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An enhanced label propagation (LP) method called GraphHop was proposed recently. It outperforms graph convolutional networks (GCNs) in the semi-supervised node classification task on various networks. Although the performance of GraphHop was explained intuitively with joint node attribute and label signal smoothening, its rigorous mathematical treatment is lacking. In this paper, we propose a label efficient regularization and propagation (LERP) framework for graph node classification, and present an alternate optimization procedure for its solution. Furthermore, we show that GraphHop only offers an approximate solution to this framework and has two drawbacks. First, it includes all nodes in the classifier training without taking the reliability of pseudo-labeled nodes into account in the label update step. Second, it provides a rough approximation to the optimum of a subproblem in the label aggregation step. Based on the LERP framework, we propose a new method, named the LERP method, to solve these two shortcomings. LERP determines reliable pseudo-labels adaptively during the alternate optimization and provides a better approximation to the optimum with computational efficiency. Theoretical convergence of LERP is guaranteed. Extensive experiments are conducted to demonstrate the effectiveness and efficiency of LERP. That is, LERP outperforms all benchmarking methods, including GraphHop, consistently on five test datasets and an object recognition task at extremely low label rates (i.e., 1, 2, 4, 8, 16, and 20 labeled samples per class).