Combinatorial optimization (CO) problems are challenging as the computation time grows exponentially with the input. Graph Neural Networks (GNNs) show promise for researchers in solving CO problems. This study investigates the effectiveness of GNNs in solving the maximum independent set (MIS) problem, inspired by the intriguing findings of Schuetz et al., and aimed to enhance this solver. Despite the promise shown by GNNs, some researchers observed discrepancies when reproducing the findings, particularly compared to the greedy algorithm, for instance. We reproduced Schuetz' Quadratic Unconstrained Binary Optimization (QUBO) unsupervised approach and explored the possibility of combining it with a supervised learning approach for solving MIS problems. While the QUBO unsupervised approach did not guarantee maximal or optimal solutions, it provided a solid first guess for post-processing techniques like greedy decoding or tree-based methods. Moreover, our findings indicated that the supervised approach could further refine the QUBO unsupervised solver, as the learned model assigned meaningful probabilities for each node as initial node features, which could then be improved with the QUBO unsupervised approach. Thus, GNNs offer a valuable method for solving CO problems by integrating learned graph structures rather than relying solely on traditional heuristic functions. This research highlights the potential of GNNs to boost solver performance by leveraging ground truth during training and using optimization functions to learn structural graph information, marking a pioneering step towards improving prediction accuracy in a non-autoregressive manner.