Given an undirected graph, the resistance distance between two nodes is the
resistance one would measure between these two nodes in an electrical network
if edges were resistors. Summing these distances over all pairs of nodes yields
the so-called Kirchhoff index of the graph, which measures its overall
connectivity. In this work, we consider Erdos-Renyi random graphs. Since the
graphs are random, their Kirchhoff indices are random variables. We give
formulas for the expected value of the Kirchhoff index and show it concentrates
around its expectation. We achieve this by studying the trace of the
pseudoinverse of the Laplacian of Erdos-Renyi graphs. For synchronization (a
class of estimation problems on graphs) our results imply that acquiring
pairwise measurements uniformly at random is a good strategy, even if only a
vanishing proportion of the measurements can be acquired
