jaem TWMS Journal of Applied and Engineering Mathematics 2146-1147 2587-1013 Işık University Press Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics) Kombinatorik ve Ayrık Matematik (Fiziksel Kombinatorik Hariç) ON STRONGLY $^*$-GRAPHS AND STRONGLY MULTIPLICATIVE GRAPHS https://orcid.org/0009-0001-6076-4411 Farid Mahmoud The British University in Egypt https://orcid.org/0009-0000-7008-5941 Mahran Ahmed Ezzat Ameen Ain Shams University https://orcid.org/0000-0003-0668-4071 Anwar Mohammed Ain Shams University https://orcid.org/0000-0003-3216-5873 Seoud Mohammed Abdel Azim Ain Shams University 04 07 2026 16 4 428 442 09 04 2025 02 09 2026 Copyright © 2010, TWMS Journal of Applied and Engineering Mathematics 2010 TWMS Journal of Applied and Engineering Mathematics

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.

 strongly $^*$-graphs strongly multiplicative Graph labeling [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. [9] Kanani, K. K., and Chhaya, T. M., (2017), Strongly multiplicative labeling of some standard graphs, Int. J. Math. Soft Comput., 7(1), 13–21. [10] Gallian, J. A., (2022), A dynamic survey of graph labeling, Electron. J. Combin., 25. [11] West, D. B., (2001), Introduction to Graph Theory, Prentice Hall, 2nd Edition, 6.