The field of graph learning has been substantially advanced by the development of deep learning models, in particular graph neural networks. However, one salient yet largely under-explored challenge is detecting Out-of-Distribution (OOD) nodes on graphs. Prevailing OOD detection techniques developed in other domains like computer vision, do not cater to the interconnected nature of graphs. This work aims to fill this gap by exploring the potential of a simple yet effective method -- OOD score propagation, which propagates OOD scores among neighboring nodes along the graph structure. This post hoc solution can be easily integrated with existing OOD scoring functions, showcasing its excellent flexibility and effectiveness in most scenarios. However, the conditions under which score propagation proves beneficial remain not fully elucidated. Our study meticulously derives these conditions and, inspired by this discovery, introduces an innovative edge augmentation strategy with theoretical guarantee. Empirical evaluations affirm the superiority of our proposed method, outperforming strong OOD detection baselines in various scenarios and settings.