We consider regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this problem, the failed node downloads the information stored at a subset of vertices of the graph for the purpose of recovering the lost data. Compared to the standard setting, regenerating codes on graphs address two additional features. The repair information is moved across the network, and the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. Accordingly, the helpers far away from the failed node may be expected to contribute less data for repair than the nodes in the neighborhood of that node. We analyze regenerating codes with nonuniform download for repair on graphs. Moreover, in the process of repair, the information moved from the helpers to the failed node may be combined through intermediate processing, reducing the repair bandwidth. We derive lower bounds for communication complexity of node repair on graphs, including repair schemes with nonuniform download and intermediate processing, and construct codes that attain these bounds. Additionally, some of the nodes may act as adversaries, introducing errors into the data moved in the network. For repair on graphs in the presence of adversarial nodes, we construct codes that support node repair and error correction in systematic nodes.