The variation of the Randić index with regard to minimum and maximumdegree

Abstract

The variation of the Randić index R′(G) of a graph G is defined by R′(G) =  uv∈E(G) 1 max{d(u),d(v)} , where d(u) is the degree of vertex u andthesummationextendsoveralledges uv of G. Let G(k,n) be the set of connected simple n-vertex graphs with minimum vertex degree k. In this paper we found in G(k,n) graphs for which the variation of the Randić index attains its minimum value. When k ≤ n 2 the extremal graphs are complete split graphs K∗ k,n−k, which have only vertices of two degrees, i.e. degree k and degree n − 1, and the number of vertices of degree k is n − k, while the number of vertices of degree n − 1 is k. For k ≥ n 2 the extremal graphs have also vertices of two degrees k and n − 1, and the number of vertices of degree k is n 2 maximumdegree. . Further, we generalized results for graphs with given