Graph Neural Networks (GNNs) have demonstrated remarkable performance across various domains, but face significant limitations in deeper architectures due to the oversmoothing problem—where node representations become increasingly indistinguishable through successive graph convolution operations. This paper proposes a novel information-theoretic approach that primarily addresses oversmoothing through Rényi entropy optimization. Our method quantifies and maximizes the diversity of node representations across network layers using kernel density estimation with Gaussian kernels. By formulating a graph-structured entropy regularization term that respects the underlying topology, we encourage networks to maintain discriminative features while preserving essential structural information. This approach integrates seamlessly with existing GNN architectures, requiring minimal modifications to the training procedure. Extensive experiments on benchmark datasets demonstrate that our Rényi entropy regularization consistently improves performance across multiple GNN variants, with particularly significant gains in deeper architectures where oversmoothing is most problematic. The results show that maintaining representation diversity through entropy maximization effectively counters the homogenization tendency of deep GNNs, establishing a principled foundation for developing more robust and expressive graph neural networks.