Research On The Distance Spectral Characterization Of Graphs

Spectral graph theory is a very important subject in graph theory. The spectrum of a graph always has a close relationship with some structure properties of the graph.Spectral graph theory mainly adopts the algebraic techniques to investigate the spectra of graphs. Which graphs are determined by their spectra is a famous question in spectral graph theory. In this paper, We mainly consider some questions on which graphs are determined by their distance spectra, and prove that some graphs are determined by their distance spectra.The main results are as follows.In the first chapter, we introduce the background of spectral graph theory and some important concepts and notations. Furthermore, we also show the problems we discussed and list the results obtained in this paper.In the second chapter, we mainly consider the distance spectral characterization of trees. We prove that two kinds of special trees—path and double star—are determined by their distance spectra. Meanwhile, we give the distance characteristic polynomial of double stars.In the third chapter, we mainly discuss the distance spectral characterization of some special graphs. We give the distance characteristic polynomial of some graphs, and show that these graphs are determined by their distance spectra.In the fourth chapter, we mainly investigate the distance spectral characterization on the second largest distance eigenvalue. We prove that the graphs with λ2(D(G)) ≤17-√3292are determined by their distance spectra.