Proper d-Lucky Number for Certain Rooted Product Graphs

Abstract

Objectives: In this study, rooted product of path with cycle, path with complete graph cycle with complete graph and complete graph with complete graphs are taken and examined for the existence of d-lucky labeling for the same. Methods: The rooted product graphs are obtained of path with cycle, path with complete graph, cycle with complete graph and complete graph with itself and then proper d-lucky numbers are obtained for the above mentioned graphs. Construction and descriptive method are used to prove the results. Findings: Proper d-lucky labeling for the said graphs are verified and proper d-lucky numbers for the same are obtained. Novelty: It involves the mathematical formulations which involves labeling the vertices of a graph in such a manner that no two adjacent vertices have the same labeling and the neighborhood sums and degree sums of the adjacent vertices are different, which gives rise to proper d-lucky labeling. Keywords: d-Lucky Labeling; d-Lucky Numbers; Proper d-Lucky Labeling; Proper d-Lucky Number; Rooted Product