Elegant Labeling of the Lotus Graph and Its Variations

Abstract

An elegant labeling f of a graph G(V,E) with ∣E(G)∣=q is an injective function f:V(G)→{0,1,2,…,q} such that each edge xy in E(G) is assigned the induced label f∗(xy)=(f(x)+f(y))mod(q+1), where all resulting edge labels are distinct and non-zero. A graph that admits an elegant labeling is called an elegant graph. Research on elegant labelings has been extensively conducted on simple graph classes such as cycle graphs and star graphs. However, for more complex graph structures like the lotus graph, studies on elegant labelings remain unavailable. The lotus graph possesses a unique symmetrical characteristic, systematically constructed from a combination of cycles and a high-degree central vertex. This symmetrical structure presents a distinct mathematical challenge to ensure that the labeling criteria are satisfied. A lotus graph visually resembles a blooming lotus petal, a distinctive structure that mathematically can only be fully realized if it meets the lower bound of p≥3. This study aims to demonstrate that the lotus graph L(p) and its two variations, namely the graph L∗(p) and the graph L(p,p), are elegant graphs for any p≥3. These two variations of the lotus graph are obtained through the deletion of pendant vertices for L∗(p) and the addition of pendant vertices for L(p,p). An exploration patterns is conducted on several simpler lotus graphs, which are subsequently generalized for p≥3, while the analytical method is utilized for formal mathematical verification. The results show that the lotus graph L(p) and its two structural variations, L∗(p) and L(p,p), are elegant graphs for any p≥3. The findings of this research are expected to expand the scope of study on graph classes that admit elegant labelings and serve as a foundation for future advanced studies regarding other structural graph variations.