QUOTIENT SQUARE SUM SP E- CORDIAL LABELING

Abstract

Graph labeling is a well-established area of graph theory in which numbers are assigned to the vertices or edges of a graph subject to certain conditions. Cordial labeling, introduced by Cahit, requires the induced vertex and edge label classes to differ by at most one, and has since given rise to numerous variants, including SP (sum–product) based E-cordial labelings. In this paper, we introduce a new variant called Quotient Square Sum SP E-Cordial Labeling, in which each edge of a graph is assigned one of two labels, and the induced label of every vertex is derived from an arithmetic function that combines the sum and product of the labels of its incident edges, taking the quotient of the sum of their squares to their sum, and reducing the result modulo two. A graph is said to be Quotient Square Sum SP E-Cordial if the resulting vertex and edge label distributions differ by at most one in each class. In this paper, we establish that several standard graph families the path , cycle , comb graph , crown graph , triangular snake , and the tadpole (dragon) graph  admit this labeling. Explicit labeling schemes are constructed for each family, and the induced vertex and edge partitions are computed and shown to satisfy the required cordiality conditions in all cases. We further extend these results to establish that the helm graph, closed helm graph, semi-total point graph of a path, semi-total point graph of a cycle, subdivision of a star graph, and the count tree graph also admit this labeling, bringing the total number of graph families treated in this paper to twelve.