Radio Mean Labeling of Certain Classes of Graphs

Abstract

This thesis provides a comprehensive investigation of radio mean labeling of connected graphs, a labeling scheme that assigns distinct positive integers to each vertex satisfying a condition involving vertex labels and distances. For a connected graph G with diameter d, a radio mean labeling is a one-to-one function and#947; from V pGq to Z `, the set of all positive integers where dpu, vq ` r and#947;puq`and#947;pvq 2 s and#283; 1 ` d, at u, v P V pGq. The inequality is called the radio mean condition. The span of and#947; is the maximum integer assigned to any v P V pGq by and#947;. Further, the radio mean number of G, denoted by rmnpGq, is the smallest span of all feasible radio mean labelings of G. The main objective is to determine the radio mean number, the minimum span of all possible radio mean labelings for a given graph. The study explores the properties and characteristics of radio mean labelings and their practical and combinatorial applications. The radio mean labeling of subdivided book graphs, paths, total graph of paths, and cycles is studied, revealing the challenges in determining the radio mean number for graphs with varying diameters. The existing literature on the radio mean number of paths and cycles are critiqued, and new upper bounds are proposed. The thesis also presents alternative methods for obtaining radio mean labelings, including an integer linear programming model and a polynomial time algorithm. Additionally, the study examines the extension of radio mean labeling from a subgraph with specific properties to its supergraph, improving the range of the radio mean number. Finally, the potential of labeled graphs in secure communication, such as modeling communication networks and generating keys for cryptography, is explored. newline