Minimum Degree Distance of Five Cyclic Graphs

Nadia Khan; Munazza Shamus; Fauzia Ghulam Hussain; Mansoor Iqbal

doi:doi:10.11648/j.pamj.20211003.13

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Minimum Degree Distance of Five Cyclic Graphs

Nadia Khan, Munazza Shamus, Fauzia Ghulam Hussain, Mansoor Iqbal

Published in Pure and Applied Mathematics Journal (Volume 10, Issue 3)

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 G_n. The subclass of connected graphs with 5 cycles are denoted by G_n⁵. The classification of graph G∈G_n⁵ depends on the number of edges and the sum of the degrees of the vertices of the graph. Any graph in G_n⁵ 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∈G_n⁵ in such a way that the resultant graph belongs to G_n⁵and 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), 2024. Published by Science Publishing Group

Keywords

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

References

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[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.

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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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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

Nadia Khan

Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
Munazza Shamus

Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
Fauzia Ghulam Hussain

Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
Mansoor Iqbal

National University of Comouter & Emerging Sciences, Islamabad, Pakistan

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

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  -

Copy | Download