Super (a, d) -P₂⨀Pₘ-Antimagic Total Labeling of Corona Product of Two Paths

Graph labeling involves mapping the elements of a graph (edges and vertices) to a set of positive integers. The concept of an anti-magic super outer labeling (a, d) -H pertains to assigning labels to the vertices and edges of a graph using natural numbers 1, 2, 3,. . . , p+q. The weights of the outer labels H form an arithmetic sequence a, a+d, a+2d,. . . , a+ (k-1) d, where 'a' represents the first term, 'd' is the common difference, and 'k' denotes the total number of outer labels, with the smallest label assigned to a vertex. This study investigates the super (a, d) -P₂⨀Pₘ-antimagic total labeling of the corona product Pₙ⨀Pₘ, where n and m are both greater than or equal to 3. We define the labeling functions for vertices and edges based on the partitioning of labels into three subsets. Using k-balanced and (k, δ) -anti balanced multisets, we demonstrate that for m being odd, Pₙ⨀Pₘ is super (9m² n+4mn+m-n+3, 1) -P₂ ⨀▒P_ (m) -antimagic, and for m being even, Pₙ⨀Pₘ is super (9m² n+4mn+m-2n+5, 3) -P₂ ⨀▒P_ (m) -antimagic. The labeling scheme is illustrated through examples. For the case when m is odd, an antimagic total labeling of P₃ ⨀▒P₃ forms a super (282, 1) - P₂ ⨀▒P_ (3) -antimagic labeling. In the case of even m, an antimagic total labeling of P₃ ⨀▒P_ (4) results in a super (483, 3) - P₂ ⨀▒P_ (4) -antimagic labeling. Both of these examples provide insights into the antimagic properties of corona products.