Extremal graphs for the geometric–arithmetic index with given minimum degree

Abstract

Let G(k,n) be the set of connected simple n-vertex graphs with minimum vertex degree k. The geometric–arithmetic index GA(G) of a graph G is defined by GA(G) =  uv 2 √ dudv du+dv , where d(u) is the degree of vertex u and the summation extends over all edges uv of G. In this paper we find for k ≥ ⌈k0⌉, with k0 = q0(n − 1), where q0 ≈ 0.088 is the unique positive root of the equation q √ q+q+3 √ q−1 =0,extremalgraphsinG(k,n)forwhich the geometric–arithmetic index attains its minimum value, or we give a lower bound. We show that when korniseven, the extremal graphs are regular graphs of degree k.