Um estudo sobre a matriz distância de grafos

Abstract

In this thesis we discuss important aspects of two families of graphs: Caterpillar and Threshold graphs. Our goal is to determine the multiplicity of the distance eigenvalues and the number of distinct distance eigenvalues for these graphs. For some graphs within these families, we are able to describe exactly the eigenvalues or locate them between better bounds than those found in the literature. We also determine a new parameter to threshold graphs which is associated with the structure as well as spectral aspects of the graph. This parameter allow us to calculate the multiplicity of some eigenvalues and the number of distinct distance eigenvalues. Finally, we verify that the threshold graphs with four or five distinct distance eigenvalues are characterized by their spectrum.