On spectra and energies of certain graphs

Abstract

Spectral graph theory is the study of the eigenvalues and eigenvectors of matrices newlineassociated with graphs. In graph spectra, we can find various relations between the newlinespectrum and the structure of a graph. It plays a vital role in solving various problems newlinein communication networks. The study of different structural properties of a graph is newlinecharacterized from the properties of eigenvalues and eigenvectors of various matrices newlinesuch as adjacency, Laplacian, signless Laplacian, normalized Laplacian, Randi´c etc. newlineThis thesis focuses on the study of the different types of spectrum and energy of newlinethe graphs obtained from various graph operations. We discuss adjacency, Laplacian, newlinenormalized Laplacian and signless Laplacian spectrum of several graph operations. newlineMoreover, infinitely many pairs of cospectral graphs are constructed with respect to newlinedifferent matrices. The construction of integral graphs is one of the interesting problem newlinein spectral graph theory. An integral graph is a graph whose spectrum consists newlineentirely of integers. The concept of integral graph was first introduced by Harary and newlineSchwenk in 1974. Here, some new families of integral graphs are obtained from our newlinegraph operations. Further, formulae for finding the number of spanning trees, the newlineKemeney s constant and the Kirchhoffs index of the resulting graphs are determined. newlineThe construction of equienergetic graphs is an important problem in spectral newlinegraph theory. In this thesis, we construct some equienergetic and Randi´c equienergetic newlinegraphs. Also, some new families of order energetic and hypoenergetic graphs