ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS

Mahmoud Farid; Ahmed Ezzat Ameen Mahran; Mohammed Anwar; Mohammed Abdel Azim Seoud

ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS

Abstract

This paper studies strongly $^{*}$-graphs as a variation of strongly multiplicative graphs. We show that every strongly multiplicative graph induces a strongly $^{*}$-graph. We establish a relationship between the upper bounds on the number of edges of strongly $^{*}$-graphs $\lambda^{*}(n)$ and strongly multiplicative graphs, and identify a condition under which the upper bound for strongly $^{*}$-graphs exceeds that of strongly multiplicative graphs, we show that this condition holds for infinitely many values of $n$. We derive explicit formulas for $ lambda^{*}(n)$. Finally, we prove the independence of several necessary conditions for graphs that do not admit strongly$^{*}$-labeling.

Keywords

References

[1] Beineke, L. W., and Hegde, S. M., (2001), Strongly multiplicative graphs, Discuss. Math. Graph Theory, 21, 63–75.
[2] Adiga, C., Ramashekar, H., and Somashekara, D., (2004), A note on strongly multiplicative graphs, Discuss. Math., 24, 81–83.
[3] Seoud, M. A., and Zid, A., (1999), Strong multiplicativity of unions and corona of paths and complete graphs, Proc. Math. Phys. Soc. Egypt, 74, 59–71.
[4] Seoud, M. A., and Mahran, A. E. A., (2015), On strongly multiplicative graphs, Ars Combinatoria, 118, 155–165.
[5] Adiga, C., and Somashekara, D., (1999/2000), Strongly*-graphs, Math. Forum, 13, 31–36.
[6] Seoud, M. A., and Mahran, A. E. A., (2012), Necessary conditions for strongly*-graphs, AKCE Int. J. Graphs Combin., 9(2), 115–122.
[7] Seoud, M. A., Roshdy, E. M., and AboShady, M. S., (2017), On strongly*-graphs, Proc. Pakistan Acad. Sci., Sect. A, 54(2), 179–195.
[8] Kafshgarzaferani, R., (2005), The maximum number of edges in a strongly multiplicative graph, Bull. Iran. Math. Soc., 31(2), 53–56.

Details

Primary Language

English

Subjects

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Journal Section

Research Article

Authors

Mahmoud Farid ^* This is me
0009-0001-6076-4411
Egypt

Ahmed Ezzat Ameen Mahran This is me
0009-0000-7008-5941
Egypt

Mohammed Anwar
0000-0003-0668-4071
Egypt

Mohammed Abdel Azim Seoud This is me
0000-0003-3216-5873
Egypt

Publication Date

April 7, 2026

Submission Date

September 4, 2025

Acceptance Date

February 9, 2026

Published in Issue

Year 2026 Volume: 16 Number: 4

IZ

https://izlik.org/JA83WH67SC

Cite

RIS / Bibtex

APA

Farid, M., Mahran, A. E. A., Anwar, M., & Seoud, M. A. A. (2026). ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS. TWMS Journal of Applied and Engineering Mathematics, 16(4), 428-442. https://izlik.org/JA83WH67SC

AMA

1.Farid M, Mahran AEA, Anwar M, Seoud MAA. ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS. JAEM. 2026;16(4):428-442. https://izlik.org/JA83WH67SC

Chicago

Farid, Mahmoud, Ahmed Ezzat Ameen Mahran, Mohammed Anwar, and Mohammed Abdel Azim Seoud. 2026. “ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16 (4): 428-42. https://izlik.org/JA83WH67SC.

EndNote

Farid M, Mahran AEA, Anwar M, Seoud MAA (April 1, 2026) ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS. TWMS Journal of Applied and Engineering Mathematics 16 4 428–442.

IEEE

[1]M. Farid, A. E. A. Mahran, M. Anwar, and M. A. A. Seoud, “ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS”, JAEM, vol. 16, no. 4, pp. 428–442, Apr. 2026, [Online]. Available: https://izlik.org/JA83WH67SC

ISNAD

Farid, Mahmoud - Mahran, Ahmed Ezzat Ameen - Anwar, Mohammed - Seoud, Mohammed Abdel Azim. “ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16/4 (April 1, 2026): 428-442. https://izlik.org/JA83WH67SC.

JAMA

1.Farid M, Mahran AEA, Anwar M, Seoud MAA. ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS. JAEM. 2026;16:428–442.

MLA

Farid, Mahmoud, et al. “ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 4, Apr. 2026, pp. 428-42, https://izlik.org/JA83WH67SC.

Vancouver

1.Mahmoud Farid, Ahmed Ezzat Ameen Mahran, Mohammed Anwar, Mohammed Abdel Azim Seoud. ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS. JAEM [Internet]. 2026 Apr. 1;16(4):428-42. Available from: https://izlik.org/JA83WH67SC