We consider elliptic problems on graphs under given loads and bilateral contact conditions. We ask the question which graph is best suited to sustain the loads and the constraints. More precisely, given a cost function we may look at a multiple node of the graph with edge degree $N$ and ask as to whether that node should be resolved into a number of node of edge degree 3, in order to decrease the cost. Thus, we are looking into the sensitivity of a graph carrying an elliptic problem with respect to changing the topology of the graph by releasing nodes with high edge degree. In order words, we are looking into the topological gradient of an elliptic problem on a graph.