Investigation of $C_m$-Supermagic Properties in the Union of $C_n$ and $C_m$ Graph Structures

Graph labeling, the assignment of numbers to the vertices or edges of a graph, finds applications in diverse fields such as network addressing, channel allocation, data mining, image processing, cryptography, and logistics. A $C_m$-supermagic labeling involves assigning integers to a graph's edges and vertices such that the labels' sum for all $C_m$ cycles equals a constant value. This paper explores the $C_m$-supermagic properties of the $Cl_{n,m}$ graph, formed by the union of a $C_n$ graph and $n$ $C_m$ graphs. It comprehensively analyzes the conditions under which $Cl_{n,m}$ exhibits $C_m$-supermagic properties and derive explicit labeling constructions. These results contribute to understanding $C_m$-supermagic graphs and their potential applications in theoretical and applied domains.