This paper proposes a novel framework for improving graph convolution operations by integrating topological persistence into the Gaussian Mixture Model Convolution (GMMConv) operator. The current works in character recognition lack the integration of topological and persistence methods that help in effectively capturing complex structural features. In this work, two persistence-augmented variants are introduced that adaptively modify Gaussian weighting functions to capture critical topologically significant features, for example connected components, loops, and cycles, in handwritten character graphs. Performance tests on the MNIST Skeleton and MNIST Superpixel datasets demonstrate that the modified Gaussian Mixture Model variant achieves superior classification performance, significantly improving accuracy, precision, and recall. Furthermore, the theoretical analysis establishes stability and optimal noise reduction, confirming the robustness and explainability of the proposed approach.