American Journal of Mathematical and Computer Modelling

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Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices

Received: Nov. 24, 2022    Accepted: Jan. 20, 2023    Published: Feb. 09, 2023
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Abstract

The graph theory plays an important role in the network analysis, social networking as well as in many engineering fields such as electrical circuits, artificial intelligence, architecture, making the design or pattern of roads, buildings, shopping mall and etc. Due to this wide range application human enjoying her life with peacefully, Graph theory creates a way for human being to connect among themselves by social network. All above applications based on graph or molecule which may be the planer, non-planer and Peterson graph or etc. Peterson graph is the most important and reasonable example of Hypo-Hamiltonian graph. In the earlier, it was found as a hypo-traceable graph (graph which has not Hamiltonian graph. Naeem et al has worked on “A Two-Connected Graph with Gallai’s Property” In his research paper he has applied the property and has found the longest path and cycle in the graph. In this research paper we will develop the 3-dimensional graph of computational molecule contains 19 vertices and will split it into three different planes (xy, xz and yz-plane), and will find the longest path, longest cycle the molecule. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

DOI 10.11648/j.ajmcm.20230801.11
Published in American Journal of Mathematical and Computer Modelling ( Volume 8, Issue 1, March 2023 )
Page(s) 1-5
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

Gallai Property, Hamiltonian Path, Hamiltonian Cycle, Hypo Hamiltonian Graph, Graph Theory, Traceable Graph

References
[1] Abdul Naeem & Ali Dino Jumani, “Graph with non-concurrent Longest Paths”. IJCSIS International Journal of ComputerScience and Information Security, VOL. 17 No. 4, April 2019.
[2] Abdul Naeem& Ali Dino Jumani. “A Planar Lattice Graph, with Empty Intersection of All Longest Path”. EngineeringMathematics. Vol. 3, No. 1, 2019, pp. 6-8. doi: 10.11648/j.engmath.20190301.12.
[3] T. Zamfirescu, Graphen, in welchen je zwei Eckpunktedurcheinenlangsten Wegvermiedenwerden, Rend. Sem. Mat. Univ. Ferrara 21 (1975), 17–24.
[4] T. Zamfirescu, L'histoire et l'´etatpr´esent des bornesconnuespourPkj, Ckj, Pkjet Ckj, Cahiers CERO 17 (1975), 427-439.
[5] P. Erdos and G. Katona (eds.), Theory of Graphs, Proc. Colloq. Tihany, Hungary, Sept. 1966, Academic Press, New York (1968).
[6] H. Walther, Uber die Nichtexistenzeines Knotenpunktes, durchdenallelangsten Wegeeines Graphengehen, J. Comb. Theory 6 (1969) 1-6.
[7] H. Walther, H. J. Voss, UberKreise in Graphen, VEBDeutscherVerlag der Wissenschaften, Berlin, 1974.
[8] T. Zamfirescu, A two-Connected Planar Graph withoutConcurrent Longest Paths, J. Combin. Theory B13 (1972) 116-121.
[9] W. Schmitz, Uber Langste Wege und Kreise in Graphen, Rend. Sem. Mat. Univ. Padova 53 (1975) 97-103.
[10] T. Zmfirescu, on longest paths and circuits in graphs, Math. Scand. 38 (1976) 211-239.
[11] T. Zamfirescue, intersecting longest paths or cycles: A shortsurvey, Analele Univ. Craiova, Seria Mat. Info. 28 (2001) 1-9.
[12] H. WALTHER, Uber die Nichtexistenzzweier Knotenpunkteeines Graphen, die allellngsten Kreisefassen, J. CombinatorialTheory 8 (1970), 330-333.
[13] B. Grunbaum, Vertices missed by longest paths or circuits, J. Comb. Theory, A 17 (1974), 31-38.
[14] W. Hatzel, Einplanarerhypohamiltonscher Graph mit 57Knoten, Math. Ann. 243 (1979), 213-216.
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  • APA Style

    Rabnawaz Mallah, Inayatullah Soomro, Sarang Latif, Dost Muhammad, Altaf Hussain. (2023). Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices. American Journal of Mathematical and Computer Modelling, 8(1), 1-5. https://doi.org/10.11648/j.ajmcm.20230801.11

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    ACS Style

    Rabnawaz Mallah; Inayatullah Soomro; Sarang Latif; Dost Muhammad; Altaf Hussain. Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices. Am. J. Math. Comput. Model. 2023, 8(1), 1-5. doi: 10.11648/j.ajmcm.20230801.11

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    AMA Style

    Rabnawaz Mallah, Inayatullah Soomro, Sarang Latif, Dost Muhammad, Altaf Hussain. Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices. Am J Math Comput Model. 2023;8(1):1-5. doi: 10.11648/j.ajmcm.20230801.11

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  • @article{10.11648/j.ajmcm.20230801.11,
      author = {Rabnawaz Mallah and Inayatullah Soomro and Sarang Latif and Dost Muhammad and Altaf Hussain},
      title = {Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices},
      journal = {American Journal of Mathematical and Computer Modelling},
      volume = {8},
      number = {1},
      pages = {1-5},
      doi = {10.11648/j.ajmcm.20230801.11},
      url = {https://doi.org/10.11648/j.ajmcm.20230801.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajmcm.20230801.11},
      abstract = {The graph theory plays an important role in the network analysis, social networking as well as in many engineering fields such as electrical circuits, artificial intelligence, architecture, making the design or pattern of roads, buildings, shopping mall and etc. Due to this wide range application human enjoying her life with peacefully, Graph theory creates a way for human being to connect among themselves by social network. All above applications based on graph or molecule which may be the planer, non-planer and Peterson graph or etc. Peterson graph is the most important and reasonable example of Hypo-Hamiltonian graph. In the earlier, it was found as a hypo-traceable graph (graph which has not Hamiltonian graph. Naeem et al has worked on “A Two-Connected Graph with Gallai’s Property” In his research paper he has applied the property and has found the longest path and cycle in the graph. In this research paper we will develop the 3-dimensional graph of computational molecule contains 19 vertices and will split it into three different planes (xy, xz and yz-plane), and will find the longest path, longest cycle the molecule. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.},
     year = {2023}
    }
    

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    T1  - Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices
    AU  - Rabnawaz Mallah
    AU  - Inayatullah Soomro
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    AU  - Dost Muhammad
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    DO  - 10.11648/j.ajmcm.20230801.11
    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
    SP  - 1
    EP  - 5
    PB  - Science Publishing Group
    SN  - 2578-8280
    UR  - https://doi.org/10.11648/j.ajmcm.20230801.11
    AB  - The graph theory plays an important role in the network analysis, social networking as well as in many engineering fields such as electrical circuits, artificial intelligence, architecture, making the design or pattern of roads, buildings, shopping mall and etc. Due to this wide range application human enjoying her life with peacefully, Graph theory creates a way for human being to connect among themselves by social network. All above applications based on graph or molecule which may be the planer, non-planer and Peterson graph or etc. Peterson graph is the most important and reasonable example of Hypo-Hamiltonian graph. In the earlier, it was found as a hypo-traceable graph (graph which has not Hamiltonian graph. Naeem et al has worked on “A Two-Connected Graph with Gallai’s Property” In his research paper he has applied the property and has found the longest path and cycle in the graph. In this research paper we will develop the 3-dimensional graph of computational molecule contains 19 vertices and will split it into three different planes (xy, xz and yz-plane), and will find the longest path, longest cycle the molecule. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.
    VL  - 8
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, Shah Abdul Latif University, Khairpur, Pakistan

  • Department of Mathematics, Shah Abdul Latif University, Khairpur, Pakistan

  • Department of Mathematics, Shah Abdul Latif University, Khairpur, Pakistan

  • Department of Mathematics, Shah Abdul Latif University, Khairpur, Pakistan

  • Department of Mathematics, Shah Abdul Latif University, Khairpur, Pakistan

  • Section