On The Spectrum Of Some Graphs

In the last few years,many researchers had a large number of studies on the spectrum of connected graphs.And on the basis of previous studies by scholars,We determined all tricyclic graphs with minimum distance eigenvalues in(-2-(?),-2],furthermore,we characterized part of graphs with exactly three nonnegative eigenvalues for adjacent matrix in this paper.Suppose that G is a simple graph with the vertex set V(G)={v1,v2,...,vn}.The distance matrix of G is denoted by D(G)=(dij)n×n,where dij is the distance between vi and vj.The least eigenvalue of D(G)is also called the least distance eigenvalue of G.Denote by A(G)=(aij)n×n the adjacency matrix of figure G,where aij=1 if vi and vj are adjacent,and otherwise aij=0.Since A(G)is symmetric we can write its eigenvalues to be ?1(G)??2(G)?...? ?N(G).we describe the main conclusions by three parts of this paper:Firstly,in the second part,we suppose that the connected graph G,its edge set and point set are E(G)=m and V(G)=n,respectively.The graph G be called tricyclic graph when E(G)and V(G)satisfied |E(G)|=|V(G)|+2.In this part,we determined all tricyclic graphs G with the least eigenvalues of the distance matrix D(G)in(-2-(?),-2]by five lemmas and a theorem.Secondly,in the first section of the third part,suppose that G is a connected graph with the edge set E(G)=m and the point set V(G)=n.G be called c-cyclic graph that if|E(G)|=|V(G)|-1+c.And G be called the tree,unicyclic graphs,bicyclic graphs and tricyclic graphs when c=0,1,2,3,k,respectively.In this section,we find all tree,unicyclic graphs,bicyclic graphs and tricyclic graphs with exactly three nonnegative eigenvalues for the adjacent matrix A(G)by four theorems.Thirdly,in the second section of the third part,suppose that G is disconnected graph.In this section,we characterize all disconnected graphs with two or three connected com-ponents with exactly three nonnegative eigenvalues for the adjacency matrix A(G)by One theorem and three corollaries.