Some recent graph labelings

Abstract

Thesis Summary Graph labeling, which is the communication between number theory and graph theory, is an assignment of numbers to the vertices of a graph or edges, subject to specific constraints. Graph labelings were first presented in the late 1960s. Many graph labelings techniques have been considered in the intervening fifty years in more than 2500 research paper. In this thesis we will introduce some basic definitions of graph theory and number theory and some basic operations on graphs as an introduction to the graph labeling in the first chapter. In the second chapter we will study graphs under a certain labeling condition which is called strongly *-graphs. We get the maximum number of edges of any graph with n vertices to be strongly *- graph and study some new families to be strongly *- graphs. Results in this chapter were published in the “Proceedings of the Pakistan Academy of Sciences”. In the third chapter we will study another type of graph labeling which is called vertex equitable labeling, we make a survey for all graphs with order at most 6 whether they are vertex equitable or not and we get the maximum number of edges of any graph with n vertices to be a vertex equitable graph. Results in this chapter are accepted for publication in “Utilitas Mathematica”. In the fourth chapter we study another type of graph labeling, which is called edge product cordial (EPC) labeling, and we present some properties of EPC graphs and make a survey on all graphs with the number of vertices not more than 6 to find out whether they are EPC or not. Finally, we study some families of graphs to be EPC or not.