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

Author

Listed:
  • Mahdi Sohrabi-Haghighat

    (Arak University)

  • Mohammadreza Rostami

    (Mahallat Institute of Higher Education)

Abstract

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 $$(GA_{1})$$ ( G A 1 ) index of G(2, n) is obtained by the following formula: $$\begin{aligned} GA_1^* = \left\{ \begin{array}{ll} n&{}\quad n \le 24, \\ \mathrm{{24}}\mathrm{{.79}}&{}\quad n = 25, \\ \frac{{4\left( {n - 2} \right) \sqrt{2\left( {n - 2} \right) } }}{n}&{}\quad n \ge 26. \\ \end{array} \right. \end{aligned}$$ G A 1 ∗ = n n ≤ 24 , 24 . 79 n = 25 , 4 n - 2 2 n - 2 n n ≥ 26 .

Suggested Citation

  • Mahdi Sohrabi-Haghighat & Mohammadreza Rostami, 2017. "The minimum value of geometric-arithmetic index of graphs with minimum degree 2," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 218-232, July.
    • Handle: RePEc:spr:jcomop:v:34:y:2017:i:1:d:10.1007_s10878-016-0062-4
    DOI: 10.1007/s10878-016-0062-4
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    References listed on IDEAS

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    1. Li, Xueliang & Liu, Bolian & Liu, Jianxi, 2010. "Complete solution to a conjecture on Randic index," European Journal of Operational Research, Elsevier, vol. 200(1), pages 9-13, January.
    2. Rodríguez, José M. & Sigarreta, José M., 2016. "Spectral properties of geometric–arithmetic index," Applied Mathematics and Computation, Elsevier, vol. 277(C), pages 142-153.
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    Cited by:

    1. Milica Milivojević Danas & Ljiljana Pavlović, 2022. "On the extremal geometric–arithmetic graphs with fixed number of vertices having minimum degree," Annals of Operations Research, Springer, vol. 316(2), pages 1257-1266, September.

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