Graph representation and learning is a fundamental problem in machine learning area. Graph Convolutional Networks (GCNs) have been recently studied and demonstrated very powerful for graph representation and learning. Graph convolution (GC) operation in GCNs can be regarded as a composition of feature aggregation and nonlinear transformation step. Existing GCs generally conduct feature aggregation on a full neighborhood set in which each node computes its representation by aggregating the feature information of all its neighbors. However, this full aggregation strategy is not guaranteed to be optimal for GCN learning and also can be affected by some graph structure noises, such as incorrect or undesired edge connections. To address these issues, we propose to integrate elastic net based selection into graph convolution and propose a novel graph elastic convolution (GeC) operation. In GeC, each node can adaptively select the optimal neighbors in its feature aggregation. The key aspect of the proposed GeC operation is that it can be formulated by a regularization framework, based on which we can derive a simple update rule to implement GeC in a self-supervised manner. Using GeC, we then present a novel GeCN for graph learning. Experimental results demonstrate the effectiveness and robustness of GeCN.