Face Magic and 1-Antimagic Labelings of Rooted Product of Path and Cycle

In graph theory, the concept of graph labeling is a significant area of research due to its both theoretical and practical implications. It provides valuable insights into the structural properties and labeling constraints of the graphs. This study focuses on the face magic labeling and 1-antimagic labeling of types (1,1,1), (1,1,0), (1,0,0), (1,0,1), (0,1,0) and (0,1,1) of the rooted product of graphs, a construction that combines two graphs, path Pn and cycle Cm, by attaching a copy of Cm to each vertex of Pn. This study establishes the theoretical foundations for face magic and d-antimagic labelings of type (a,b,c) in the rooted product of graphs, providing detailed criteria and methodologies.