Graph Energy and Estrada Index Through Graph Labeling

Abstract

ABSTRACT newlineGraph labeling is one of the fascinating areas of graph theory with wide newlineranging applications. Graph labelings were first introduced in the 1960s where newlinethe vertices and edges are assigned real values or subsets of a set subject to certain newlineconditions. An enormous body of literature has grown around graph labeling in newlinethe last four decades. Labeled graphs provide mathematical models for a broad newlinerange of applications. An earnest attempt has been made through this thesis newlineto contribute substantially towards the investigation of the new concept, Even newlineHarmonious Labeling (EHL), exploring its properties and labeling schemes. newlineAnother interesting labeling system called strongly quotient labeling has been newlinemade the focal point of investigation and discussions have shown new results. newlineAlgebraic graph theory deals with the interrelation between Graph newlineTheory and Algebra. Results of Algebra are used to solve problems in Graph newlineTheory and vice-versa. Some of the important problems in algebraic graph the- newlineory are Matrix completion problems, minimum rank problems and problems in newlinespectra of graphs. Spectral graph theory is the study of relations between the newlinestructure of a graph and the spectra of certain matrices associated with the graph. newlineRecently, a tremendous research activity has occurred in Graph energy. One of newlinethe remarkable chemical applications of Graph energy is based on Graph eigen- newlinevalues. newlineTaking on such new trends in research on Graph Energy, Estrada in- newlinedex and Graph eigenvalues, the H-eigenvalues, the bounds on Harary energy, newlineHarary estrada index with respect to the Harary matrix for strongly quotient newlinegraphs (SQG) have been dealt with in great discussion in this research and ar- newlinerived at findings are enlisted in details. The study has also developed better newlinebounds and analysed the several variations of them. Extending the concept fur- newlinether to strongly quotient labeling, the research has estimated the Laplacian newlineeigenvalues, the bounds on Laplacian energy and Laplacian estrada index through newlineLaplacian matrix for stro