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