On Generalized θ-graph Whose Second Largest Eigenvalue Does Not Exceed 1

The main results of this paper can be divided into two parts. The first part, all generalized 0-graphs whose second largest eigenvalue does not exceed 1 have been determined. The second part, allθn,k(T)-graphs whose second largest eigenvalue does not exceed 1 have been determined.In chapter one, we introduce the background, terminology.In chapter two, we introduce some important results. In [2], Cvetkovic asked if it was possible to determine all the graphs whose second largest eigenvalue does not exceed 1. In [7]t D.Cao and Y.Hong determine all graphs without isolate vertices with the property 0 <λ2 < 3/1. In [14], M.Petrovic give all graphs with the propertyλ2 < 2 - 1. In [6], D.Cvetkovic and S.Simic obtained some important property of the graphs satisfyingλ2 <5-1/2. In [11], A.Neumaier determined all the trees withλ2 < 1. In [12], M.Petrovic determined all the bipartite graphs withλ2 < l(and therefore all trees). In [16], GuangHui Xu determined all the unicyclic graphs withλ2 < 1. In [9], ShuGuang Guo determined all the bicyclic graphs withλ2 < 1.Chapter three is devoted to studying generalizedθ-graphs whose second largest eigenvalue does not exceed 1. In this chapter, we first prove two generalizedθ-graphs have the property thatλ2 = 1, i.e, the second large eigenvalue is equal to 1. Then we determine all generalizedλ-graphs whose second largest eigenvalue does not exceed 1.Chapter four is devoted to studyingθn,k(T)-graphs whose second largest eigenvalue does not exceed 1. In this chapter, we first prove fourθn,k(T)-graphs have the property thatλ2 < 1, i.e, the second large eigenvalue does not exceed 1. Then we determine allθn,k(T)-graphs whose second largest eigenvalue does not exceed 1.