Reflexive Edge Strength in Certain Graphs with Dominant Vertex
Keywords:
Edge irregular reflexive k-labeling, Reflexive edge strength, Book graph, Triangular book graph, Jahangir graph, Helm graph
Abstract
Consider a basic, connected graph G with an edge set of $E(G)$ and a vertex set of $V(G)$. The functions $f_e$ and $f_v$, which take $k=max\{k_e, 2k_v\}$, from the edge set to the first $k_e$ natural number and the non-negative even number up to $2k_v$, respectively, are the components of total $k$-labeling. An \textit{edge irregular reflexive $k$ labeling} of the graph $G$ is the total $k$-labeling, if for every two different edges $x_1x_2$ and $x_1'x_2'$ of $G$, $wt(x_1x_2) \neq wt(x_1'x_2')$, where $wt(x_1x_2)=f_v(x_1)+f_e(x_1x_2)+f_v(x_2)$. The reflexive edge strength of graph $G$ is defined as the minimal $k$ for graph $G$ with an edge irregular reflexive $k$-labeling; it is denoted by $res(G)$. The $res(G)$, where $G$ are the book, triangular book, Jahangir, and helm graphs, was found in this work.References
1. Agustin, I. H. Utoyo, I. Dafik, Venkatachalam, M. 2020. ”Edge irregular reflexive labeling of some tree graphs”. Journal of Physics:
Conference Series. Volume 1543. https:
10.1088/1742-6596/1543/1/012008.
2. Tanna, D. Ryan, J. and Semaniˇcov´a-Feˇnovˇc´ıkov´a A. 2017. ”Reflexive Edge Irregular Labelling of Prisms and Wheels”. Australas. J.
Combin, 69:394-401.
3. G. Chartrand, L. Lesniak, P. Zhang, Graphs and Digraphs, sixth ed., Taylor and Francis Group, Boca Raton, New York, 2016.
4. Joseph A. Gallian. 2017. A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatorics, 20:1-432.
5. G. Chartrand, M.S. Jacobson, J.Lehel, O.R. Oellermann, S. Ruiz, F. Saba, Irregular Networks, Congr. Numer. 64(1988) 187-192.
6. M. Baca, Jendrol’ S, Miller M and Ryan J, 2007 On Irregular Total Labelings, Discrete Math 307 pp 1378-1388.
7. Ali Ahmad and M. Ba˘ca. 2013. On Vertex Irregular Total Labelings. Ars Combinatoria, 112:129-139.
8. S. Bhavanari, S. Devanabolna, and M. Bhavanari. 2016. Star Number of A Graph. Research Journal of Science and IT Management,
05(11):18-22.
9. I. Tarawneh, R. Hasni, and A. Ahmad. 2016. On the Edge Irregularity Strength of Corona Product of Cycle with Isolated Vertices.
AKCE International Journal of Graphs and Combinatorics, 13:213-217.
10. F. Ashraf, M. Baˇca, M. Lascs´akov´a and A. Semaniˇcov´a-Feˇnovˇc´ikov´a. 2017. On H-Irregularity Strength of Graphs. Discussiones
Mathematicae: Graph Theory, 37:1067-1078.
11. Slamin, Dafik, and W. Winnona. 2011. Total Vertex Irregularity Strength of the Disjoint Union of Sun Graphs. The Electronic Journal
of Combinatorics, 2012:1-9.
12. Ika Hesti Agustin, Dafik, Marsidi, Ermita Rizki Albirri. 2016. On the total H-irregularity strength of graphs: A new notion. Journal
of Physics: Conference Series, Vol 855.
13. M. Baˇca, M. Irfan, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ikov´a, and D. Tanna. 2017. On Edge Irregular Reflexive Labellings for the
Generalized Friendship Graphs. Mathematics, 5(67):1-11.
14. D. Tanna, J. Ryan, and A. Semaniˇcov´a-Feˇnovˇc´ikov´a. 2017. Edge Irregular Reflexive Labeling of Prisms and Wheels. Australasian
Journal of Combinatorics, 69(3):394-401.
15. M. Baˇca, M. Irfan, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ikov´a, and D. Tanna. Note on edge irregular reflexive labelings of graphs. AKCE
International Journal of Graphs and Combinatorics , 16 (2019) 145–157.
16. M. Venkatachalam, J. V. Vivin, and K. Kaliraj. 2012. Harmonious Coloring on Double Star Graph Families. Tamkang Journal of
Mathematics, 43(2):153-158.
17. P. Ghosh and A. Pal. 2015. Some Results of Labeling on Broom Graph. Journal of Advances in Mathematics, 9(9):3055-3061.
18. Indriati, D., Widodo, and Rosyida, I. 2020. ”A Quantum Chaotic Image Cryptosystem and Its Application in IoT Secure
Communication”. in Journal of Physics. Vol. 1489, pp. 1-6.
19. Baˇca, M., Irfan, M., Ryan, J. Semaniˇcov´a-Feˇnovˇc´ikov´a, A., Tanna, D. 2019. ”Note on edge irregular reflexive labelings of graphs”.
AKCE International Journal of Graphs and Combinatorics. Volume 16. https:
doi.org/10.1016/j.akcej.2018.01.013.
20. Zhang, X., Ibrahim, M. Bokhary, S. A. U. H. and Siddiqui, M. K. 2018. ”Edge Irregular Reflexive Labeling for the Disjoint Union of
Gear Graphs and Prism Graphs”. in MDPI: Mathematics. Vol. 6, pp. 1-10.
21. Yoonga, K. K., Irfanb, M. Tarawehc, I. and Ahmadd, A. 2021. ”On the edge irregular reflexive labeling of corona product of graphs
with path”. in AKCE International Journal of Graphs and Combinatorics. Vol. 18, pp. 53–59.
22. Guirao, J. L. G., Ahmad, S., Siddiqui, M.K. Ibrahim, M. 2018. ”Edge Irregular Reflexive Labeling for Disjoint Union of Generalized
Petersen Graph”. Mathematics. Volume 6(304). https:
10.3390/math6120304.
23. Ibrahim, M., Majeed, S., Siddiqui, M.K. 2020. ”Edge Irregular Reflexive Labeling for Star, Double Star and Caterpillar Graphs”.
TWMS J. App. and Eng. Math. Volume 10(3). https:
10.3390/math6120304.
24. Baˇca, M., Irfan, M. , Ryan, J., Semaniˇcov´a-Feˇnovˇc´ikov´a, A., Tanna, D. 2017. ”On Edge Irregular Reflexive Labellings for the
Generalized Friendship Graphs”. Mathematics. Volume 5(67). https:
10.3390/math5040067.
25. Ke, Y., Khan, M. J. A. Ibrahim, M. and Siddiqui, M. K. 2021. ”On edge irregular reflexive labeling for Cartesian product of two
graphs”. in Eur. Phys. J. Plus. Vol. 6, pp. 1-13.
26. Agustin I. H., Dafik, Utoyo M. I., Slamin, Venkatachalam M. 2021. ”The Reflexive Edge Strength On Some Almost Regular Graphs”.
Heliyon. Volume 7. https://doi.org/10.1016/j.heliyon.2021.e06991
27. Galian, J,A. A. 2007. ”Dynamic Survey of Graph Labeling”. in Electronic Journal of Combinatorics, Dynamic Survey. 14: DS6.
Conference Series. Volume 1543. https:
10.1088/1742-6596/1543/1/012008.
2. Tanna, D. Ryan, J. and Semaniˇcov´a-Feˇnovˇc´ıkov´a A. 2017. ”Reflexive Edge Irregular Labelling of Prisms and Wheels”. Australas. J.
Combin, 69:394-401.
3. G. Chartrand, L. Lesniak, P. Zhang, Graphs and Digraphs, sixth ed., Taylor and Francis Group, Boca Raton, New York, 2016.
4. Joseph A. Gallian. 2017. A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatorics, 20:1-432.
5. G. Chartrand, M.S. Jacobson, J.Lehel, O.R. Oellermann, S. Ruiz, F. Saba, Irregular Networks, Congr. Numer. 64(1988) 187-192.
6. M. Baca, Jendrol’ S, Miller M and Ryan J, 2007 On Irregular Total Labelings, Discrete Math 307 pp 1378-1388.
7. Ali Ahmad and M. Ba˘ca. 2013. On Vertex Irregular Total Labelings. Ars Combinatoria, 112:129-139.
8. S. Bhavanari, S. Devanabolna, and M. Bhavanari. 2016. Star Number of A Graph. Research Journal of Science and IT Management,
05(11):18-22.
9. I. Tarawneh, R. Hasni, and A. Ahmad. 2016. On the Edge Irregularity Strength of Corona Product of Cycle with Isolated Vertices.
AKCE International Journal of Graphs and Combinatorics, 13:213-217.
10. F. Ashraf, M. Baˇca, M. Lascs´akov´a and A. Semaniˇcov´a-Feˇnovˇc´ikov´a. 2017. On H-Irregularity Strength of Graphs. Discussiones
Mathematicae: Graph Theory, 37:1067-1078.
11. Slamin, Dafik, and W. Winnona. 2011. Total Vertex Irregularity Strength of the Disjoint Union of Sun Graphs. The Electronic Journal
of Combinatorics, 2012:1-9.
12. Ika Hesti Agustin, Dafik, Marsidi, Ermita Rizki Albirri. 2016. On the total H-irregularity strength of graphs: A new notion. Journal
of Physics: Conference Series, Vol 855.
13. M. Baˇca, M. Irfan, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ikov´a, and D. Tanna. 2017. On Edge Irregular Reflexive Labellings for the
Generalized Friendship Graphs. Mathematics, 5(67):1-11.
14. D. Tanna, J. Ryan, and A. Semaniˇcov´a-Feˇnovˇc´ikov´a. 2017. Edge Irregular Reflexive Labeling of Prisms and Wheels. Australasian
Journal of Combinatorics, 69(3):394-401.
15. M. Baˇca, M. Irfan, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ikov´a, and D. Tanna. Note on edge irregular reflexive labelings of graphs. AKCE
International Journal of Graphs and Combinatorics , 16 (2019) 145–157.
16. M. Venkatachalam, J. V. Vivin, and K. Kaliraj. 2012. Harmonious Coloring on Double Star Graph Families. Tamkang Journal of
Mathematics, 43(2):153-158.
17. P. Ghosh and A. Pal. 2015. Some Results of Labeling on Broom Graph. Journal of Advances in Mathematics, 9(9):3055-3061.
18. Indriati, D., Widodo, and Rosyida, I. 2020. ”A Quantum Chaotic Image Cryptosystem and Its Application in IoT Secure
Communication”. in Journal of Physics. Vol. 1489, pp. 1-6.
19. Baˇca, M., Irfan, M., Ryan, J. Semaniˇcov´a-Feˇnovˇc´ikov´a, A., Tanna, D. 2019. ”Note on edge irregular reflexive labelings of graphs”.
AKCE International Journal of Graphs and Combinatorics. Volume 16. https:
doi.org/10.1016/j.akcej.2018.01.013.
20. Zhang, X., Ibrahim, M. Bokhary, S. A. U. H. and Siddiqui, M. K. 2018. ”Edge Irregular Reflexive Labeling for the Disjoint Union of
Gear Graphs and Prism Graphs”. in MDPI: Mathematics. Vol. 6, pp. 1-10.
21. Yoonga, K. K., Irfanb, M. Tarawehc, I. and Ahmadd, A. 2021. ”On the edge irregular reflexive labeling of corona product of graphs
with path”. in AKCE International Journal of Graphs and Combinatorics. Vol. 18, pp. 53–59.
22. Guirao, J. L. G., Ahmad, S., Siddiqui, M.K. Ibrahim, M. 2018. ”Edge Irregular Reflexive Labeling for Disjoint Union of Generalized
Petersen Graph”. Mathematics. Volume 6(304). https:
10.3390/math6120304.
23. Ibrahim, M., Majeed, S., Siddiqui, M.K. 2020. ”Edge Irregular Reflexive Labeling for Star, Double Star and Caterpillar Graphs”.
TWMS J. App. and Eng. Math. Volume 10(3). https:
10.3390/math6120304.
24. Baˇca, M., Irfan, M. , Ryan, J., Semaniˇcov´a-Feˇnovˇc´ikov´a, A., Tanna, D. 2017. ”On Edge Irregular Reflexive Labellings for the
Generalized Friendship Graphs”. Mathematics. Volume 5(67). https:
10.3390/math5040067.
25. Ke, Y., Khan, M. J. A. Ibrahim, M. and Siddiqui, M. K. 2021. ”On edge irregular reflexive labeling for Cartesian product of two
graphs”. in Eur. Phys. J. Plus. Vol. 6, pp. 1-13.
26. Agustin I. H., Dafik, Utoyo M. I., Slamin, Venkatachalam M. 2021. ”The Reflexive Edge Strength On Some Almost Regular Graphs”.
Heliyon. Volume 7. https://doi.org/10.1016/j.heliyon.2021.e06991
27. Galian, J,A. A. 2007. ”Dynamic Survey of Graph Labeling”. in Electronic Journal of Combinatorics, Dynamic Survey. 14: DS6.
Published
2025-03-28
How to Cite
Marsidi, Dafik, Susanto, Kristiana, A. I., Agustin, I. H., & Venkatachalam, M. (2025). Reflexive Edge Strength in Certain Graphs with Dominant Vertex. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-2210
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Section
Research Articles
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