Contra harmonicmean labeling of graphs

Abstract

A graph G = (V,E) consists of a set of objects V = {v1, v2, . . . , vn} newlinecalled vertices and another set E = {e1, e2, . . . , en}, whose elements newlineare called edges, such that each edge ek is identified with an ordered newlinepair (vi, vj) of vertices. newlineGraph labeling problem is one of the most distinguished areas of re- newlinesearch in graph theory. Most graph labeling methods trace their origin newlineintroduced by Rosa in 1967. Several labelings like Prime labeling, Se- newlinequential labeling , Antimagic labeling, Indexable, Strong indexable, newlineGeometric mean labeling, Harmonic mean labeling, etc., were intro- newlineduced by many authors. Mean and Harmonic mean labeling are sig- newlinenificant for the present study. Based on this, Contra Harmonic mean newlinelabeling of graphs is introduced. A Contra Harmonic mean of two newlinepositive integers x and y is x2+y2 newlinex+y . newlineChapter 1, deals with the basic definitions and results needed in the newlinesequel. newlineIn chapter 2, Contra Harmonic mean labeling of graphs are introduced. newlineA Contra Harmonic mean labeling of a graph G with p vertices and q newlineedges is a function f : V (G) ! {0, 1, . . . , q} such that distinct vertices newlinereceive distinct numbers and {l[f(u)]2+[f(v)]2 newlinef(u)+f(v) m or j[f(u)]2+[f(v)]2 newlinef(u)+f(v) k : uv 2 newlineE(G)} = {1, 2, 3, . . . , q}. A graph is a Contra Harmonic mean graph newlineif it has a Contra Harmonic mean labeling. Here, graphs that allow newlineContra Harmonic mean labeling are investigated. newlineContra Harmonic mean labeling of graphs obtained from standard newlinegraphs are discussed in chapter 3. newlineIn chapter 4, a detailed study is made on Contra Harmonic mean la- newlinebeling of disconnected and subdivided graphs. newlineChapter 5 introduces Super Contra Harmonic mean labeling of graphs. newlineThe existence of Super Contra Harmonic labeling for some standard newlinegraphs are investigated. newline