Let $Z^{LMO}$ be the 3-manifold invariant of [36]. It is shown that $Z^{LMO}(M) = 1$, if the first Betti number of $M, b_1(M)$, is greater than 3. If $b_1(M) = 3$, then $Z{LMO}(M)$ is completely determined by the cohomology ring of M. A relation of $Z^{LMO}$ with the Rozansky-Witten invariants $Z_X^{RW} [M]$ is established at a physical level of rigour. We show that $Z_X^{RW} [M]$ satisfies appropriate connected sum properties suggesting that the generalized Casson invariant ought to be computable from the LMO invariant.