Graphs With Exactly Two Main Eigenvalues And Integral Graph

Let G= (V, E) be a graph with n vertices and A(G) be its adjacency matrix. The family of eigenvaluesλ1(G)≥λ2(G)≥…≥λn of A(G) is the spectrum of the graph G and the largest eigenvalueλ1(G) of A(G) is also called the index of G. A graph G is called integral if the spectrum of its adjacency matrix has only integral eigenvalues.In this paper, we study that graphs with exactly two main eigenvalues and integral graph. We mainly discuss that some proposition of the graphs with exactly two main eigenvalues and few cycles connected graphs with with exactly two main eigenvalues are determined, integral graphs with exactly two main eigenvalues and index 3 are determined, some integral trees with index 4 are determined by Godsil lemma. The paper is consist of six chapters.In chapter one. we introduce some definitions of graph and list some results of graph spectra, which will been used in this paper.In chapter two, we give and proof some proposition of the graphs with exactly two main eigenvalues. As an application, we obtain all connected unicyclic graphs with exactly two main eigenvalues.In chapter three, all connected bicyclic graphs with exactly two main eigenval-ues are determined,which are infinite but there are only 11 different forms.In chapter four, all connected tricyclic graphs with exactly two main eigenvalues are determined, which are infinite, but there are only 48 different forms.In chapter five, we show that there are exactly 25 connected integral graphs with exactly two main eigenvalues and index 3.In chapter six, all integral trees with index 4 andλ≠±3 in the spectral are determined. which are infinite but there are only 18 different constructions.