SACS | Topological Characterization of the Third Type of Triangular Hex-derived Networks

Published in Volume XXXI, Issue 2, 2021, pages 145-161, doi: 10.7561/SACS.2021.2.145

Authors: S. Das, S. Rai

Abstract

A topological index is a numerical quantity that defines a chemical descriptor to report several physical, biological and chemical properties of a chemical structure. In recent literature, various degree-based topological indices of a molecular structure are easily calculated by deriving a M-polynomial of that structure. In this paper, we first determine the expression of a M-polynomial of the triangular Hex-derived network of type three of dimension n and then obtain the corresponding degree-based topological indices from the closed form of M-polynomial. In addition, we use Maple software to represent the M-polynomial and the concerned degree-based topological indices pictorially for different dimensions.

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References

[1] Umar Ali, Yasir Ahmad, and Muhammad Shoaib Sardar. On 3-total edge product cordial labeling of tadpole, book and flower graphs. Open Journal of Mathematical Sciences, 4(1):48-55, 2020. doi:10.30538/oms2020.0093.

[2] Dragan Amic, Drago Beslo, Bono Lucic, Sonja Nikolic, and Nenad Trinajstic. The vertex-connectivity index revisited. Journal of Chemical Information and Computer Sciences, 38(5):819-822, 1998. doi:10.1021/ci980039b.

[3] Alexandru T. Balaban. Chemical Applications of Graph Theory. Mathematical Chemistry. Academic Press, 1976.

[4] Alexandru T. Balaban. Highly discriminating distance-based topological index. Chemical Physics Letters, 89(5):399-404, 1982. doi:10.1016/0009-2614(82)80009-2.

[5] Bela Bollobas and P Erdos. Graphs of extremal weights. Ars Combinatoria, 50:225-233, 1998.

[6] Shibsankar Das and Shikha Rai. M-polynomial and related degree-based topological indices of the third type of chain hex-derived network. Malaya Journal of Matematik (MJM), 8(4):1842-1850, 2020. doi:10.26637/MJM0804/0085.

[7] Shibsankar Das and Shikha Rai. M-polynomial and related degree-based topological indices of the third type of hex-derived network. Nanosystems: Physics, Chemistry, Mathematics, 11(3):267-274, 2020. doi:10.17586/2220-8054-2020-11-3-267-274.

[8] Hanyuan Deng, Jianguang Yang, and Fangli Xia. A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes. Computers & Mathematics with Applications, 61(10):3017-3023, 2011. doi:10.1016/j.camwa.2011.03.089.

[9] Emeric Deutsch and Sandi Klavzar. M-polynomial and degree-based topological indices. Iranian Journal of Mathematical Chemistry, 6(2):93-102, 2015. doi:10.22052/ijmc.2015.10106.

[10] Ernesto Estrada. Randic index, irregularity and complex biomolecular networks. Acta Chimica Slovenica, 57(3):597-603, 2010.

[11] Odile Favaron, Maryvonne Maheo, and J.-F. Sacle. Some eigenvalue properties in graphs (conjectures of Graffiti-II). Discrete Mathematics, 111(1-3):197-220, 1993. doi:10.1016/0012-365X(93)90156-N.

[12] Boris Furtula, Ante Graovac, and Damir Vukicevic. Augmented Zagreb index. Journal of Mathematical Chemistry, 48(2):370-380, 2010.

[13] Ramon Garcia-Domenech, Jorge Galvez, Jesus V. de Julian-Ortiz, and Lionello Pogliani. Some new trends in chemical graph theory. Chemical Reviews, 108(3):1127-1169, 2008. doi:10.1021/cr0780006.

[14] I. Gutman and N. Trinajstic. Graph theory and molecular orbitals. total pi-electron energy of alternant hydrocarbons. Chemical Physics Letters, 17(4):535-538, 1972. doi:10.1016/0009-2614(72)85099-1.

[15] Ivan Gutman. The acyclic polynomial of a graph. Publications de I’Institut Mathematique, 22(42):63-69, 1977. URL: http://eudml.org/doc/255036.

[16] Ivan Gutman. Degree-based topological indices. Croatica Chemica Acta, 86(4):351-361, 2013. doi:10.5562/cca2294.

[17] Richard Hammack, Wilfried Imrich, and Sandi Klavzar. Handbook of Product Graphs. CRC Press, Inc., Boca Raton, FL, USA, 2nd edition, 2011.

[18] Shin Min Kang, Waqas Nazeer, Manzoor Ahmad Zahid, Abdul Rauf Nizami, Adnan Aslam, and Mobeen Munir. M-polynomials and topological indices of hex-derived networks. Open Physics, 16(1):394-403, 2018. doi:10.1515/phys-2018-0054.

[19] Padmakar V. Khadikar, Narayan V. Deshpande, Prabhakar P. Kale, Andrey Dobrynin, Ivan Gutman, and Gyula Domotor. The Szeged index and an analogy with the Wiener index. Journal of Chemical Information and Computer Sciences, 35(3):547-550, 1995. doi:10.1021/ci00025a024.

[20] Young Chel Kwun, Mobeen Munir, Waqas Nazeer, Shazia Raque, and Shin Min Kang. M-polynomials and topological indices of V-phenylenic nanotubes and nanotori. Scientific Reports, 7(1):1-9, 2017. doi:10.1038/s41598-017-08309-y.

[21] Jonathan L. Gross, Jay Yellen, and Ping Zhang. Handbook of Graph Theory. Discrete Mathematics and Its Applications. Chapman and Hall/CRC, 2nd edition, 2013.

[22] Paul Manuel, Rajan Bharati, Indra Rajasingh, and Chris Monica M. On minimum metric dimension of honeycomb networks. Journal of Discrete Algorithms, 6(1):20-27, 2008. doi:10.1016/j.jda.2006.09.002.

[23] Ante Milicevic, Sonja Nikolic, and Nenad Trinajstic. On reformulated Zagreb indices. Molecular Diversity, 8:393-399, 2004. doi:10.1016/j.dam.2011.09.021.

[24] Mobeen Munir, Waqas Nazeer, Shazia Raque, and Shin Min Kang. M-polynomial and degree-based topological indices of polyhex nanotubes. Symmetry, 8(12):149, 2016. doi:10.3390/sym8120149.

[25] HM Nagesh and VR Girish. On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph. Open Journal of Mathematical Sciences, 4(1):470-475, 2020. doi:10.30538/oms2020.0137.

[26] F. Garcia Nocetti, Ivan Stojmenovic, and Jingyuan Zhang. Addressing and routing in hexagonal networks with applications for tracking mobile users and connection rerouting in cellular networks. IEEE Transactions on Parallel and Distributed Systems, 13(9):963-971, 2002. doi:10.1109/TPDS.2002.1036069.

[27] Muhammad Numan, Saad Ihsan Butt, and Amir Taimur. Super cyclic antimagic covering for some families of graphs. Open Journal of Mathematical Sciences, 5(1):27-33, 2021. doi:10.30538/oms2021.0142.

[28] K. Pattabiraman. Degree and distance based topological indices of graphs. Electronic Notes in Discrete Mathematics, 63:145-159, 2017. doi:10.1016/j.endm.2017.11.009.

[29] Shikha Rai and Shibsankar Das. M-polynomial and degree-based topological indices of subdivided chain hex-derived network of type 3. In 1st International Conference on Advanced Network Technologies and Intelligent Computing (ANTIC-2021), December 17-18, 2021, Communications in Computer and Information Science (CCIS) series. Springer, 2021 (Accepted).

[30] F. Simon Raj and Amalanathan George. On the metric dimension of HDN 3 and PHDN 3. In 2017 IEEE International Conference on Power, Control, Signals and Instrumentation Engineering (ICPCSI), pages 1333-1336, 2017. doi:10.1109/ICPCSI.2017.8391927.

[31] Milan Randic. Characterization of molecular branching. Journal of the American Chemical Society, 97(23):6609-6615, 1975. doi:10.1021/ja00856a001.

[32] Muhammad Riaz, Wei Gao, and A Qudair Baig. M-polynomials and degree-based topological indices of some families of convex polytopes. Open Journal of Mathematical Sciences, 2(1):18-28, 2018. doi:10.30538/oms2018.0014.

[33] Jelena Sedlar, Dragan Stevanovic, and Alexander Vasilyev. On the inverse sum indeg index. Discrete Applied Mathematics, 184:202-212, 2015. doi:10.1016/j.dam.2014.11.013.

[34] Afshan Tabassum, Muhammad Awais Umar, Muzamil Perveen, and Abdul Raheem. Antimagicness of subdivided fans. Open Journal of Mathematical Sciences, 4(1):18-22, 2020. doi:10.30538/oms2020.0089.

[35] Nenad Trinajstic. Chemical Graph Theory. Mathematical Chemistry Series. CRC Press, 2nd edition, 1992.

[36] Damir Vukicevic and Marija Gasperov. Bond additive modeling 1.Adriatic indices. Croatica Chemica Acta, 83(3):243-260, 2010.

[37] Chang-Cheng Wei, Haidar Ali, Muhammad Ahsan Binyamin, Muhammad Nawaz Naeem, and Jia-Bao Liu. Computing degree based topological properties of third type of hex-derived networks. Mathematics, 7(4):368, 2019. doi:10.3390/math7040368.

[38] Douglas B. West. Introduction to Graph Theory. Prentice Hall, 2nd edition, September 2000.

Bibtex

@article{sacscuza:das21tcttthn,
  title={Topological Characterization of the Third Type of Triangular Hex-derived Networks},
  author={S. Das, S. Rai},
  journal={Scientific Annals of Computer Science},
  volume={31},
  number={2},
  organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
  year={2021},
  pages={145-161},
  publisher={Alexandru Ioan Cuza University Press, Ia\c si},
  doi={10.7561/SACS.2021.2.145}
}