Liouville Product Cordial Labeling of Certain Graphs

Graph labeling provides a structured approach for assigning numerical values to the elements of a graph in order to study its underlying properties. Motivated by the interaction between number theory and graph theory, this paper introduces a new labeling scheme called Liouville Product Cordial Labeling based on the Liouville function, which assigns the values −1 and 1. In this labeling, the value of each edge is obtained as the product of the Liouville values associated with its end vertices. A graph is said to satisfy Liouville Product Cordial Labeling if the counts of edges labeled −1 and 1 differ by at most one. In this paper, the existence of this newly developed labeling for various families of graphs, such as Wheel, Tortoise graph, Star, Bistar, Twig graph, Sparkler graph, and Lobster graph, has been examined, and the conditions under which such labeling can be constructed have been identified. The proposed approach extends cordial labeling concepts through a multiplicative framework and enriches the study of number-theoretic graph labelings.