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)=Σuv2√du dv/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, 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. © 2013 Elsevier B.V. All rights reserved.