Sandwiching the (generalized) Randić index

The well-known Randić index of a graph GG is defined as R(G)=∑(du⋅dv)−1/2R(G)=∑(du⋅dv)−1/2, where the sum is taken over all edges uv∈E(G)uv∈E(G) and dudu and dvdv denote the degrees of uu and vv, respectively. Recently, it was found useful to use its simplified modification: R′(G)=∑(max{du,dv})−1R′(G)=∑(max{du,dv})−1, which represents a lower bound for the Randić index. In this paper we introduce generalizations of R′R′ and its counterpart, R″R″, defined as Rα′(G)=∑min{duα,dvα} and Rα″(G)=∑max{duα,dvα}, for any real number αα. Clearly, the former is a lower bound for the generalized Randić index, and the latter is its upper bound. We study extremal values of Rα′ and Rα″, and present extremal graphs within the classes of connected graphs and trees. We conclude the paper with several problems.