Face Bimagic and 1-Antimagic Labelings of Rooted Product of Particular Graph Classes

Graph labeling constitutes a significant area of research in graph theory, encompassing a wide variety of labeling methods and their multifaceted applications. This work specifically concentrates on the assignment of labels to a planar graph. It establishes the theoretical framework for face bimagic and 1-antimagic labelings of type (1, 0, 0) and (1, 0, 1) for the rooted product of P_n and K_2,m, outlining comprehensive criteria and methodologies. The paper also explores the behavior of the associated magic constant in various configurations. Through constructive examples and parameter variations, we demonstrate how label assignments affect the constancy or variability of sums across faces of a given graph, thus offering deeper insights into the labeling characteristics of planar graphs.