Some Results On Quasi-harmonic Graph

The theory of graphs spectra is a crossing area between graph theory and algebra. Itis also a branch of algebraic graph theory. In the way of main eigenvalues, Hagos gavethe sufcient and necessary condition of the graphs with exactly two main eigenvalues of agraph. Afterwards, many in our country began to study the question of how to characterizegraphs with exactly two main eigenvalues. Hou and Tian characterized Trees and Unicyclicgraphs with exactly two main eigenvalues of a graph. Zhu, Hu and Li characterized Bicyclicgraphs with exactly two main eigenvalues of a graph, at the same time Shi also gave thisconclusion in an other paper, but it is so pity that those papers all nelgected the case ofb <0. Huang and Yin find it and solve this question, in Completely characterizations ofunicyclic and bicyclic graphs with exactly two main eigenvalues. If there is unique contentsa, b and A2j aAj b=0, then we call it G(a, b), denoted by G(a, b)In this paper, we focus on λ-quasi-harmonic graph. The main results are as follows:In the first chapter, introduction, we give the basic definitions, symbols and notationsabout graphs spectra. In the first section, we gave the main eigenvalues polynomial, the no-tations about λ-harmonic graph, λ-quasi-harmonic graph, H(k1, k2). In the second section,we look back the study of the theory of graph spectrum, especially, the study about themain eigenvalues of a graph, and introduce the present situation and history, and give themain results of this thesis, i.e., λ-harmonic graph.In the second chapter, in the first section, we give the sufcient√and necessa√ry condition ofλ-quasi-harmonic graph, and its two main eigenvalues are λ1=λ, λ2=λ and λ≥2.In the second section, we give the other quality of λ-quasi-harmonic graph. In the thirdsection, we characterized some of λ-quasi-harmonic graph. In the last section, we get someresults about the line and complement of λ-quasi-harmonic graph.In the last chapter, we give some results about G(1, b) graph.