Enumeration of the Gutman and Schultz indices in the random polygonal chains

Wanlin Zhu; Minglei Fang; Xianya Geng; Wanlin Zhu; Minglei Fang; Xianya Geng

doi:10.3934/mbe.2022506

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 10826-10845. doi: 10.3934/mbe.2022506

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Enumeration of the Gutman and Schultz indices in the random polygonal chains

School of mathematics and big data, Anhui University of Science and Technology, 232001 Huainan, China

Received: 28 June 2022 Revised: 14 July 2022 Accepted: 18 July 2022 Published: 29 July 2022

The Gutman index and Schultz index of a connected graph are degree-distance-based topological indices. In this paper, we devoted to establish the explicit analytical expressions for the simple formulae of the expected values of the Gutman and Schultz indices in a random polygonal. Based on these results above, we get the extremal values and average values of Gunman and Schultz indices of all polygonal chains.
- Gutman index,
- Schultz index,
- random polygonal chains,
- expected value,
- extremal value,
- average value
Citation: Wanlin Zhu, Minglei Fang, Xianya Geng. Enumeration of the Gutman and Schultz indices in the random polygonal chains[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 10826-10845. doi: 10.3934/mbe.2022506

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Abstract

The Gutman index and Schultz index of a connected graph are degree-distance-based topological indices. In this paper, we devoted to establish the explicit analytical expressions for the simple formulae of the expected values of the Gutman and Schultz indices in a random polygonal. Based on these results above, we get the extremal values and average values of Gunman and Schultz indices of all polygonal chains.

References

[1]	A. I. Pavlyuchko, E. V. Vasiliev, L. A. Gribov, Quantum chemical estimation of the overtone contribution to the IR spectra of hydrocarbon halogen derivatives, J. Struct. Chem., (2010), 1045–1051. https://doi.org/10.1007/s10947-010-0161-5 doi: 10.1007/s10947-010-0161-5
[2]	D. R. Flower, On the properties of bit string-based measures of chemical similarity, J. Chem. Inf. Comput. Sci., 38 (1998), 379–386. https://doi.org/10.1021/ci970437z doi: 10.1021/ci970437z
[3]	H. L. Donald, M. A. Whitehead, Molecular geometry and bond energy. III. Cyclooctatetraene and related compounds, J. Am. Chem. Soc., 91 (1969), 238–242. https://doi.org/10.1021/ja01030a003 doi: 10.1021/ja01030a003
[4]	E. Estrada, D. Bonchev, P. Zhang, Chemical Graph Theory, Discrete Math. Appl., (2013), 1–24. https://doi.org/10.1201/b16132-92
[5]	J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, New York, 2008.
[6]	F. Buckley, F. Harary, Distance in Graphs, Addison-Wesley, Reading, 1989. https://doi.org/10.1007/978-0-8176-4789-6-3
[7]	H. 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. https://doi.org/10.1246/bcsj.44.2332 doi: 10.1246/bcsj.44.2332
[8]	M. Garavelli, F. Bernardi, A. Cembran, Cyclooctatetraene Computational Photo- and Thermal Chemistry: A reactivity model for conjugated hydrocarbons, J. Am. Chem. Soc., 124 (2002), 13770–13789. https://doi.org/10.1021/ja020741v doi: 10.1021/ja020741v
[9]	R. C. Entringer, D. E. Jackson, D. A. Snyder, Distance in graphs, Czechoslovak Math. J., 26 (1976), 283–296.
[10]	A. L. Chen, F. J. Zhang, Wiener index and perfect matchings in random phenylene chains, MATCH Commun. Math. Comput. Chem., 61 (2009), 623–630. https://doi.org/10.1117/12.676730 doi: 10.1117/12.676730
[11]	H. Deng, Wiener indices of spiro and polyphenyl hexagonal chains, Math. Computer Model., 55 (2012), 634–644. https://doi.org/10.1016/j.mcm.2011.08.037 doi: 10.1016/j.mcm.2011.08.037
[12]	L. Ma, H. Bian, B. J. Liu, H. Z. Yu, The expected values of the Wiener indices in the random phenylene and spiro chains, Ars Combin., 130 (2017), 267–274.
[13]	Q. N. Zhou, L. G. Wang, Y. Lu, Wiener index and Harary index on Hamilton-connected graphs with large minimum degree, Discrete Appl. Math., 247 (2018), 180–185. https://doi.org/10.1016/j.dam.2018.03.063 doi: 10.1016/j.dam.2018.03.063
[14]	Wiener, Structrual determination of paraffin boiling points, J. Am. Chem. Soc.,, 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005
[15]	S. Mukwembi, S. Munyira, MunyiraDegree distance and minimum degree, Bull. Aust. Math. Soc., 87 (2013), 255–271. https://doi.org/10.1017/S0004972712000354 doi: 10.1017/S0004972712000354
[16]	S. L. Wei, W. C. Shiu, Enumeration of Wiener indices in random polygonal chains, J. Math. Anal. Appl., 469 (2018), 537–548. https://doi.org/10.1016/j.jmaa.2018.09.027 doi: 10.1016/j.jmaa.2018.09.027
[17]	W. Yang, F. Zhang, Wiener index in random polyphenyl chains, MATCH Commun. Math. Comput. Chem., 68 (2012), 371–376. https://doi.org/10.1155/2012/128492 doi: 10.1155/2012/128492
[18]	L. L. Zhang, Q. S. Li, S. C. Li, M. J. Zhang, The expected values for the Schultz index, Gutman index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain, J. Discrete Appl. Math., 282 (2020), 243–256. https://doi.org/10.1016/j.dam.2019.11.007 doi: 10.1016/j.dam.2019.11.007
[19]	J. F. Qi, M. L. Fang, X. Y. Geng, The expected value for the Wiener index in the Random Spiro Chains, Polycycl. Aromat. Comp.. https://doi.org/10.1080/10406638.2022.2038218
[20]	A. Heydari, On the modified Schultz index of $C_{4}C_{8}(S)$ nanotubes and nanotorus, Digest. J. Nanomater. Biostruct., 5 (2010), 51–56. https://doi.org/10.1063/1.3279788 doi: 10.1063/1.3279788
[21]	M. R. Farahani, Hosoya, Schultz, modified Schultz polynomials and their topological indices of benzene molecules: First members of polycyclic aromatic hydrocarbons (PAHs), Int. J. Theor. Chem., 1 (2013), 9–16.
[22]	Z. Xiao, S. Chen, The modified Schultz index of armchair polyhex nanotubes, J. Comput. Theor. Nanosci., 6 (2009), 1109–1114. https://doi.org/10.1166/jctn.2009.1150 doi: 10.1166/jctn.2009.1150
[23]	G. H. Huang, M. J. Kuang, H. Y. Deng, The expected values of Kirchhoff indices in the random polyphenyl and spiro chains, Ars Math. Contemp., 9 (2015), 197–207. https://doi.org/10.26493/1855-3974.458.7b0 doi: 10.26493/1855-3974.458.7b0
[24]	W. C. Evans, D. Evans, Hydrocarbons and derivatives, in Trease and Evans' Pharmacognosy (eds. W. C. Evans and D. Evans), (2009) 173–193. https://doi.org/10.1016/B978-0-7020-2933-2.00019-8
[25]	W. B. Person, G. C. Pimentel, K. S. Pitzer, The Structure of Cyclooctatetraene, J. Am. Chem. Soc., 74 (1952), 3437–3438. https://doi.org/10.1021/ja01133a524 doi: 10.1021/ja01133a524
[26]	E. Booth, G. Strobel, B. Knighton, J. Sears, B. Geary, R. Avci, A rapid column technique for trapping and collecting of volatile fungal hydrocarbons and hydrocarbon derivatives, Biotechnol. Letters., (2011), 1963–1972. https://doi.org/10.1007/s10529-011-0660-2
[27]	S. Chen, Modified Schultz index of zig-zag polyhex nanotubes, J. Comput. Theor. Nanosci., 6 (2009), 1499–1503. https://doi.org/10.1166/jctn.2009.1201 doi: 10.1166/jctn.2009.1201
[28]	P. Zhao, B. Zhao, X. Chen, Y. Bai, Two classes of chains with maximal and minimal total $\pi-electron$ energy, MATCH Commun. Math. Comput. Chem., 62 (2009), 525–536.
[29]	X. Chen, B. Zhao, P. Zhao, Six-membered ring spiro chains with extremal Merrifild-Simmons index and Hosoya index, MATCH Commun. Math. Comput. Chem., 62 (2009), 657–665. https://doi.org/10.1111/j.1467-9892.2008.00605 doi: 10.1111/j.1467-9892.2008.00605
[30]	Y. Bai, B. Zhao, P. Zhao, Extremal Merrifield-Simmons index and Hosoya index of polyphenyl chains, MATCH Commun. Math. Comput. Chem., 62 (2009), 649–656. https://doi.org/10.1111/j.1467-9892.2008.00605.x doi: 10.1111/j.1467-9892.2008.00605.x
[31]	I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci., 34 (1994), 1087–1089. https://doi.org/10.1021/ci00021a009 doi: 10.1021/ci00021a009
[32]	R. J. Schwamm, M. D. Anker, M. Lein, R. vs. Addition: The reaction of an Aluminyl Anion with 1, 3, 5, 7-Cyclooctatetraene, J. Chem., 58 (2019), 1489–1493. https://doi.org/10.1002/ange.201811675 doi: 10.1002/ange.201811675
[33]	R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
[34]	G. Luthe, J. A. Jacobus, L. W. Robertson, Receptor interactions by polybrominated diphenyl ethers versus polychlobrinated biphenyls: a theoretical structure-activity assessment, Environ. Toxicol. Pharm., 25 (2008), 202–210. https://doi.org/10.1016/j.etap.2007.10.017 doi: 10.1016/j.etap.2007.10.017
[35]	M. Traetteberg, G. Hagen, S. J. Cyvin, IV. 1, 3, 5, 7-Cyclooctatetraene, Zeitschrift Für Naturforschung B., 25 (1970), 134–138. https://doi.org/10.1515/znb-1970-0201
[36]	N. Milas, J. Nolan, Jr. Petrus, H. L. Otto, Notes-Ozonization of Cyclooctatetraene, J. Org. Chem., 23 (1958), 624–625. https://doi.org/10.1021/jo01098a611 doi: 10.1021/jo01098a611
[37]	W. B. Person, G. C. Pimentel, K. S. Pitzer, The Structure of Cyclooctatetraene, J. Am. Chem. Soc., 74 (1952), 3437–3438. https://doi.org/10.1021/ja01133a524 doi: 10.1021/ja01133a524
[38]	F. S. Mathews, W. N. Lipscomb, The structure of Silver Cyclooctatetraene Nitrate, J. Phys. Chem., 63 (1959), 845–850. https://doi.org/10.1021/j150576a017 doi: 10.1021/j150576a017
[39]	R. C. Tendick, J. T. O'Beck, M. R. Nimmo, T. Y. Francis Record, Hydrocarbon conversion system and method with a plurality of sources of compressed oxygen-containing gas, Free Patents Online, (2002).
[40]	Q. R. Li, Q. Yang, H. Yin, S. Yang, Analysis of by-products from improved Ullmann reaction using TOFMS and GCTOFMS, J. Univ. Sci. Technol. China., 34 (2004), 335–341. https://doi.org10.2174/0929866043478455 doi: 10.2174/0929866043478455
[41]	S. Tepavcevic, A. T. Wroble, M. Bissen, D. J. Wallace, Y. Choi, L. Hanley, Photoemission studies of polythiophene and polyphenyl films produced via surface polymerization by ion-assisted deposition, J. Phys. Chem. B., 109 (2005), 7134–7140. https://doi.org/10.1021/jp0451445 doi: 10.1021/jp0451445

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