Edge-Irregular Reflexive Strength of Non-Planar Graphs

Symmetry in non-planar graphs is a fundamental concept that enhances understanding, simplifies analyses, and has practical implications in diverse fields such as science, engineering, and mathematics. A total κ-labeling for a graph Gνˇ is composed of two labeling: one is an edge labeling Υe:E(Gνˇ)→{1,2,3,…,κe} and the other is a vertex labeling Υv:V(Gνˇ)→{0,2,4,…,2κv}, where κ=max{κe,2κv}. The weight of an edge under reflexive labeling is defined as wt(pq)=Υv(p)+Υe(pq)+Υv(q)∀e=pq. The total κ−labeling is said to be an edge-irregular reflexive κ−labeling, if for every two edges eδ and ej, the weights are distinct. The lowest value of κ for which the graph Gνˇ has an irregular reflexive edge κ−labeling is called the reflexive edge strength of Gνˇ, denoted as res(Gνˇ). res(Gνˇ) captures irregularity while preserving reflexivity by quantifying edge-weight variability in reflexive graphs. In this article, we are interested in determining the tight lower bound for non-planar prisms, cross prisms, cross-particle modified graphs, and cross-particle glowing graphs under reflexive labeling.