Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons

Alphonse Emadak; Robert Martin Nemba; Leonel Tsassi Djoumessi; Jules Cesar Epee Ndongue

doi:doi:10.11648/j.sjc.20190703.11

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Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons

Alphonse Emadak, Robert Martin Nemba, Leonel Tsassi Djoumessi, Jules Cesar Epee Ndongue

Published in Science Journal of Chemistry (Volume 7, Issue 3)

Received: 1 June 2019 Accepted: 9 July 2019 Published: 22 July 2019

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Abstract

A mathematical process of enumeration of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons is given by using a method which consisted of the excision of the radical center in order to obtain a conjugated hydrocarbon having a 12-annulene core ring appended to one or more polybenzenoïd fragments, the partition of the resulting conjugated hydrocarbon into smaller independent resonant circuits. Polyhex monoradical skeletons are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoïd fragments. Mathematical formulas are established for three cases of assemblies, one edge linear assemblies, two edges linear assemblies with two linear polybenzenoid fragments located at right angles θ = 60°, 120° or 180° and three edges linear assemblies with three linear polybenzenoid fragments located at right angles θ= 60°, 120° or 180°. Results show that the number of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons increases with the number of fused benzenoids according to results compiled in the table for h equal 1, 2, 3 and 4. Difficulty to represent graphs when the number of structures increases with the number of benzenoids cannot be neglected.

Published in	Science Journal of Chemistry (Volume 7, Issue 3)
DOI	10.11648/j.sjc.20190703.11
Page(s)	56-61
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Polyhex, Enumeration, Kekule, Monoradicals, Linear Assemblies

References

[1]	R. Tosic, D. Masulovic, I. Stojmenovic, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, “Enumeration of Polyhex Hydrocarbons to h=17,” J. Chem. Inf. Comput. Sci., vol. 35, pp. 181–187, 1995.
[2]	N. Trinajstic, S. Nikolic, J. V. Knop, W. R. Müller, and K. Szymanski, Computational Chemical Graph Theory, Ellis Horwood, New York, London 1991.
[3]	N. Trinajstic, Chemical Graph Theory (2nd ed.), CRC, Boca Raton, London, Tokyo 1992.
[4]	I. Gutman and S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer, Berlin 1989.
[5]	I. Gutman and J. Cioslowski, Publ. Inst. Math. Beograd 42, 21 (1987).
[6]	I. Gutman, Match 13, 173 (1982).
[7]	S. J. Cyvin and I. Gutman, Z. Naturforsch. 41a, 1079 (1986).
[8]	B. N. Cyvin, J. Brunvoll, S. J. Cyvin, and I. Gutman, Match 21, 301 (1986).
[9]	S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Z. Naturforsch. 41a, 1429 (1986).
[10]	S. J. Cyvin, Match 20, 165 (1986).
[11]	R. Chen and S. J. Cyvin, Match 22, 175 (1987).
[12]	A. T. Balaban, J. Brun voll, B. N. Cyvin, and S. J. Cyvin, Tetrahedron 44, 221 (1988).
[13]	F. Zhang, R. Chen, and X. Guo, Graphs Comb. 1, 383 (1985).
[14]	A. V. Kostochka, Proc. 30. Int. Wiss. Koll. TH Ilmenau F, 49 (1985).
[15]	S. J. Cyvin and I. Gutman, Kekule Structures in Benzenoid Hydrocarbons (Lecture Notes in Chemistry 46), Springer, Berlin 1988.
[16]	I. Gutman and S. J. Cyvin (eds.), Topics Curr. Chem. (Advances in the Theory of Benzenoid Hydrocarbons) 153 (1990).
[17]	I. Gutman (ed.), Topics Curr. Chem. (Advances in the Theory of Benzenoid Hydrocarbons II) 162 (1992).
[18]	P. John, Z. Phys. Chem. 271, 941 (1990).
[19]	A. T. Balaban, J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Tetrahedron 44, 221 (1988).
[20]	R. Bellman and S. Dreyfus, Applied Dynamic Programming, Princeton Press, Princeton, New Jersey 1962.
[21]	H. Hosoya, Topics Curr. Chem. 153, 273 (1990).
[22]	I. Gutman, J. Serb. Chem. Soc. 11, 607 (1988).
[23]	A. T. Balaban, Pure and Appl. Chem. 65, 1 (1993).
[24]	A. T. Balaban, X. Liu, S. J. Cyvin, and D. J. Klein, Benzenoids with Maximum Kekule Structure Counts for Given Number of Hexagons, J. Chem. Inf. Comput. Sei. 33, 429 (1993).
[25]	R. Bellman, Dynamic Programming, Princeton Press, Princeton, New Jersey 1957.

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APA Style

Alphonse Emadak, Robert Martin Nemba, Leonel Tsassi Djoumessi, Jules Cesar Epee Ndongue. (2019). Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons. Science Journal of Chemistry, 7(3), 56-61. https://doi.org/10.11648/j.sjc.20190703.11

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ACS Style

Alphonse Emadak; Robert Martin Nemba; Leonel Tsassi Djoumessi; Jules Cesar Epee Ndongue. Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons. Sci. J. Chem. 2019, 7(3), 56-61. doi: 10.11648/j.sjc.20190703.11

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AMA Style

Alphonse Emadak, Robert Martin Nemba, Leonel Tsassi Djoumessi, Jules Cesar Epee Ndongue. Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons. Sci J Chem. 2019;7(3):56-61. doi: 10.11648/j.sjc.20190703.11

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@article{10.11648/j.sjc.20190703.11,
  author = {Alphonse Emadak and Robert Martin Nemba and Leonel Tsassi Djoumessi and Jules Cesar Epee Ndongue},
  title = {Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons},
  journal = {Science Journal of Chemistry},
  volume = {7},
  number = {3},
  pages = {56-61},
  doi = {10.11648/j.sjc.20190703.11},
  url = {https://doi.org/10.11648/j.sjc.20190703.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20190703.11},
  abstract = {A mathematical process of enumeration of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons is given by using a method which consisted of the excision of the radical center  in order to obtain a conjugated hydrocarbon having a 12-annulene core ring appended to one or more polybenzenoïd fragments, the partition of the resulting conjugated hydrocarbon into smaller independent resonant circuits. Polyhex monoradical skeletons are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoïd fragments. Mathematical formulas are established for three cases of assemblies, one edge linear assemblies, two edges linear assemblies with two linear polybenzenoid fragments located at right angles θ = 60°, 120° or 180° and three edges linear assemblies with three linear polybenzenoid fragments located at right angles θ= 60°, 120° or 180°. Results show that the number of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons increases with the number of fused benzenoids according to results compiled in the table for h equal 1, 2, 3 and 4. Difficulty to represent graphs when the number of structures increases with the number of benzenoids cannot be neglected.},
 year = {2019}
}

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TY  - JOUR
T1  - Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons
AU  - Alphonse Emadak
AU  - Robert Martin Nemba
AU  - Leonel Tsassi Djoumessi
AU  - Jules Cesar Epee Ndongue
Y1  - 2019/07/22
PY  - 2019
N1  - https://doi.org/10.11648/j.sjc.20190703.11
DO  - 10.11648/j.sjc.20190703.11
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 56
EP  - 61
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20190703.11
AB  - A mathematical process of enumeration of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons is given by using a method which consisted of the excision of the radical center  in order to obtain a conjugated hydrocarbon having a 12-annulene core ring appended to one or more polybenzenoïd fragments, the partition of the resulting conjugated hydrocarbon into smaller independent resonant circuits. Polyhex monoradical skeletons are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoïd fragments. Mathematical formulas are established for three cases of assemblies, one edge linear assemblies, two edges linear assemblies with two linear polybenzenoid fragments located at right angles θ = 60°, 120° or 180° and three edges linear assemblies with three linear polybenzenoid fragments located at right angles θ= 60°, 120° or 180°. Results show that the number of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons increases with the number of fused benzenoids according to results compiled in the table for h equal 1, 2, 3 and 4. Difficulty to represent graphs when the number of structures increases with the number of benzenoids cannot be neglected.
VL  - 7
IS  - 3
ER  -

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Author Information

Alphonse Emadak

Department of Inorganic Chemistry, Faculty of Science, University of Yaounde I, Yaounde, Cameroon
Robert Martin Nemba

Department of Inorganic Chemistry, Faculty of Science, University of Yaounde I, Yaounde, Cameroon
Leonel Tsassi Djoumessi

Department of Inorganic Chemistry, Faculty of Science, University of Yaounde I, Yaounde, Cameroon
Jules Cesar Epee Ndongue

Department of Inorganic Chemistry, Faculty of Science, University of Yaounde I, Yaounde, Cameroon

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APA Style

Alphonse Emadak, Robert Martin Nemba, Leonel Tsassi Djoumessi, Jules Cesar Epee Ndongue. (2019). Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons. Science Journal of Chemistry, 7(3), 56-61. https://doi.org/10.11648/j.sjc.20190703.11

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ACS Style

Alphonse Emadak; Robert Martin Nemba; Leonel Tsassi Djoumessi; Jules Cesar Epee Ndongue. Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons. Sci. J. Chem. 2019, 7(3), 56-61. doi: 10.11648/j.sjc.20190703.11

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AMA Style

Alphonse Emadak, Robert Martin Nemba, Leonel Tsassi Djoumessi, Jules Cesar Epee Ndongue. Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons. Sci J Chem. 2019;7(3):56-61. doi: 10.11648/j.sjc.20190703.11

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@article{10.11648/j.sjc.20190703.11,
  author = {Alphonse Emadak and Robert Martin Nemba and Leonel Tsassi Djoumessi and Jules Cesar Epee Ndongue},
  title = {Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons},
  journal = {Science Journal of Chemistry},
  volume = {7},
  number = {3},
  pages = {56-61},
  doi = {10.11648/j.sjc.20190703.11},
  url = {https://doi.org/10.11648/j.sjc.20190703.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20190703.11},
  abstract = {A mathematical process of enumeration of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons is given by using a method which consisted of the excision of the radical center  in order to obtain a conjugated hydrocarbon having a 12-annulene core ring appended to one or more polybenzenoïd fragments, the partition of the resulting conjugated hydrocarbon into smaller independent resonant circuits. Polyhex monoradical skeletons are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoïd fragments. Mathematical formulas are established for three cases of assemblies, one edge linear assemblies, two edges linear assemblies with two linear polybenzenoid fragments located at right angles θ = 60°, 120° or 180° and three edges linear assemblies with three linear polybenzenoid fragments located at right angles θ= 60°, 120° or 180°. Results show that the number of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons increases with the number of fused benzenoids according to results compiled in the table for h equal 1, 2, 3 and 4. Difficulty to represent graphs when the number of structures increases with the number of benzenoids cannot be neglected.},
 year = {2019}
}

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TY  - JOUR
T1  - Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons
AU  - Alphonse Emadak
AU  - Robert Martin Nemba
AU  - Leonel Tsassi Djoumessi
AU  - Jules Cesar Epee Ndongue
Y1  - 2019/07/22
PY  - 2019
N1  - https://doi.org/10.11648/j.sjc.20190703.11
DO  - 10.11648/j.sjc.20190703.11
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 56
EP  - 61
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20190703.11
AB  - A mathematical process of enumeration of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons is given by using a method which consisted of the excision of the radical center  in order to obtain a conjugated hydrocarbon having a 12-annulene core ring appended to one or more polybenzenoïd fragments, the partition of the resulting conjugated hydrocarbon into smaller independent resonant circuits. Polyhex monoradical skeletons are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoïd fragments. Mathematical formulas are established for three cases of assemblies, one edge linear assemblies, two edges linear assemblies with two linear polybenzenoid fragments located at right angles θ = 60°, 120° or 180° and three edges linear assemblies with three linear polybenzenoid fragments located at right angles θ= 60°, 120° or 180°. Results show that the number of Kekule structures of any polybenzenoïd monoradical having only linear assemblies of benzenoid hydrocarbons increases with the number of fused benzenoids according to results compiled in the table for h equal 1, 2, 3 and 4. Difficulty to represent graphs when the number of structures increases with the number of benzenoids cannot be neglected.
VL  - 7
IS  - 3
ER  -

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