Which generalized Randić indices are suitable measures of molecular branching?

Molecular branching is a very important notion, because it influences many physicochemical properties of chemical compounds. However, there is no consensus on how to measure branching. Nevertheless two requirements seem to be obvious: star is the most branched graph and path is the least branched graph. Every measure of branching should have these two graphs as extremal graphs. In this paper we restrict our attention to chemical trees (i.e. simple connected graphs with maximal degree at most 4), hence we have only one requirement that the path be an extremal graph. Here, we show that the generalized Randić index R p ( G ) = ∑ u v ∈ E ( G ) ( d u d v ) p is a suitable measure for branching if and only if p ∈ [ λ , 0 ) ∪ ( 0 , λ ′ ) where λ is the solution of the equation 2 x + 6 x + 1 2 ⋅ 1 2 x + 1 4 ⋅ 1 6 x − 11 4 ⋅ 4 x = 0 in the interval ( − 0.793 , − 0.792 ) and λ ′ is the positive solution of the equation 3 ⋅ 3 x − 2 ⋅ 2 x − 4 x = 0 . These results include the solution of the problem proposed by Clark and Gutman.