Document Type : Original Research Article


1 Department of Mathematics and Statistics, The University of Lahore, Lahore, 54000, Pakistan

2 Department of Health and Physical Education, Lahore College for Women University, Pakistan

3 MCS, National University of Science and Technology, Islamabad, Pakistan

4 Department of Applied Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran

5 Faculty of Education, Van Yuzuncu Yıl University, Van, Turkey


Chemical graph theory is related to the structure of different chemical compounds. A chemical graph represents the molecule of the substance. Chemical graph theory provides the connection between the real number and the different physical, chemical, and biological properties of the chemical species. By implementing the mathematical tools, a chemical graph is converted into a real number. This number can have the predicating ability about the properties of the molecule. In this article, we find some topological indices via M-polynomial for the Starphene graph.

Graphical Abstract

Calculating the topological indices of Starphene graph via M-polynomial approach


Main Subjects


In theoretical chemistry, topological indices have received substantial attention. The topological directories enable us to understand easily the different structural properties of the chemical substance. So, topological index has a key role, to demonstrate the chemical structure. These indices are obtained by using mathematical tools and are used to explain a molecule under testing.

Topological indices are worked out from their definition; however, these can also be deliberate by resources of their M-polynomial. Table 2 shows some important degree-based topological indices. M-polynomial is also a graph representative mathematical object. With the support of M-polynomial, we compute relatively a lot of degree dependent topological invariant that are represented in Table 3.

For a graph G, the well-known M-polynomial is defined as [6]:



Where δ = min {dv|v V(G)}, Δ = max {dv|v V(G)} and mij(G) is the number of edges vuE(G) such that {dv ; du} = {i; j}. Table 3 shows some well-known degree-based topological directories computed by the use of M-Polynomial. M-polynomial of various graphs has been previously introduced [1, 3-5,10,12, 14-30]. In this paper, we have computed M-polynomial and topological directories of St(l,p,r).

Figures 1 and 2 show a Starphene St(l,p,r) which can be considered as a configuration acquired by merging three linear polyenes of length l, p and r, respectively.


TABLE 2 Degree-based topological indices

Where operator used are defined as

M-Polynomial of Starphene graph

Theorem 3.1. If Starphene is denoted by    St(l,p,r) then for l,p,r ≥ 3, M-polynomial of St(l,p,r) is M[St(l,p,r),a, b] = 9a2b2 + (4(l+p+r)-18)a2b3  + (l+p+r)a3b3.

Proof. Let St(l,p,r) be a Starphene then via Table 1 besides Figure 2, the edge partition of St(l,p,r) is

E2;2(St(l,p,r))={e=uv∈St(l,p,r)du=2;dv=2} →|E2;2St(l,p,r)|=9

E2;3(St(l,p,r))={e=uv∈St(l,p,r):du=2;dv=3} →|E2;3St(l,p,r)|=(4(l+p+r)-18)

E3;3(St(l,p,r))={e=uv∈St(l,p,r):du=3;dv=3} →|E3;3St(l,p,r)|=(l+p+r)

The following outcome comes by applying the interpretation of M-polynomial

Topological directories of Starphene

Theorem 4.1. Let St (l; p; z) be a Starphene:

M[St(l,p,r);a;b]=9a2b2+(4(l+p+r)-18)a2b3 +(l+p+r)a3b3

Proof. Let M[St(l,p,r);a;b]=9a2b2+(4(l+p+r)-18)a2b3+(l+p+r)a3b3

The first Zagreb index


A graphical sketch of topological indices of St(l,p,r) is shown in Figure 4. By means of graphs, we hold up the performance of the topological indices along different parameters. Although the graphs look to be identical, they in fact have distinct gradients.


In the current article, we worked out a closed-form of M-polynomial for the graph Starphene and then we derived numerous degree-based topological directories as well. Topological indices help to reduce the number of experiments. These topological indices can help to understand more about the biological, chemical, and physical characteristics of a molecule. The topological index has a significant role that represents the chemical structure of a molecule to a real number. It is used to express the molecule which is under deliberation. These results are very helpful in estimating the physico-chemical properties for these chemical structures.


The authors extend their real appreciation to the reviewers for their insightful comments and technical suggestions to enhance quality of the article.


Farkhanda Afzal:

Mohammad Reza Farahani:

Murat Cancan:


How to cite this article: Faryal Chaudhry, Sumera Sattar, Muhammad Ehsan, Farkhanda Afzal*, Mohammad Reza Farahani, Murat Cancan. Calculating the topological indices of starphene graph via M-polynomial approach. Eurasian Chemical Communications, 2021, 3(10), 656-664. Link:


Copyright © 2021 by SPC (Sami Publishing Company) + is an open access article distributed under the Creative Commons Attribution License(CC BY)  license  (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[1] D. Afzal, S. Hussain, M. Aldemir, M. Farahani, F. Afzal, Eurasian Chem. Commun., 2020, 2, 1117-1125. [crossref], [Google Scholar], [Publisher]
[2] A.T. Balaban, Chem. Phys. Let., 1982, 89, 399-404. [crossref], [Google Scholar], [Publisher]
[3] 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]  
[4] F. Chaudhry, M. Ehsan, F. Afzal, M.R. Farahani, M. Cancan, I. Ciftci, Eurasian Chem. Commun., 2021, 3, 146-153. [crossref], [Google Scholar], [Publisher
[5] F. Chaudhry, M. Ehsan, F. Afzal, M.R. Farahani, M. Cancan, I. Ciftci, Eurasian Chem. Commun., 2021, 3, 103-109. [crossref], [Google Scholar], [Publisher
[6] E. Deutsch, S. Klavžar, Iran. J. Math. Chem., 2015, 6, 93-102. [crossref], [Google Scholar], [Publisher
[7] S. Fajtlowicz, Congr, 1987, 60, 189-197.             
[8] B. Furtula, A. Graovac, D. Vukičević, J. Math. Chem., 2010, 48, 370-380. [crossref], [Google Scholar], [Publisher
[9] I. Gutman, N. Trinajstić, Chem. Phys. Let., 1972, 17, 535-538. [crossref], [Google Scholar], [Publisher
[10] 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
[11] M. Randić, J. American Chem. Soc., 1975, 97, 6609-6615. [crossref], [Google Scholar], [Publisher
[12] D.Y. Shin, S. Hussain, F. Afzal, C. Park, D. Afzal, M.R. Farahani, Front. Chem., 2021, 8, 613873-613877. [crossref], [Google Scholar], [Publisher
[13] J. Holec, B. Cogliati, J. Lawrence, A. Berdonces-Layunta, P. Herrero, Y. Nagata, M. Banasiewicz, Angew. Chem., 2021, 60, 7752-7758. [crossref], [Google Scholar], [Publisher
[14] M. Cancan, S. Ediz, M.R. Farahani, Eurasian Chem. Commun., 2020, 2, 641-645. [crossref], [Google Scholar], [Publisher]   
[15] A.Q. Baig, M. Naeem, W. Gao, J.B. Liu, Eurasian Chem. Commun., 2020, 2, 634-640. [crossref], [Google Scholar], [Publisher]    
[16] F. Afzal, M.A. Razaq, D. Afzal, S. Hameed, Eurasian Chem. Commun., 2020, 2, 652-662. [crossref], [Google Scholar], [Publisher]     
[17] M. Alaeiyan, C. Natarajan, G. Sathiamoorthy, M.R. Farahani, Eurasian Chem. Commun., 2020, 2, 646-651. [crossref], [Google Scholar], [Publisher]     
[18] M. Imran, S.A. Bokhary, S. Manzoor, M.K. Siddiqui. Eurasian Chem. Communm., 2020, 2, 680-687. [crossref], [Google Scholar], [Publisher]     
[19] Z. Ahmad, M. Naseem, M.K. Jamil, Sh. Wang, M.F. Nadeem, Eurasian Chem. Commun., 2020, 2, 712-721. [crossref], [Google Scholar], [Publisher]     
[20] Z. Ahmad, M. Naseem, M.K. Jamil, M.K. Siddiqui, M.F. Nadeem, Eurasian Chem. Commun., 2020, 2, 663-671. [crossref], [Google Scholar], [Publisher]     
[21]   A. Alsinai, A. Alwardi, M.R. Farahaini, N.D. Soner, Eurasian Chem. Commun., 2021, 3, 219-226.  [crossref], [Google Scholar], [Publisher]     
[22]   A. Alsinai, A. Alwardi, N.D. Soner. J. D. Math. Sci. Cryp., 2021, 24, 307-324. [crossref], [Google Scholar], [Publisher]     
[23] F. Afzal, S. Hussain, D. Afzal, S. Razaq, J. Inf. Opt. Sci., 2020, 41, 1061-1076. [crossref], [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] S. Hussain F. Afzal, D. Afzal, M.R. Farahani, M. Cancan, S. Ediz, Eurasian Chem. Commun., 2021, 3, 180-186. [crossref], [Google Scholar], [Publisher]     
[30] S. Hussain, F. Afzal, D. Afzal, M. Cancan, S, Ediz, M.R. Farahani, J. Dis. Math. Sci. Cryp., 2021, 24, 415-426. [crossref], [Google Scholar], [Publisher]