Computing the Hosoya and the Merrifield-Simmons Indices of Two Special Benzenoid Systems

Document Type : Research Paper


1 Faculty of Engineering and Natural Sciences, Department of Mathematics, Bursa Technical University, 16320 Bursa, Turkey

2 Faculty of Arts and Science, Department of Mathematics, Bursa Uludag University, 16059 Bursa, Turkey


Gutman et al. gave some relations for computing the Hosoya indices of two special benzenoid systems R_n and P_n. In this paper, we compute the Hosoya index and Merrifield-Simmons index of R_n and P_n by means of introducing four vectors for each benzenoid system and index. As a result, we compute the Hosoya index and the Merrifield-Simmons index of R_n and P_n by means of a product of a certain matrix of degree n and a certain vector.



    1. Alishahi and S. H. Shalmaee, On the edge eccentric and modified edge eccentric connectivity indices of linear benzenoid chains and double hexagonal chains, J. Mol. Struct. 1204 (2020) 127446.
    2. Cruz, C. A. Marín and J. Rada, Computing the Hosoya Index of Catacondensed Hexagonal Systems, MATCH Commun. Math. Comput. Chem. 77 (2017) 749-764.
    3. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    4. Gutman and S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer-Verlag, Berlin, 1989.
    5. Gutman, N. Kolaković, A. Graovac and D. Babić, A method for calculation of the Hosoya index of polymers, Srudies Phys. Theor. Chem.63 (1989) 141-154.
    6. Gutman, Extremal hexagonal chains, J. Math. Chem. 12 (1993) 197-210.
    7. Gultekin and B. Sahin, Some Relations between Kekulé Structure and Morgan-Voyce Polynomials, Iranian J. Math. Chem. 8 (2) (2017) 221-229.
    8. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 44 (1971) 2332-2339.
    9. Huang, M. Kuang and H. Deng, The expected values of Hosoya index and Merrifield-Simmons index in a random polyphenylene chain, J. Comb. Optim. 32 (2016) 550-562.
    10. E. Merrifield and H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989.
    11. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1) (1982) 16-21.
    12. Rada, Vertex-degree-based topological indices of hexagonal systems with equal number of edges, Appl. Math. Comput. 296 (2017) 270-276.
    13. Rada, R. Cruz and I. Gutman, Vertex-degree-based topological indices of catacondensed hexagonal systems, Chem. Phys. Lett. 572 (2013) 154-157.
    14. Ren and F. Zhang, Double hexagonal chains with minimal total π-electron energy, J. Math. Chem. 42 (4) (2007) 1041-1056.
    15. Ren and F. Zhang, Extremal double hexagonal chains with respect to k- matchings and k-independent sets, Discrete Appl. Math. 155 (17) (2007) 2269-2281.
    16. Ren and F. Zhang, Double hexagonal chains with maximal Hosoya index and minimal Merrifield-Simmons index, J. Math. Chem. 42 (4) (2007) 679-690.
    17. C. Shiu, Extremal Hosoya index and Merrifield-Simmons index of hexagonal spiders, Discrete Appl. Math. 156 (15) (2008) 2978-2985.
    18. Wagner and I. Gutman, Maxima and Minima of the Hosoya Index and Merrifield-Simmons: A survey of results and techniques, Acta Appl. Math. 112 (2010) 323-346.
    19. Wagner and H. Wang, Introduction to Chemical Graph Theory, CRC Press, Taylor-Francis, Boca Raton, FL, 2018.
    20. Wei and S. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield-Simmons index, Discrete Appl. Math. 271 (2019) 205-217.
    21. -J. Xu, Q.-H. He, S. Zhou and W. H. Chan, Hosoya Polynomials of Random Benzenoid Chains, Iranian J. Math. Chem. 7 (1) (2016) 29-38.