompted by [8]; in particular, as a very special case of our results, we shall prove the conjecture published in [8].

The exact formulae appearing in the enumeration of labelled graphs in general, and in the enumeration of k-coloured labelled graphs in particular, tend to be very pleasing, especially because of the functional equations relating them.

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D1 in the inverse image of N under p. The choice of D1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ1. Let π : D → D1 be the fiber bundle obtained by exponentiating the normal bundle of D1 in D. Let μ be the volume form of D1.

A t-cap in a geometry is a set of t points no three of which are collinear. A (t, k)-cap is a set of t points, no k + 1 of which are collinear.

It has been shown in [3] that any Desarguesian PG(2n, q2) is a disjoint union of (q2n+l – l)/(q – 1) (q2n+l – l)/(q + l)-caps. These caps were obtained as intersections of 2n Hermitian Varieties of a certain kind; the intersection of 2n + 1 such varieties was empty. Furthermore, the caps in question constituted the ‘large points” of a PG(2n, q), with the incidence relation defined in a natural way.

It seemed at the time that nothing similar could be said about odd-dimensional projective geometries, if only because |PG(2n – 1, q)| ∤ |PG(2n – l, q2)|.