Title: Stable, Efficient, and Flexible Monotone Operator Implicit Graph Neural Networks. Abstract: Implicit graph neural networks (IGNNs) that solve a fixed-point equilibrium equation for representation learning can learn the long-range dependencies (LRD) in the underlying graphs and show remarkable performance for various graph learning tasks. However, the expressivity of IGNNs is limited by the constraints for their well-posedness guarantee. Moreover, when IGNNs become effective for learning LRD, their eigenvalues converge to the value that slows down the convergence, and their performance is unstable across different tasks. In this paper, we provide a new well-posedness condition of IGNNs leveraging monotone operator theory. The new well-posedness characterization informs us to design effective parameterizations to improve the accuracy, efficiency, and stability of IGNNs. Leveraging accelerated operator splitting schemes and graph diffusion convolution, we design efficient and flexible implementations of monotone operator IGNNs that are significantly faster and more accurate than existing IGNNs.