A Study of Edge Cordial Graphs on Harmonious Graphs in Graph Theory: Exploring Ek-cordial labeling and its applications in graph theory

Abstract

This study consists of Special emphasis has been given to Edge cordial graphs, Friendly indexset of graphs, Harmonious graphs, Difference cordial graphs and vertex - graphs. Yilmaz and Cahitintroduced a weaker version of edge graceful called edge cordial labeling and also defined a new graphlabeling technique called Ek – cordial labeling. This Ek – cordial labeling is the generalization of eadgecordial labeling. In this study it is proved that the following graphs are Ek – cordial graphs. The labelingor valuation or numbering of a simple graph G is a one-to-one mapping from its vertex set (edge set) intoa set of non-negative integers which induces an assignment of labels to the edges (vertices) of G. Graphlabeling is one of the fascinating areas of graph theory with wide range of applications. Graph labelingproblems were formulated in the mid 1960‘s from a long standing conjecture of Ringel and a paper byRosa. Even the structure of graceful trees is not completely known till today. Ringel conjectured that alltrees are graceful. This conjecture remains unsolved. labeling of a graph G with q edges known as ‗β-valuation‘ which is an injection of the set of its vertices into the set of integers {0, 1, 2, …, q} such thatthe value of its edges are the numbers from 1 to q, the values of an edge being the positive differencebetween the numbers assigned to its end vertices. Golomb called such labeling graceful.