On the General Randic Index for Certain Families of Trees

The general Randić index \(w_\alpha(G)\) of a graph \(G\) is the sum of the weights \(( d_G(u) d_G(v))^\alpha\) of all edges \(uv\) of \(G\). We give bounds for \(w_{-1}(T)\) when \(T\) is a tree of order \(n\). We also show that \(lim_{n\to\infty} f(n)/n\) exists, and give bounds for the limit, where \(f(n) = \max\{w_{-1}(T): T\) is a tree of order \(n\)}. Finally, we find the expected value and variance of \(w_\alpha(T)\) for certain families of trees.