Web of Science (Emerging Sources Citation Index), ISC

Document Type : Original Research Article

Authors

1 Department of Mathematics College of Computer Sciences and Mathematics, University of Kufa, Najaf, Iraq

2 Ministry of Education, Dhi Qar Education Directorate. Dhi Qar Centre, Iraq

Abstract

The geometric arithmetic  index is one of the recently most investigated degree-based molecular structure descriptors that have applications in chemistry. For a graph , the  index is defined as , where  denotes the degree of a vertex  in . In this paper, we obtain the general formulas of geometric arithmetic  index for two types of trees graphs. Also, we provide the general formulas to the bicyclic graphs after attaching branches of  for all vertices. Finally, we computed the geometric arithmetic  index to the molecular graph representing the molecular structure of the bicyclic chemical graphs.

Graphical Abstract

The geometric arithmetic (GA) index of certain trees and bicyclic graphs

Keywords

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-------------------------------------------------------------------------------

 

 

-------------------------------------------------------------------------------

 

 

 

 

[1] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH Verlag GmbH, 2000. [crossref], [Google Scholar], [Publisher]
[2] M. Randić, J. Am. Chem. Soc., 1975, 97, 6609-6615. [crossref], [Google Scholar], [Publisher]
[3] B. Furtula, A. Graovac, D. Vukičević, Discret. Appl. Math., 2009, 157, 2828-2835. [crossref], [Google Scholar], [Publisher]
[4] I. Gutman, O.E. Polansky, Graph Theory and Molecular Orbitals. In: Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986, 42-45. [crossref], [Google Scholar], [Publisher]
[5] J. Devillers, A.T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPR, (Ed), Gordon & Breach Science Publishers: Amsterdam, 1999.  
[6] D. Vukičević, B. Furtula, J. Math. Chem., 2009, 46, 1369-1376. [crossref], [Google Scholar], [Publisher]
[7] K.C. Das, I. Gutman, B. Furtula, MATCH Commun. Math. Comput. Chem., 2011, 65, 595-644. [Pdf], [Google Scholar], [Publisher]
[8] K.C. Das, MATCH Commun. Math. Comput. Chem, 2010, 64, 619-630. [Pdf], [Google Scholar], [Publisher]
[9] K.C. Das, I. Gutman, B. Furtula, Discret. Appl. Math., 2011, 159, 2030-2037. [crossref], [Google Scholar], [Publisher]
[10] K.C. Das, N. Trinajstić, Chem. Phys. Lett., 2010, 497, 149-151. [crossref], [Google Scholar], [Publisher]
[11] T. Divnić, M. Milivojević, L. Pavlović, Discret. Appl. Math., 2014, 162, 386-390. [crossref], [Google Scholar], [Publisher]
[12] Z. Du, B. Zhou, N. Trinajstić, MATCH Commun. Math. Comput. Chem., 2011, 66, 681-697. [Pdf], [Google Scholar], [Publisher]
[13] M. Mogharrab, MATCH Commun. Math. Comput. Chem., 2011 65, 33-38. [Pdf], [Google Scholar], [Publisher]
[14] Y. Yuan, B. Zhou, N. Trinajstić, J. Math. Chem., 2010, 47, 833-841. [crossref], [Google Scholar], [Publisher]
[15] C.C. Wei, H. Ali, M.A. Binyamin, M.N. Naeem, J.B. Liu, Mathematics, 2019, 7, 1-22. [crossref], [Google Scholar], [Publisher]
[16] H. Ali, M.A. Binyamin, M.K. Shafiq, W. Gao, Mathematics, 2019, 7, 1-17. [crossref], [Google Scholar], [Publisher]
[17] B. Zhou, R. Xing, Zeitschrift für Naturforsch. A, 2011, 66, 61-66. [crossref], [Google Scholar], [Publisher]
[18] X. Zhang, M.K. Siddiqui, M. Naeem, Symmetry, 2018, 10, 1-12.
[19] N. Chidambaram, S. Mohandoss, X. Yu, X. Zhang, AIMS Math., 2020, 5, 6521-6536. [crossref], [Google Scholar], [Publisher]
[20] J. Chen, The atom-bond connectivity index of graphs, Ph.D. thesis, Xiamen University, (in Chinese), 2011.
[21] J.S. Chen, X.F. Guo, Appl. Math. J. Chinese Univ., 2012, 27, 243-252. [crossref], [Google Scholar], [Publisher]
[22] M.A. Mohammed, K.A. Atan, A. M. Khalaf, M.R.M. Said, R. Hasni, Bridgi. Res. End., 2016, 2015, 263-267.
[23] S.A. Hosseini, M.B. Ahmadi, I. Gutman, MATCH Commun. Math. Comput. Chem., 2014, 71, 5-20. [Pdf], [Google Scholar], [Publisher]
[24] M.R. Farahani, Acta Chim. Slov., 2012, 59, 965-968. [Pdf], [Google Scholar], [Publisher]
[25] M.R. Farahani, World Appl. Sci. J., 2012, 20, 1248-1251. [crossref], [Google Scholar], [Publisher]
[26] M.R. Farahani, W. Gao, J. Chem. Pharm. Res., 2015 7, 535-539. [Pdf], [Google Scholar], [Publisher]
[27] M.R. Farahani, J. Chem. Acta., 2013, 2, 22-25.  [Google Scholar], [Publisher]
[28] W. Gao, M.R. Farahani, J. Nanotech. 2016, 2016, Article ID 3129561, 1-6. [crossref], [Google Scholar], [Publisher]
[29] D. Afzal, S. Hussain, M. Aldemir, M. Farahani, F. Afzal, Eurasian Chem. Commun., 2020, 2, 1117-1125. [crossref], [Google Scholar], [Publisher]  
[30] F. Chaudhry, M. Ehsan, D. Afzal, M.R. Farahani, M. Cancan, E. Ediz, J. Dis. Math. Sci. Cryp., 2021, 24, 401-414. [crossref], [Google Scholar], [Publisher]  
[31] F. Chaudhry, M. Ehsan, F. Afzal, M.R. Farahani, M. Cancan, I. Ciftci, Eurasian Chem. Commun., 2021, 3, 146-153. [crossref], [Google Scholar], [Publisher]   
[32] F. Chaudhry, M. Ehsan, F. Afzal, M. Farahani, M. Cancan, I. Ciftci, Eurasian Chem. Commun., 2021, 3, 103-109. [crossref], [Google Scholar], [Publisher]   
[33] A.J.M. Khalaf, S. Hussain, D. Afzal, F. Afzal, A. Maqbool, J. Dis. Math. Sci. Cryp., 2020, 23, 1217-1237. [crossref], [Google Scholar], [Publisher]   
[34] D.Y. Shin, S. Hussain, F. Afzal, C. Park, D. Afzal, M.R. Farahani, Front. Chem., 2021, 8, 613873-613877. [crossref], [Google Scholar], [Publisher]
[35] M. Cancan, S. Ediz, M.R. Farahani, Eurasian Chem. Commun., 2020, 2, 641-645. [crossref], [Google Scholar], [Publisher]    
[36] A.Q. Baig, M. Naeem, W. Gao, J.B. Liu, Eurasian Chem. Commun., 2020, 2, 634-640. [crossref], [Google Scholar], [Publisher]  
[37] F. Afzal, M.A. Razaq, D. Afzal, S. Hameed, Eurasian Chem. Commun., 2020, 2, 652-662. [crossref], [Google Scholar], [Publisher]  
[38] M. Alaeiyan, C. Natarajan, G. Sathiamoorthy, M.R. Farahani, Eurasian Chem. Commun., 2020, 2, 646-651. [crossref], [Google Scholar], [Publisher]  
[39] M. Imran, S.A. Bokhary, S. Manzoor, M.K. Siddiqui, Eurasian Chem. Communm., 2020, 2, 680-687. [crossref], [Google Scholar], [Publisher]  
[40] A. Alsinai, A. Alwardi, M.R. Farahani, N.D. Soner, Eurasian Chem. Commun., 2021, 3, 219-226. [crossref], [Google Scholar], [Publisher]  
[41] A. Alsinai, A. Alwardi, N.D. Soner. J. D. Math. Sci. Cryp., 2021, 24, 307-324. [crossref], [Google Scholar], [Publisher]  
[42] F. Afzal, S. Hussain, D. Afzal, S. Razaq, J. Inf. Opt. Sci., 2020, 41, 1061-1076. [crossref], [Google Scholar], [Publisher]