Graph-based semi-supervised learning (GSSL) studies the problem where in addition to a set of data points with few available labels, there also exists a graph structure that describes the underlying relationship between data items. In practice, structure uncertainty often occurs in graphs when edges exist between data with different labels, which may further results in prediction uncertainty of labels. Considering that Gaussian process generalizes well with few labels and can naturally model uncertainty, in this paper, we propose an Uncertainty aware Graph Gaussian Process based approach (UaGGP) for GSSL. UaGGP exploits the prediction uncertainty and label smooth regularization to guide each other during learning. To further subdue the effect of irrelevant neighbors, UaGGP also aggregates the clean representation in the original space and the learned representation. Experiments on benchmarks demonstrate the effectiveness of the proposed approach.