Abstract
The geometric–arithmetic index GA of a graph is defined as sum of weights of all edges of graph. The weight of one edge is quotient of the geometric and arithmetic mean of degrees of its end vertices. The predictive power of GA for physico-chemical properties is somewhat better than the predictive power of other connectivity indices. Let G(k, n) be the set of connected simple n-vertex graphs with minimum vertex degree k. In this paper we characterized graphs on which GA index attains minimum value, when the number of vertices of minimum degree k is \(n-1\) and \(n-2\). We also gave a conjecture about the structure of the extremal graphs on which this index attains its minimum value and lower bound for this index where k is less or equal to \(q_0\), and \(q_0\) is approximately 0.0874. For k greater or equal to \(q_0\) and k or n are even, extremal graphs in this set for which GA index attains its minimum value, are regular graphs of degree k.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aouchiche, M., & Hansen, P. (2007). On a conjecture about the Randić index. Discrete Mathematics, 307(2), 262–265.
Bianchi, M., Cornaro, A., Palacios, J. L., & Torriero, A. (2019). Lower bounds for the geometric-arithmetic index of graphs with pendant and fully connected vertices. Discrete Applied Mathematics, 257, 53–59.
Caporossi, G., Gutman, I., & Hansen, P. (1999). Variable neighborhood search for extremal graphs. 4. Chemical trees with extremal connectivity index. Computers & Chemistry, 23, 469–477.
Das, K., Gutman, I., & Furtula, B. (2011). Survey on geometric-arithmetic indices of graphs. MATCH-Communications in Mathematical and in Computer Chemistry, 65, 595–644.
Deng, H., Elumalai, S., & Balachandran, S. (2018). Maximum and second maximum geometric-arithmetic index of tricyclic graphs. MATCH-Communications in Mathematical and in Computer Chemistry, 79, 467–475.
Divnić, T., Milivojević, M., & Pavlović, Lj. (2014). Extremal graphs for the geometric-arithmetic index with given minimum degree. Discrete Applied Mathematics, 162, 386–390.
Du, Z., Zhou, B., & Trinajsitć, N. (2011). On geometric-arithmetic indices of (Molecular) trees, unicyclic graphs and bicyclic graphs. MATCH-Communications in Mathematical and in Computer Chemistry, 66, 681–697.
Fath-Tabar, G., Furtula, B., & Gutman, I. (2010). A new geometric-arithmetic index. Journal of Mathematical Chemistry, 47(1), 477–486.
Ghorbani, M., & Khaki, A. (2010). A note on the fourth version on the geometric-arithmetic index. Optoelectronics and Advanced Materials - Rapid Communications, 4, 2212–2215.
Graovac, A., Ghorbani, M., & Hosseinzadeh, M. A. (2011). Computing fifth geometric-arithmetic index for nanostar dendrimers. Journal of Mathematical Nanoscience, 1, 33–42.
Harary, F. (1962). The Maximum Connectivity of a Graph. Proceedings of the National Academy of Sciences of the United States of America, 48, 1142–1146.
Martínez, A., & Rodríguez, J. M. (2018). New lower bounds for the geometric-arithmetic index. MATCH-Communications in Mathematical and in Computer Chemistry, 79, 467–475.
Milovanović, I. Ž, Milovanović, E. I., & Matejić, M. M. (2018). On upper bounds for the geometric-arithmetic topological index. MATCH-Communications in Mathematical and in Computer Chemistry, 80, 109–127.
Mogharrab, M., & Fath-Tabar, G. (2011). Some bounds on \(GA_1\) index of graphs. MATCH-Communications in Mathematical and in Computer Chemistry, 65, 33–38.
Sohrabi-Haghighat, M., & Rostami, M. (2015). Using linear programming to find the extremal graphs with minimum degree 1 with respect to Geometric-Arithmetic index. Applied mathematics in Engineering, Management and Technology, 3(1), 534–539.
Sohrabi-Haghighat, M., & Rostami, M. (2017). The minimum value of geometric-arithmetic index of graphs with minimu degree 2. Journal of Combinatorial Optimization, 34(1), 218–232.
Tomescu, I., Marinescu-Ghemeci, R., & Mihai, G. (2009). On dense graphs having minimum Randić index. The Romanian Journal of Information Science and Technology (ROMJIST), 12(4), 455–465.
Vukičević, D., & Furtula, B. (2009). Topological index based on the ratios of geometrical and arithmetical means of end -vertex degrees of edges. Journal of Mathematical Chemistry, 46(2), 1369–1376.
Wilczek, P. (2018). New geometric-arithmetic indices. MATCH-Communications in Mathematical and in Computer Chemistry, 79, 5–54.
Yuan, Y., Zhou, B., & Trinajstić, N. (2010). On geometric-arithmetic index. Journal of Mathematical Chemistry, 47(2), 833–841.
Zhou, B., Gutman, I., Furtula, B., & Du, Z. (2009). On two types of geometric-arithmetic index. Chemical Physics Letters, 482, 153–155.
Acknowledgements
This research was supported by Serbian Ministry for Education and Science, Grant No. 174033 “Graph Theory and Mathematical Programming with Applications to Chemistry and Computing”.
Funding
We declare that we have no financial interest.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We declare that we have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Milivojević Danas, M., Pavlović, L. On the extremal geometric–arithmetic graphs with fixed number of vertices having minimum degree. Ann Oper Res 316, 1257–1266 (2022). https://doi.org/10.1007/s10479-022-04778-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-022-04778-1