On the extremal geometric–arithmetic graphs with fixed number of vertices having minimum degree

Abstract

The geometric–arithmetic index GA of a graph is defined as sum of weights of all edges of graph. The weight of one edge is quotient of the geometric and arithmetic mean of degrees of its end vertices. The predictive power of GA for physico-chemical properties is somewhat better than the predictive power of other connectivity indices. Let G(k, n) be the set of connected simple n-vertex graphs with minimum vertex degree k. In this paper we characterized graphs on which GA index attains minimum value, when the number of vertices of minimum degree k is n- 1 and n- 2. We also gave a conjecture about the structure of the extremal graphs on which this index attains its minimum value and lower bound for this index where k is less or equal to q, and q is approximately 0.0874. For k greater or equal to q and k or n are even, extremal graphs in this set for which GA index attains its minimum value, are regular graphs of degree k.