We have performed a molecular-dynamics simulation study of the influence of attractive dispersion interactions on the phase behavior of an ensemble of rigid bead-necklace molecules by gradually scaling the bead-bead interaction from a purely repulsive potential to a full Lennard-Jones potential. The system consisted of 600 bead-necklace molecules each composed of 11 interaction centers (beads). For all values of the attractive potential investigated the system manifested a nematic phase and a smectic A phase and both the isotropic-nematic and nematic-smectic A transitions were found to be weakly first order. The Gibbs-Duhem integration method was employed to trace the isotropic-nematic and nematic-smectic A(P,T) coexistence curves over the range of attractive interactions. With increasing attraction, the temperature-pressure range over which each phase was stable deceased to the advantage of the more ordered coexisting phase with a consequent narrowing of (P,T) space for the mesophases. Systems with weak attractions manifested transition strengths (entropy of transition) for the isotropic-nematic and nematic-smectic A transitions that were only weakly dependent upon temperature-pressure, while systems with stronger attractions exhibited transition strengths that decreased with increasing temperature. The phase behavior and thermodynamics of the isotropic-nematic and nematic-smectic A phase transitions for the systems with relatively strong dispersion interactions were found to be more representative of the behavior of real liquid crystalline materials than systems dominated by repulsive interactions.