  We consider topological indices I that are sums of f(deg(u)) f(deg(v)), where
{u,v} are adjacent vertices and f is a function. The Randi{\'c} connectivity
index or the Zagreb group index are examples for indices of this kind. In
earlier work on topological indices that are sums of independent random
variables, we identified the correlation between I and the edge set of the
molecular graph as the main cause for correlated indices. We prove a necessary
and sufficient condition for I having zero covariance with the edge set.
