Emerging trends in distribution grids such as sector integration and high penetration of distributed energy resources increase uncertainty and volatility of the power system. Under such conditions, a full state acquisition is necessary for optimal and safe operation particularly. Given the ability of neural networks to handle incomplete data sets and real-time operation, they have been preferred compared to conventional Gauss-Newton approaches. However, most of the proposed approaches are not related to the physical structure of the grid. Since electrical grids can be described as a graph, the integration of graph structures into the neural network is a consequential step. This paper proposes a Graph Convolutional Network-based approach to state estimation that integrates the graph structure of the grid directly into the filter matrix of the Graph Convolutional Network. Throughout this study, the proposed approach is evaluated using the IEEE 37-feeder test system.