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Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks

Received: 7 October 2021     Accepted: 1 November 2021     Published: 10 November 2021
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Abstract

Let d be a positive real number. An L(1,d)-labeling of a graph G is an assignment of nonnegative real numbers to the vertices of G such that the adjacent vertices are assigned two different numbers (labels) whose difference is at least one, and the difference between numbers (labels) for any two distance-two vertices is at least d. The minimum range of labels over all L(1,d)-labelings of a graph G is called the L(1,d)-labeling number of G, denoted by λ(1,d) (G). The L(1,d)-labeling with d≥1 of graph arose from the code assignment problem of computer wireless network and the L(1,d)-labeling with 0(1,d) (G), is the minimum σ such that there exists a circular σ-L(1,d)-labeling of G. In this paper, the code assignment of 3-D computer wireless network is abstracted as the circular L(1,d)-labeling of book graph, and the authors determined the circular L(1,d)-labeling numbers of book graph for any positive real number d≥2 basing on the properties and constructions of book graphs.

Published in International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 4)
DOI 10.11648/j.ijssam.20210604.12
Page(s) 120-124
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

Circular L(1,d)-labeling, Book Graph, Code Assignment

References
[1] Calamoneri T. (2006). The L(h,k)-Labelling Problem: A Survey and Annotated Bibliography [J]. The Computer Journal. 49 (5), 585-608.
[2] Yeh R K. (2005). A survey on labeling graphs with a condition at distance two [J]. Discrete Math. 306, 1217-1231.
[3] Griggs J R, Jin X T. (2007). Recent Progress in Mathematics and Engineering on Optimal Graph Labellings with Distance Conditions [J]. Journal of Combinatorial Optimization. 14 (2-3), 249-257.
[4] Jin X T, Yeh R K. (2005). Graph distance-dependent labeling related to code assignment in computer networks [J]. Naval Research Logistics. 52 (2): 159-164.
[5] Niu, Q. (2007). L(j,k)-labeling of graph and edge span [D]. M. Phil. Thesis. Southeast University. Nanjing, China.
[6] Griggs J R, Jin X T. (2005). Optimal channel assignments for lattices with conditions at distance two [J]. In Proc. 5th Int. Workshop on algorithms for wireless Mobile, Ad Hoc and Sensor Networks (W-MAN ’05), Denver, Colorado, April 4-8. IEEE Computer Society.
[7] Shiu W C, Wu Q. (2013). L(j,k)-labeling number of direct product of path and cycle [J]. Acta Mathematica Sinica. 29 (8): 1437–1448.
[8] Wu Q, Shiu W C. (2017). L(j,k)-labeling numbers of square of paths [J]. AKCE International Journal of Graphs and Combinatorics. 14 (3): 307–316.
[9] Wu Q. (2018). L(j,k)-labeling number of generalized Petersen graph [J]. IOP Conference Series: Materials Science and Engineering. 466: 012084.
[10] Wu Q. (2018). L(j,k)-labeling number of Cactus graph [J]. IOP Conference Series: Materials Science and Engineering. 466: 012082.
[11] Liu L, Wu Q.(2020). L(1,2)-labeling numbers on square of cycles [J]. AKCE International Journal of Graphs and Combinatorics. 17 (3): 915-919.
[12] Rao W L, Wu Q, Li Y. (2020). The L(1,2)-labeling numbers of Cartesian product of three paths [J]. Journal of Tianjin University of Technology and Education. 30 (3): 57-63.
[13] Heuvel J, Leese R A, Shepherd M A. (1998). Graph labeling and radio channel assignment [J]. Journal of Graph Theory. 29 (4): 263-283.
[14] Wu Q, Lin W S. (2010). Circular L(j,k)-labeling numbers of trees and products of graphs [J]. Southeast University. 26 (1): 142-145.
[15] Wu Q, Shiu W C, Sun P K. (2014). Circular L(j,k)-labeling number of direct product of path and cycle [J]. Journal of Combinatorial Optimization. 27 (2): 355-368.
[16] Wu Q, Shiu W C. (2017). Circular L(j,k)-labeling number of square of paths [J]. Journal of Combinatorics and Number Theory. 9 (1): 41-46.
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  • APA Style

    Yu Guo, Qiong Wu. (2021). Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks. International Journal of Systems Science and Applied Mathematics, 6(4), 120-124. https://doi.org/10.11648/j.ijssam.20210604.12

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    ACS Style

    Yu Guo; Qiong Wu. Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks. Int. J. Syst. Sci. Appl. Math. 2021, 6(4), 120-124. doi: 10.11648/j.ijssam.20210604.12

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    AMA Style

    Yu Guo, Qiong Wu. Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks. Int J Syst Sci Appl Math. 2021;6(4):120-124. doi: 10.11648/j.ijssam.20210604.12

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  • @article{10.11648/j.ijssam.20210604.12,
      author = {Yu Guo and Qiong Wu},
      title = {Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {6},
      number = {4},
      pages = {120-124},
      doi = {10.11648/j.ijssam.20210604.12},
      url = {https://doi.org/10.11648/j.ijssam.20210604.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210604.12},
      abstract = {Let d be a positive real number. An L(1,d)-labeling of a graph G is an assignment of nonnegative real numbers to the vertices of G such that the adjacent vertices are assigned two different numbers (labels) whose difference is at least one, and the difference between numbers (labels) for any two distance-two vertices is at least d. The minimum range of labels over all L(1,d)-labelings of a graph G is called the L(1,d)-labeling number of G, denoted by λ(1,d) (G). The L(1,d)-labeling with d≥1 of graph arose from the code assignment problem of computer wireless network and the L(1,d)-labeling with 0(1,d) (G), is the minimum σ such that there exists a circular σ-L(1,d)-labeling of G. In this paper, the code assignment of 3-D computer wireless network is abstracted as the circular L(1,d)-labeling of book graph, and the authors determined the circular L(1,d)-labeling numbers of book graph for any positive real number d≥2 basing on the properties and constructions of book graphs.},
     year = {2021}
    }
    

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    T1  - Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks
    AU  - Yu Guo
    AU  - Qiong Wu
    Y1  - 2021/11/10
    PY  - 2021
    N1  - https://doi.org/10.11648/j.ijssam.20210604.12
    DO  - 10.11648/j.ijssam.20210604.12
    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
    SP  - 120
    EP  - 124
    PB  - Science Publishing Group
    SN  - 2575-5803
    UR  - https://doi.org/10.11648/j.ijssam.20210604.12
    AB  - Let d be a positive real number. An L(1,d)-labeling of a graph G is an assignment of nonnegative real numbers to the vertices of G such that the adjacent vertices are assigned two different numbers (labels) whose difference is at least one, and the difference between numbers (labels) for any two distance-two vertices is at least d. The minimum range of labels over all L(1,d)-labelings of a graph G is called the L(1,d)-labeling number of G, denoted by λ(1,d) (G). The L(1,d)-labeling with d≥1 of graph arose from the code assignment problem of computer wireless network and the L(1,d)-labeling with 0(1,d) (G), is the minimum σ such that there exists a circular σ-L(1,d)-labeling of G. In this paper, the code assignment of 3-D computer wireless network is abstracted as the circular L(1,d)-labeling of book graph, and the authors determined the circular L(1,d)-labeling numbers of book graph for any positive real number d≥2 basing on the properties and constructions of book graphs.
    VL  - 6
    IS  - 4
    ER  - 

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Author Information
  • Department of Computational Science, School of Science, Tianjin University of Technology and Education, Tianjin, China

  • Department of Computational Science, School of Science, Tianjin University of Technology and Education, Tianjin, China

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