Sandwiching the (generalized) Randić index

The well-known Randic index of a graph G is defined as R ( G ) = ? ( d u ? d v ) - 1 / 2 , where the sum is taken over all edges u v ? E ( G ) and d u and d v denote the degrees of u and v , respectively. Recently, it was found useful to use its simplified modification: R ' ( G ) = ? ( max { d u , d v } ) - 1 , which represents a lower bound for the Randic index. In this paper we introduce generalizations of R ' and its counterpart, R ? , defined as R α ' ( G ) = ? min { d u α , d v α } and R α ? ( G ) = ? max { d u α , d v α } , for any real number α . Clearly, the former is a lower bound for the generalized Randic index, and the latter is its upper bound. We study extremal values of R α ' and R α ? , and present extremal graphs within the classes of connected graphs and trees. We conclude the paper with several problems.