Graphs provide a versatile framework for modeling complex, non-euclidean relationships across a wide range of domains. To capture both structural topology and node features, Graph Neural Networks (GNNs) have emerged as a method of choice. However, as GNNs grow deeper, they suffer from oversmoothing. In spectral terms, oversmoothing corresponds to collapse of the embedding energy into the lowest-frequency components of the graph Laplacian, thereby degrading the network’s discriminative power. To address this, we propose \(\mathcal {E}^2\) -GNN (Eigenspectrum Enhanced Graph Neural Network)—a framework that mitigates oversmoothing by explicitly preserving spectral diversity during message passing. At its core is a novel class of Chebyshev polynomial filters augmented with Cheeger’s inequality-based regularization, which sustains both the spectral gap and feature variance even in deeper layers. We provide a theoretical analysis demonstrating that our regularized filters retain essential eigenspectral properties of the graph Laplacian, offering a grounded solution to oversmoothing. Extensive experiments across diverse graph datasets demonstrate that \(\mathcal {E}^2\) -GNN consistently outperforms strong baselines. Our model achieves accuracies of 90.58%, 81.87%, 94.59%, 95.68%, 78.71%, 76.97%, 64.89% on Cora, Citeseer, Amazon-Photo, Amazon-Computers, Squirrel, Chameleon, and USA-Air, along with 99.23% F1 score on PPI and 83.03% AUC on Elliptic respectively.