Graph Convolutional Networks (GCNs) have emerged as powerful tools for learning on graph-structured data, yet the behavior of dropout in these models remains poorly understood. This paper presents a comprehensive theoretical analysis of dropout in GCNs, revealing that its primary role differs fundamentally from standard neural networks - preventing oversmoothing rather than co-adaptation. We demonstrate that dropout in GCNs creates dimension-specific stochastic sub-graphs, leading to a form of structural regularization not present in standard neural networks. Our analysis shows that dropout effects are inherently degree-dependent, resulting in adaptive regularization that considers the topological importance of nodes. We provide new insights into dropout's role in mitigating oversmoothing and derive novel generalization bounds that account for graph-specific dropout effects. Furthermore, we analyze the synergistic interaction between dropout and batch normalization in GCNs, uncovering a mechanism that enhances overall regularization. Our theoretical findings are validated through extensive experiments on both node-level and graph-level tasks across 14 datasets. Notably, GCN with dropout and batch normalization outperforms state-of-the-art methods on several benchmarks, demonstrating the practical impact of our theoretical insights.