Graph Composite Labeling techniques and Practical Applications

In this study, the intriguing concept of composite labeling within the realm of graph theory is explored. A composite labeling, represented by a bijection denoted as f, involves mapping elements from both the vertex set $V(H)$ and the edge set $E(H)$ of a graph H to integers ranging from 1 to $m+n$. A key condition imposed on this mapping is that the greatest common divisor (gcd) of $f(u v)$ and $f(v w)$ must not equal 1, where $u, v$ and w are vertices in $V(H)$. The research primarily focuses on the Star graph $k_{1, n}$ and its applications in diverse fields such as networking, blockchain, and online commerce. It is demonstrated that composite labeling is applicable to the Star graph, opening up intriguing possibilities for its use in practical scenarios. Furthermore, the exploration extends to other graph structures, including the Crown graph $\left(C_{n} \times K_{1}\right)$, the Comb graph $\left(P_{n} \times K_{1}\right)$, the Bistar graph $\left(B_{n \times n}\right)$, the join sum of two copies of Cycle $\left(C_{n}\right)$, the one-point union of six copies of $P_{4}$, along with Caterpillar trees and the Flower graph $f_{(n \times m)}$. It is established that composite labeling is a versatile concept that can be employed in various graph types. This study not only enhances the understanding of composite labeling within graph theory but also highlights its practical relevance in multiple domains. The anticipation is that these findings will inspire further research into this fascinating area, uncovering new applications and insights that can benefit both theoretical graph theory and practical network analysis.