The identification of potential interactions between drugs and target proteins is crucial in pharmaceutical sciences. The experimental validation of interactions in genomic drug discovery is laborious and expensive; hence, there is a need for efficient and accurate in-silico techniques which can predict potential drug-target interactions to narrow down the search space for experimental verification. In this work, we propose a new framework, namely, Multi-Graph Regularized Nuclear Norm Minimization, which predicts the interactions between drugs and target proteins from three inputs: known drug-target interaction network, similarities over drugs and those over targets. The proposed method focuses on finding a low-rank interaction matrix that is structured by the proximities of drugs and targets encoded by graphs. Previous works on Drug Target Interaction (DTI) prediction have shown that incorporating drug and target similarities helps in learning the data manifold better by preserving the local geometries of the original data. But, there is no clear consensus on which kind and what combination of similarities would best assist the prediction task. Hence, we propose to use various multiple drug-drug similarities and target-target similarities as multiple graph Laplacian (over drugs/targets) regularization terms to capture the proximities exhaustively. Extensive cross-validation experiments on four benchmark datasets using standard evaluation metrics (AUPR and AUC) show that the proposed algorithm improves the predictive performance and outperforms recent state-of-the-art computational methods by a large margin. Software is publicly available at https://github.com/aanchalMongia/MGRNNMforDTI.