The Covariance of Topological Indices that Depend on the Degree of a Vertex

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.