Graph convolutional networks (GCNs) are vulnerable to perturbations of the graph structure that are either random, or, adversarially designed. The perturbed links modify the graph neighborhoods, which critically affects the performance of GCNs in semi-supervised learning (SSL) tasks. Aiming at robustifying GCNs conditioned on the perturbed graph, the present paper generates multiple auxiliary graphs, each having its binary 0-1 edge weights flip values with probabilities designed to enhance robustness. The resultant edge-dithered auxiliary graphs are leveraged by an adaptive (A)GCN that performs SSL. Robustness is enabled through learnable graph-combining weights along with suitable regularizers. Relative to GCN, the novel AGCN achieves markedly improved performance in tests with noisy inputs, graph perturbations, and state-of-the-art adversarial attacks. Further experiments with protein interaction networks showcase the competitive performance of AGCN for SSL over multiple graphs.