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Minimum Degree Distance of Five Cyclic Graphs

Received: 26 May 2021     Accepted: 28 July 2021     Published: 4 August 2021
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Abstract

Let G be a connected graph with n vertices. Then the class of connected graphs having n vertices is denoted by Gn. The subclass of connected graphs with 5 cycles are denoted by Gn5. The classification of graph GGn5 depends on the number of edges and the sum of the degrees of the vertices of the graph. Any graph in Gn5 contains five linearly independent cycles having at least n+3 edges and the sum of degrees of vertices of 5-cyclic must be equal to twice of n+4. In this paper, minimum degree distance of class of five cyclic connected graph is investigated. To find minimum degree distance of a graph some transformations T have been defined. These transformation have been applied on the graph GGn5 in such a way that the resultant graph belongs to Gn5 and also degree distance of T(G) is always must be less than G. For n=5, the five 5-cyclic graph has minimum degree distance 78 and the minimum degree distance of 5-cyclic graphs having six vertices is 124. In case of n greater than 6, a general formula for minimum degree distance is investigated. In this paper, we proved that the minimum degree distance of connected 5 cyclic graphs is 3n2+13n-62 by using transformations, for n≥7.

Published in Pure and Applied Mathematics Journal (Volume 10, Issue 3)
DOI 10.11648/j.pamj.20211003.13
Page(s) 84-88
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), 2021. Published by Science Publishing Group

Keywords

Weiner Index, Graphical Sequence, Degree Distance, Five Cyclic Graphs

References
[1] A. A. Dobrynin and A. A. Kochetova. “A Degree distance of a graph: A degree analogue of the Wiener index”. J. Chem. Inform. Comput. Sci., 34 (1994), 1082-1086.
[2] I. Gutman. “Selected properties of the Schultz molecular topological index”. J. Chem. Inform. Comput. Sci}, 34 (1994), 1087-1089.
[3] J. W. Moon. “Counting Labelled Trees”. Canadian Mathematical Monographs}, Vol. 1, W. Clowes and Sons, London and Beccles, (1970).
[4] J. K. Senior. “Partitions and their representative graphs”. Amer. J. Math., 73 (1951), 663-689.
[5] I. Tomescu. “Some extremal properties of the degree distance of a graph”. Discrete Appl. Math., 98 (1999), 159-163.
[6] A. I. Tomescu. “Note on unicyclic and bicyclic graphs having minimum degree distance”. Discrete Appl. Math., 156 (2008), 125-130.
[7] W. Zhu. “A note on tricyclic graphs with minimum degree distance”. Discrete. Math. Algorithms and applications. Vol 3, No. 1 (2011) 25-32.
[8] H. Hosoya. “Topological index A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons”. Bull. Chem. Soc Jpn., 4 (1971), 2332-2339.
[9] N. Khan, M. T. Rahim, Z. Raza. “A note on the degree distance of connected 4-cycle graph.” Utilitas Mathematica 93 (2014), 109-116.
[10] M. Schocker. “On degree sequences of graphs with given cyclomatic number”. Publ. Inst. Math, (N. S) 69 (2001), 34− 40.
Cite This Article
  • APA Style

    Nadia Khan, Munazza Shamus, Fauzia Ghulam Hussain, Mansoor Iqbal. (2021). Minimum Degree Distance of Five Cyclic Graphs. Pure and Applied Mathematics Journal, 10(3), 84-88. https://doi.org/10.11648/j.pamj.20211003.13

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    ACS Style

    Nadia Khan; Munazza Shamus; Fauzia Ghulam Hussain; Mansoor Iqbal. Minimum Degree Distance of Five Cyclic Graphs. Pure Appl. Math. J. 2021, 10(3), 84-88. doi: 10.11648/j.pamj.20211003.13

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    AMA Style

    Nadia Khan, Munazza Shamus, Fauzia Ghulam Hussain, Mansoor Iqbal. Minimum Degree Distance of Five Cyclic Graphs. Pure Appl Math J. 2021;10(3):84-88. doi: 10.11648/j.pamj.20211003.13

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  • @article{10.11648/j.pamj.20211003.13,
      author = {Nadia Khan and Munazza Shamus and Fauzia Ghulam Hussain and Mansoor Iqbal},
      title = {Minimum Degree Distance of Five Cyclic Graphs},
      journal = {Pure and Applied Mathematics Journal},
      volume = {10},
      number = {3},
      pages = {84-88},
      doi = {10.11648/j.pamj.20211003.13},
      url = {https://doi.org/10.11648/j.pamj.20211003.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211003.13},
      abstract = {Let G be a connected graph with n vertices. Then the class of connected graphs having n vertices is denoted by Gn. The subclass of connected graphs with 5 cycles are denoted by Gn5. The classification of graph G∈Gn5 depends on the number of edges and the sum of the degrees of the vertices of the graph. Any graph in Gn5 contains five linearly independent cycles having at least n+3 edges and the sum of degrees of vertices of 5-cyclic must be equal to twice of n+4. In this paper, minimum degree distance of class of five cyclic connected graph is investigated. To find minimum degree distance of a graph some transformations T have been defined. These transformation have been applied on the graph G∈Gn5 in such a way that the resultant graph belongs to Gn5 and also degree distance of T(G) is always must be less than G. For n=5, the five 5-cyclic graph has minimum degree distance 78 and the minimum degree distance of 5-cyclic graphs having six vertices is 124. In case of n greater than 6, a general formula for minimum degree distance is investigated. In this paper, we proved that the minimum degree distance of connected 5 cyclic graphs is 3n2+13n-62 by using transformations, for n≥7.},
     year = {2021}
    }
    

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  • TY  - JOUR
    T1  - Minimum Degree Distance of Five Cyclic Graphs
    AU  - Nadia Khan
    AU  - Munazza Shamus
    AU  - Fauzia Ghulam Hussain
    AU  - Mansoor Iqbal
    Y1  - 2021/08/04
    PY  - 2021
    N1  - https://doi.org/10.11648/j.pamj.20211003.13
    DO  - 10.11648/j.pamj.20211003.13
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 84
    EP  - 88
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20211003.13
    AB  - Let G be a connected graph with n vertices. Then the class of connected graphs having n vertices is denoted by Gn. The subclass of connected graphs with 5 cycles are denoted by Gn5. The classification of graph G∈Gn5 depends on the number of edges and the sum of the degrees of the vertices of the graph. Any graph in Gn5 contains five linearly independent cycles having at least n+3 edges and the sum of degrees of vertices of 5-cyclic must be equal to twice of n+4. In this paper, minimum degree distance of class of five cyclic connected graph is investigated. To find minimum degree distance of a graph some transformations T have been defined. These transformation have been applied on the graph G∈Gn5 in such a way that the resultant graph belongs to Gn5 and also degree distance of T(G) is always must be less than G. For n=5, the five 5-cyclic graph has minimum degree distance 78 and the minimum degree distance of 5-cyclic graphs having six vertices is 124. In case of n greater than 6, a general formula for minimum degree distance is investigated. In this paper, we proved that the minimum degree distance of connected 5 cyclic graphs is 3n2+13n-62 by using transformations, for n≥7.
    VL  - 10
    IS  - 3
    ER  - 

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Author Information
  • Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

  • Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

  • Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

  • National University of Comouter & Emerging Sciences, Islamabad, Pakistan

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