Advanced Studies of Metric Sets of Graphs

Abstract

By a graph G = (V;E); we mean a finite undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n = jV (G)j and m = jE(G)j respectively. In this thesis we introduce three new graph parameters which are as follows: Uniform number of a graph.2. Detour homometric number of a graph. 3. Strong homometric number of a graph. First, we present the basic definitions and theorems which are needed for the subsequent chapters. A detailed literature review related to the detour distance in graphs is given in this thesis. Secondly, we initiate the study of uniform sets and the uniform number of a graph. We present some basic results on this parameter and determine the uniform number for several classes of cyclic and acyclic graphs. We characterize the Hamilton-connected graphs and star graphs with and(G) = 1: We also investigate the behavior of the newly introduced graph parameter uniform number of a graph with other existing graph parameters such as domination number, clique number, independence number and the chromatic number of a graph. The third chapter introduces the new graph parameter, Detour homometric sets and Detour homometric number of a graph and determine the Detour homometric number for several classes of cyclic graphs. The fourth chapter introduces the new graph parameter, Strong homometric sets and Strong homometric number of a graph and determines the Strong homometric number for several classes of cyclic graphs. Finally, we give a summary of the thesis and scope for further work. newline