The minimum value of geometric-arithmetic index of graphs with minimum degree 2

The geometric-arithmetic index was introduced in the chemical graph theory and it has shown to be applicable. The aim of this paper is to obtain the extremal graphs with respect to the geometric-arithmetic index among all graphs with minimum degree 2. Let G(2, n) be the set of connected simple graphs on n vertices with minimum degree 2. We use linear programming formulation and prove that the minimum value of the first geometric-arithmetic ((GA_{1})) index of G(2, n) is obtained by the following formula:

$$begin{aligned} GA_1^* = left{ begin{array}{ll} n&{}quad n le 24, mathrm{{24}}mathrm{{.79}}&{}quad n = 25, frac{{4left( {n - 2} right) sqrt{2left( {n - 2} right) } }}{n}&{}quad n ge 26. end{array} right. end{aligned}$$