Note on two generalizations of the Randić index

For a given graph G, the well-known Randić index of G, introduced by Milan Randić in 1975, is defined as R ( G ) = ¿ u v ¿ E ( G ) ( d u d v ) - 1 / 2 , where the sum is taken over all edges uv and du denotes the degrees of u. Bollobás and Erdös generalized this index by replacing - 1 / 2 with any real number α, which is called the general Randić index. Dvořák et¿al. introduced a modified version of Randić index: R ' ( G ) = ¿ u v ¿ E ( G ) ( max { d u , d v } ) - 1 . Based on this, recently, Knor et¿al. introduced two generalizations: R α ' ( G ) = ¿ u v ¿ E ( G ) min { d u α , d v α } and R α ' ' ( G ) = ¿ u v ¿ E ( G ) max { d u α , d v α } , for any real number α. Clearly, the former is a lower bound for the general Randić index, and the latter is its upper bound. Knor et¿al. studied extremal values of R α ' ( G ) and R α ' ' ( G ) and concluded some open problems. In this paper, we consider the open problems and give some comments and results. Some results for chemical trees are obtained.