Deep generative models have recently emerged as a promising de novo drug design method. In this respect, deep generative conditional variational autoencoder (CVAE) models are a powerful approach for generating novel molecules with desired drug-like properties. However, molecular graph-based models with disentanglement and multivariate explicit latent conditioning have not been fully elucidated. To address this, we proposed a molecular-graph $\beta$-CVAE model for de novo drug design. Here, we empirically tuned the value of disentanglement and assessed its ability to generate molecules with optimised univariate- or-multivariate properties. In particular, we optimised the octanol-water partition coefficient (ClogP), molar refractivity (CMR), quantitative estimate of drug-likeness (QED), and synthetic accessibility score (SAS). Results suggest that a lower $\beta$ value increases the uniqueness of generated molecules (exploration). Univariate optimisation results showed our model generated molecular property averages of ClogP = 41.07% $\pm$ 0.01% and CMR 66.76% $\pm$ 0.01% by the Ghose filter. Multivariate property optimisation results showed that our model generated an average of 30.07% $\pm$ 0.01% molecules for both desired properties. Furthermore, our model improved the QED and SAS (exploitation) of molecules generated. Together, these results suggest that the $\beta$-CVAE could balance exploration and exploitation through disentanglement and is a promising model for de novo drug design, thus providing a basis for future studies.