Extremal graphs for the Randić index when minimum, maximum degrees and order of graphs are odd

Abstract

Let be the set of connected simple n-vertex graphs with minimum vertex degree and maximum vertex degree . The Randić index of a graph G is defined by , where is the degree of vertex u and the summation extends over all edges uv of G. In this paper, we find for , and k, m, n are odd, extremal graphs in for which the Randić index attains its minimum value. We show that the extremal graphs have vertices of degree k, m and , the number of vertices of degree is one and the number of vertices of degree k is as close to as possible.