  The problems of maximizing the spectral radius and the number of spanning
trees in a class of bipartite graphs with certain degree constraints are
considered. In both the problems, the optimal graph is conjectured to be a
Ferrers graph. Known results towards the resolution of the conjectures are
described. We give yet another proof of a formula due to Ehrenborg and van
Willigenburg for the number of spanning trees in a Ferrers graph. The main tool
is a result which gives several necessary and sufficient conditions under which
the removal of an edge in a graph does not affect the resistance distance
between the end-vertices of another edge.
