  Message passing graph neural networks (GNNs) are a popular learning
architectures for graph-structured data. However, one problem GNNs experience
is oversquashing, where a GNN has difficulty sending information between
distant nodes. Understanding and mitigating oversquashing has recently received
significant attention from the research community. In this paper, we continue
this line of work by analyzing oversquashing through the lens of the effective
resistance between nodes in the input graph. Effective resistance intuitively
captures the ``strength'' of connection between two nodes by paths in the
graph, and has a rich literature spanning many areas of graph theory. We
propose to use total effective resistance as a bound of the total amount of
oversquashing in a graph and provide theoretical justification for its use. We
further develop an algorithm to identify edges to be added to an input graph to
minimize the total effective resistance, thereby alleviating oversquashing. We
provide empirical evidence of the effectiveness of our total effective
resistance based rewiring strategies for improving the performance of GNNs.
