Local Metric Dimension of Certain Classes of Circulant Networks

V. Jude Annie Cynthia; M. Ramya; S. Prabhu

doi:10.20965/jaciii.2023.p0554

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JACIII Vol.27 No.4 pp. 554-560

doi: 10.20965/jaciii.2023.p0554

(2023)

Research Paper:

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Local Metric Dimension of Certain Classes of Circulant Networks

V. Jude Annie Cynthia^* , M. Ramya^,, and S. Prabhu^

^*Department of Mathematics, Stella Maris College
17 Cathedral Road, Chennai, Tamil Nadu 600086, India

^**Department of Mathematics, Chevalier T. Thomas Elizabeth College for Women
16 St. Mary’s Road, Maryland, Sembium, Perambur, Chennai, Tamil Nadu 600011, India

^***Department of Mathematics, Rajalakshmi Engineering College
Bangalore Highway, Thandalam, Chennai 602105, India

Received:

July 16, 2022

Accepted:

February 28, 2023

Published:

July 20, 2023

Keywords:

local metric basis, local metric dimension, Harary graph, Toeplitz network, ILLIAC network

Abstract

Let G(V,E) be a graph with a set of vertices V and a set of edges E. Then, a minimum subset W_l of V is said to be a local metric basis of G if for any two adjacent vertices u,v∈V∖W_l there exists a vertex w∈W_l such that d(u,w) ≠ d(v,w). The cardinality of a local metric basis is referred to as the local metric dimension of the graph G denoted by β_l(G). In this paper, we investigate the local metric dimensions of certain circulant-related architectures such as Harary graphs H_k,n with even k or n, Toeplitz networks, and ILLIAC networks.

Cite this article as:

V. Cynthia, M. Ramya, and S. Prabhu, “Local Metric Dimension of Certain Classes of Circulant Networks,” J. Adv. Comput. Intell. Intell. Inform., Vol.27 No.4, pp. 554-560, 2023.

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References

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[2] F. Harary and R. A. Melter, “On the metric dimension of a graph,” Ars Combinatoria, Vol.2, pp. 191-195, 1976.
[3] F. Okamoto, L. Crosse, B. Phinezy, P. Zhang, and Kalamazoo, “The local metric dimension of graphs,” Mathematica Bohemica, Vol.135, No.3, pp. 239-255, 2010.
[4] D. Kuziak and I. G. Yero, “Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results,” arXiv:2107.04877, 2021.
[5] V. J. A. Cynthia and M. Ramya, “The Local Metric Dimension of Cyclic Split Graph,” Annals of Pure and Applied Mathematics, Vol.8, No.2, pp. 201-205, 2014.
[6] V. J. A. Cynthia and M. Ramya, “Local Metric Dimension of Torus Networks,” Int. J. of Pure and Applied Mathematics, Vol.120, No.7, pp. 225-232, 2018.
[7] V. J. A. Cynthia and M. Ramya, “Local Metric Dimension of Cube Connected Cycle Derived Networks,” Int. J. of Pure and Applied Mathematics, Vol.118, No.23, pp. 449-455, 2018.
[8] V. J. A. Cynthia and M. Ramya, “Local Metric Dimension of Mesh Related Architectures,” Int. J. of Pure and Applied Mathematics, Vol.109, No.8, pp. 185-168, 2016.
[9] V. J. A. Cynthia and Fancy V. F., “On Local Metric Dimension of Kautz Network,” Int. J. of Pure and Applied Mathematics, Vol.118, No.23, pp. 119-126, 2018.
[10] V. J. A. Cynthia and Fancy V. F., “Local Metric Dimension of Certain Networks,” Int. J. of Pure and Applied Mathematics, Vol.120, No.7, pp. 69-77, 2018.
[11] V. J. A. Cynthia and V. F. Fancy, “Local Metric Dimension of Some Windmill Graphs and Sunlet Graphs,” Advances and Applications in Discrete Mathematics, Vol.27, No.1, pp. 73-84, 2021. http://doi.org/10.17654/DM027010073
[12] Fancy V. F. and V. J. A. Cynthia, “Local Metric Dimension of Wheel Related Graphs,” Int. J. of Mathematics and Computer Science, Vol.16, No.4, pp. 1303-1315, 2021.
[13] Y. Tian, W. Song, L. Chen, Y. Sung, J. Kwak, and S. Sun, “A Fast Spatial Clustering Method for Sparse LiDAR Point Clouds Using GPU Programming,” Sensors, Vol.20, No.8, Article No.2309, 2020. https://doi.org/10.3390/s20082309
[14] V. J. A. Cynthia and Fancy V. F., “Local Landmarks in Circulant Graph,” Int. J. of Mathematics Trends and Technology, Vol.53, No.3, pp. 249-253, 2018.
[15] M. v. Steen, “Graph Theory and Complex Networks,” 2010.
[16] N. Deo and M. S. Krishnamoorthy, “Toeplitz Networks and Their Properties,” IEEE Trans. on Circuits and Systems, Vol.36, No.8, pp. 1089-1092, 1989. https://doi.org/10.1109/31.192418
[17] J. Xu, “Topological Structure and Analysis of Interconnection Networks,” Springer New York, NY, 2013. https://doi.org/10.1007/978-1-4757-3387-7
[18] A. Borchert and S. Gosselin, “The metric dimension of circulant graphs and Cayley hypergraphs,” Util. Math, Vol.106, pp. 125-147, 2018.
[19] B. Rajan, I. Rajasingh, P. Venugopal, and C. Monica, “Minimum Metric Dimension of ILLIAC Networks,” Ars Combinatoria, Vol.117, pp. 95-103, 2014.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] P. J. Slater, “Leaves of trees,” Cong. Number., Vol.14, pp.549-559, 1975.

[2] [2] F. Harary and R. A. Melter, “On the metric dimension of a graph,” Ars Combinatoria, Vol.2, pp. 191-195, 1976.

[3] [3] F. Okamoto, L. Crosse, B. Phinezy, P. Zhang, and Kalamazoo, “The local metric dimension of graphs,” Mathematica Bohemica, Vol.135, No.3, pp. 239-255, 2010.

[4] [4] D. Kuziak and I. G. Yero, “Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results,” arXiv:2107.04877, 2021.

[5] [5] V. J. A. Cynthia and M. Ramya, “The Local Metric Dimension of Cyclic Split Graph,” Annals of Pure and Applied Mathematics, Vol.8, No.2, pp. 201-205, 2014.

[6] [6] V. J. A. Cynthia and M. Ramya, “Local Metric Dimension of Torus Networks,” Int. J. of Pure and Applied Mathematics, Vol.120, No.7, pp. 225-232, 2018.

[7] [7] V. J. A. Cynthia and M. Ramya, “Local Metric Dimension of Cube Connected Cycle Derived Networks,” Int. J. of Pure and Applied Mathematics, Vol.118, No.23, pp. 449-455, 2018.

[8] [8] V. J. A. Cynthia and M. Ramya, “Local Metric Dimension of Mesh Related Architectures,” Int. J. of Pure and Applied Mathematics, Vol.109, No.8, pp. 185-168, 2016.

[9] [9] V. J. A. Cynthia and Fancy V. F., “On Local Metric Dimension of Kautz Network,” Int. J. of Pure and Applied Mathematics, Vol.118, No.23, pp. 119-126, 2018.

[10] [10] V. J. A. Cynthia and Fancy V. F., “Local Metric Dimension of Certain Networks,” Int. J. of Pure and Applied Mathematics, Vol.120, No.7, pp. 69-77, 2018.

[11] [11] V. J. A. Cynthia and V. F. Fancy, “Local Metric Dimension of Some Windmill Graphs and Sunlet Graphs,” Advances and Applications in Discrete Mathematics, Vol.27, No.1, pp. 73-84, 2021. http://doi.org/10.17654/DM027010073

[12] [12] Fancy V. F. and V. J. A. Cynthia, “Local Metric Dimension of Wheel Related Graphs,” Int. J. of Mathematics and Computer Science, Vol.16, No.4, pp. 1303-1315, 2021.

[13] [13] Y. Tian, W. Song, L. Chen, Y. Sung, J. Kwak, and S. Sun, “A Fast Spatial Clustering Method for Sparse LiDAR Point Clouds Using GPU Programming,” Sensors, Vol.20, No.8, Article No.2309, 2020. https://doi.org/10.3390/s20082309

[14] [14] V. J. A. Cynthia and Fancy V. F., “Local Landmarks in Circulant Graph,” Int. J. of Mathematics Trends and Technology, Vol.53, No.3, pp. 249-253, 2018.

[15] [15] M. v. Steen, “Graph Theory and Complex Networks,” 2010.

[16] [16] N. Deo and M. S. Krishnamoorthy, “Toeplitz Networks and Their Properties,” IEEE Trans. on Circuits and Systems, Vol.36, No.8, pp. 1089-1092, 1989. https://doi.org/10.1109/31.192418

[17] [17] J. Xu, “Topological Structure and Analysis of Interconnection Networks,” Springer New York, NY, 2013. https://doi.org/10.1007/978-1-4757-3387-7

[18] [18] A. Borchert and S. Gosselin, “The metric dimension of circulant graphs and Cayley hypergraphs,” Util. Math, Vol.106, pp. 125-147, 2018.

[19] [19] B. Rajan, I. Rajasingh, P. Venugopal, and C. Monica, “Minimum Metric Dimension of ILLIAC Networks,” Ars Combinatoria, Vol.117, pp. 95-103, 2014.

Local Metric Dimension of Certain Classes of Circulant Networks

V. Jude Annie Cynthia* , M. Ramya*,**, and S. Prabhu***

V. Jude Annie Cynthia^* , M. Ramya^,, and S. Prabhu^