On the Metric Dimension for Snowflake Graph

Muhammad Rafif Fajri; Luthfi Hadiyan  Fajri; Jamaluddin  Ashari; Abdurrahman Abdurrahman; Alifaziz  Arsyad

doi:10.24036/eksakta/vol23-iss04/348

EKSAKTA: Berkala Ilmiah Bidang MIPA

Open Access

On the Metric Dimension for Snowflake Graph

Muhammad Rafif Fajri ¹

, Luthfi Hadiyan Fajri ¹

, Jamaluddin Ashari ²

, Abdurrahman Abdurrahman ³

, Alifaziz Arsyad ¹

¹ Department of Mathematics and Data Science, Faculty of Mathematics and Natural Science (FMIPA), Universitas Andalas, Padang, Indonesia

² Department of Mathematics, Faculty of Mathematics and Natural Science (FMIPA), Universitas Negeri Padang, Padang, Indonesia

³ Department of Electrical Engineering, Faculty of Electrical Engineering and Computer Science, Chien Hsin University of Science and Technologyï¼ˆå¥è¡Œç§‘æŠ€å¤§å¸), Taoyuan, Taiwan.

Published 2022-12-30

Vol. 23 No. 04 (2022) PAGES 284-299

DOI 10.24036/eksakta/vol23-iss04/348

Views / Downloads 275 202

Download PDF

Abstract

The concept of metric dimension is derived from the resolving set of a graph, that is measure the diameter among vertices in a graph. For its usefulness in diverse fields, it is interesting to find the metric dimension of various classes of graphs. In this paper, we introduce two new graphs, namely snowflake graph and generalized snowflake graph. After we construct these graphs, aided with a lemma about the lower bound of the metric dimension on a graph that has leaves, and manually recognized the pattern, we found that dim(Snow) = 24 and dim(Snow(n,a,b,c)) = n(a+c+1).

References

1

Yu, W., Liu, Z., & Bao, X. (2020). New LP relaxations for minimum cycle/path/tree cover problems. Theoretical Computer Science, 803, 71-81.

2

Slater, P. J. (1975). Leaves of trees. Congr. Numer, 14(549â€“559), 37.

3

Zubrilina, N. (2018). On the edge dimension of a graph. Discrete Mathematics, 341(7), 2083-2088

4

Melter, F. H., & Harary, F. (1976). On the metric dimension of a graph. Ars Combin, 2, 191â€“195.

5

Chartrand, G., Eroh, L., Johnson, M. A., & Oellermann, O. R. (2000). Resolvability in graphs and the metric dimension of a graph. Discrete Appl Math (1979), 105(1â€“3), 99â€“113.

6

BaÄa, M., Baskoro, E. T., Salman, A. N. M., Saputro, S. W., & Suprijanto, D. (2011). The metric dimension of regular bipartite graphs. Bulletin mathÃ©matique de la SociÃ©tÃ© des Sciences MathÃ©matiques de Roumanie, 15â€“28.

7

BaÄa, M., Baskoro, E. T., Salman, A. N. M., Saputro, S. W., & Suprijanto, D. (2011). The metric dimension of regular bipartite graphs. Bulletin mathÃ©matique de la SociÃ©tÃ© des Sciences MathÃ©matiques de Roumanie, 15â€“28.

8

Singh, P., Sharma, S., Sharma, S. K., Bhat, V. K., Singh, P., Sharma, S., Sharma, S. K., & Bhat, V. K. (2021). Metric dimension and edge metric dimension of windmill graphs. AIMS Mathematics 2021 9:9138, 6(9), 9138â€“9153.

9

Wu, J., Wang, L., & Yang, W. (2022). Learning to compute the metric dimension of graphs. Appl Math Comput, 432, 127350.

10

Akhter, S., & Farooq, R. (2019). Metric dimension of fullerene graphs. Electronic Journal of Graph Theory and Applications (EJGTA), 7(1), 91â€“103.

11

Bensmail, J., Mc Inerney, F., & Nisse, N. (2020). Metric dimension: From graphs to oriented graphs. Discrete Appl Math (1979).

12

Sooryanarayana, B., Kunikullaya, S., & Swamy, N. N. (2019). Metric dimension of generalized wheels. Arab Journal of Mathematical Sciences, 25(2), 131â€“144.

13

KlavÅ¾ar, S., Rahbarnia, F., & Tavakoli, M. (2021). Some binary products and integer linear programming for k-metric dimension of graphs. Appl Math Comput, 409, 126420.

14

Jiang, Z., & Polyanskii, N. (2019). On the metric dimension of Cartesian powers of a graph. J Comb Theory Ser A, 165, 1â€“14.

15

Shao, Z., Sheikholeslami, S. M., Wu, P., & Liu, J. B. (2018). The Metric Dimension of Some Generalized Petersen Graphs. Discrete Dyn Nat Soc, 2018.

16

Kelenc, A., Tratnik, N., & Yero, I. G. (2018). Uniquely identifying the edges of a graph: The edge metric dimension. Discrete Appl Math (1979), 251, 204â€“220.

17

Okamoto, F., Phinezy, B., & Zhang, P. (2010). The local metric dimension of a graph. Mathematica Bohemica, 135(3), 239â€“255.

18

Knor, M., Å krekovski, R., & Yero, I. G. (2022). A note on the metric and edge metric dimensions of 2-connected graphs. Discrete Appl Math (1979), 319, 454â€“460

19

Sharma, K., Bhat, V. K., & Sharma, S. K. (2022). Edge Metric Dimension and Edge Basis of One-Heptagonal Carbon Nanocone Networks. IEEE Access, 10, 29558â€“29566.

20

KlavÅ¾ar, S., & Tavakoli, M. (2020). Edge metric dimensions via hierarchical product and integer linear programming. Optimization Letters, 15(6), 1993â€“2003.

21

Sedlar, J., & Å krekovski, R. (2022). Vertex and edge metric dimensions of cacti. Discrete Appl Math (1979), 320, 126â€“139.

22

Sedlar, J., & Å krekovski, R. (2022). Vertex and edge metric dimensions of unicyclic graphs. Discrete Appl Math (1979), 314, 81â€“92.

23

FilipoviÄ‡, V., Kartelj, A., & Kratica, J. (2019). Edge Metric Dimension of Some Generalized Petersen Graphs. Results in Mathematics, 74(4), 1â€“15.

24

Peterin, I., & Yero, I. G. (2019). Edge Metric Dimension of Some Graph Operations. Bulletin of the Malaysian Mathematical Sciences Society, 43(3), 2465â€“2477.

25

Saputro, S. W., Simanjuntak, R., Uttunggadewa, S., Assiyatun, H., Baskoro, E. T., Salman, A. N. M., & BaÄa, M. (2013). The metric dimension of the lexicographic product of graphs. Discrete Math, 313(9), 1045â€“1051.

26

KlavÅ¾ar, S., & Tavakoli, M. (2020). Local metric dimension of graphs: Generalized hierarchical products and some applications. Appl Math Comput, 364, 124676.

27

Abrishami, G., Henning, M. A., & Tavakoli, M. (2022). Local metric dimension for graphs with small clique numbers. Discrete Math, 345(4), 112763.

28

Raza, H., & Ji, Y. (2020). Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2). Front Phys, 8, 211.

29

Raza, H., Liu, J. B., & Qu, S. (2020). On Mixed Metric Dimension of Rotationally Symmetric Graphs. IEEE Access, 8, 11560â€“11569.

30

M Imran Bhat, S. P. (2019). On strong metric dimension of zero-divisor graphs of rings. Korean J. Math, 27, 563â€“580.

31

Ma, X., Feng, M., & Wang, K. (2018). The strong metric dimension of the power graph of a finite group. Discrete Appl. Math., 239, 159â€“164.

32

Nikandish, R., Nikmehr, M. J., & Bakhtyiari, M. (2021). Metric and Strong Metric Dimension in Cozero-Divisor Graphs. Mediterranean Journal of Mathematics, 18(3), 1â€“12.

33

Sedlar, J., & Å krekovski, R. (2022). Metric dimensions vs cyclomatic number of graphs with minimum degree at least two. Appl Math Comput, 427, 127147.

34

Yi, E. (2022). On the edge dimension and the fractional edge dimension of graphs. Discrete Appl Math (1979).

35

Sharma, S. K., Raza, H., & Bhat, V. K. (2021). Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone. Front Phys, 9.

36

Susilowati, L., Saâ€™adah, I., Fauziyyah, R. Z., Erfanian, A., & Slamin. (2020). The dominant metric dimension of graphs. Heliyon, 6(3), e03633.

37

Susilowati, L., Slamin, R. A., & Rosfiana, A. (2019). The complement metric dimension of graphs and its operations. Int. J. Civ. Eng. Technol, 10(3), 2386â€“2396.

38

Wei, M., Yue, J., & Chen, L. (2022). The effect of vertex and edge deletion on the edge metric dimension of graphs. J Comb Optim, 44(1), 331â€“342

39

Geneson, J., Kaustav, S., & Labelle, A. (2022). Extremal results for graphs of bounded metric dimension. Discrete Appl Math (1979), 309, 123â€“129.

40

Mashkaria, S., Ã“dor, G., & Thiran, P. (2022). On the robustness of the metric dimension of grid graphs to adding a single edge. Discrete Appl Math (1979), 316, 1â€“27.

41

Lawson, J. E. (2009). Hands-on Science: Light, Physical Science (matter)-Chapter 5: The Colors of Light. Portage & Main Press. Retrieved, 6â€“28.

42

Cabarkapa, A., & Djokic, L. (2019). Importance of the color of light for the illumination of urban squares. Color Research & Application, 44(3), 446-453.

43

Chartrand, G., Lesniak, L., & Zhang, P. (2015). Graphs & Digraphs. Chapman and Hall/CRC.

44

Grier, Z., Soddu, M. F., Kenyatta, N., Odame, S. A., Sanders, J., Wright, L., & Anselmi, F. (2018). A low-cost do-it-yourself microscope kit for hands-on science education. In Optics Education and Outreach V (Vol. 10741, pp. 133-148). SPIE.

On the Metric Dimension for Snowflake Graph

Abstract

References

How to Cite

Most read articles by the same author(s)

On the Metric Dimension for Snowflake Graph

Abstract

References

How to Cite

Cite Article

Article Update

Most read articles by the same author(s)