We introduce a "derandomized'' analogue of graph squaring. This operation
increases the connectivity of the graph (as measured by the second eigenvalue)
almost as well as squaring the graph does, yet only increases the degree of the
graph by a constant factor, instead of squaring the degree.
One application of this product is an alternative proof of Reingold's recent breakthrough result that S-T Connectivity in Undirected Graphs can be solved in deterministic logspace.