Fibonacci Range Labeling with Special Emphasis to the Golden Ratio pi 1 618

Abstract

ABSTRACT newlineWe define a graph G is said to be Fibonacci range labeling if we label the vertices newlinex and#8712; V with distinct labels f : V(G)and#8594;{ f2, f3, f4, ..., fp+1} (where f1, f2, f3, ..., fp+1 are newlinefibonacci numbers) such that, when the edge e = (and#945;,and#946;) is labeled with f and#8727;(e = and#945;and#946;) = and#8968; newlinef (and#945;)2+f (and#946;)2 newlinef (and#945;)+f (and#946;) and#8969; or f and#8727;(e = and#945;and#946;) = and#8970; f (and#945;)2+f (and#946;)2 newlinef (and#945;)+f (and#946;) and#8971;, then the resulting edge labels are distinct. newlineAlso, the ratio of each edge to the subsequent edge is in the form of the golden ratio newlinegiven by Rt(Ei,i+1) = Ei+1 newlineEi newlineand#8776; and#968; (for larger i), where Rt(Ei,i+1) is the ratio of the resulting newlineinduced edges (e1,e2), (e2,e3), ..., (enand#8722;1,en) and and#968; = 1.618 known as the golden ratio. newlineIn this PhD thesis, we investigate some common graphs for Fibonacci range labeling and newlinein addition, we looked into direct product and union of graphs, vertex switching of a newlinegraphs and shell-flower related graphs. Various theorems, corollaries, lemma, remark, newlineproposition and examples are provided in each and every section as evidence in support newlineof our thesis. newline