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 d u 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. [ABSTRACT FROM AUTHOR]