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)=∑uv2dudvdu+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 qq+q+3q−1=0, extremal graphs in G(k,n) for which the geometric–arithmetic index attains its minimum value, or we give a lower bound. We show that when k or n is even, the extremal graphs are regular graphs of degree k.