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

Year 2026, Volume: 16 Issue: 4 , 428 - 442 , 07.04.2026

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

https://izlik.org/JA83WH67SC

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

strongly $^*$-graphs , strongly multiplicative , Graph labeling

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