Note on two generalizations of the Randić index

摘要

For a given graph G, the well-known Randić index of G, introduced by Milan Randić in 1975, is defined as R(G)=∑uv∈E(G)(dudv)−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)=∑uv∈E(G)(max{du,dv})−1. Based on this, recently, Knor et al. introduced two generalizations: Rα′(G)=∑uv∈E(G)min{duα,dvα}andRα′′(G)=∑uv∈E(G)max{duα,dvα},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 Rα′(G) and Rα′′(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.