One of the many interesting conjectures proposed by S. M. Ulam in (5) can be stated as follows:

If G and H are two graphs with p points vi and ui respectively (p ⩾ 3) such that for all i, G — vi is isomorphic with H — ui then G and H are themselves isomorphic.

P. J. Kelly (3) has shown this to be true for trees. The conjecture is, of course, not true for p = 2, but Kelly has verified by exhaustion that it holds for all of the other graphs with at most six points. Harary and Palmer (2) found the same to be true of the seven-point graphs.

In (1) Harary reformulated the conjecture as a problem of reconstructing G from its subgraphs G — vi and derived several of the invariants of G from the collection G — vi.

Among the unsolved problems in graphical enumeration listed in (4) is included the determination of the number of graphs and digraphs with a given partition. Parthasarathy (9) has developed a formulation for counting graphs with a given partition by making a suitable modification of the method given in (2) for the enumeration of graphs. We present here an analogous modification that leads to a formula for the number of digraphs with a given partition. Not surprisingly, the main combinatorial device for this purpose is provided by the classical theorem due to Pólya.

Two complementary techniques are used to study the electrical transport properties related to the use of diamonds as materials for ionizing radiation detectors. Transient photoconductivity using soft x-rays is used to probe the first few microns of the material, while ionizing particle-excited conductivity is used to probe the entire bulk of the material (1 millimeter). Both techniques measure the mean drift distance of free carriers, or the collection distance d. In addition, transient photoconductivity is able to extract the lifetimes and mobilities of the excited carriers. The collection distance measured by the two methods are in agreement, suggesting the material is homogeneous. At an applied field of 10 kV/cm, d is 25 to 30 microns, and, up to a field of 25 kV/cm, d has not saturated. The lifetime varies between 100 and 600 ps, and the mobility varies between 1000 and 4000 cm2/V-s, the range due to natural variations from sample to sample. The primary defects limiting the lifetime are believed to be nitrogen impurities and dislocations.

The electrical properties associated with carrier mobility, μ, and lifetime, τ, have been investigated for the chemical vapor deposited (CVD) diamond films using charged particle-induced conductivity and time resolved transient photo-induced conductivity. The collection distance, d, the average distance which electron and hole depart when driven by an applied electric field E, was measured by both methods. The collection distance is related to the carrier mobility and lifetime by d = μEτ Our measurements show that the collection distance increases linearly with sample thickness for CVD diamond films. The collection distance at the growth side of the CVD diamond film is comparable to that of single crystal natural type IIa diamond; at the substrate side of the film, the collection distance is near zero. No saturation of the collection distance is observed for film thickness up to 500 microns.