174 Y. Zou et al. / Linear Algebra and its Applications 320 (2000) 173–182If T isatree**of**ordern, **the**nT is bipartite, and its **eigenvalues** satisfy **the** relationλ i (T ) =−λ n−i+1 (T ) (i = 1, 2,...,n). So, it suffices to study those **eigenvalues**λ k (T ) for 1 k [n/2]. In this paper we always assume that 1 k [n/2].An interesting unsolved problem in **the** study **of** **the** spectra **of** **trees** is to find “**the**best possible upper bound” for **the** **kth** **eigenvalues** **of** **trees** **of** order n.Ino**the**rwords,letand letΓ n ={T | T is a tree **of** order n},¯λ k (n) = max{λ k (T ) | T ∈ Γ n }(1 k [n/2]).Then, **the** above problem asks to determine **the** function ¯λ k (n) and (if possible) findatreeT ∈ Γ n **with** λ k (T ) = ¯λ k (n).There have been considerable attempts in studying this problem, and **the** remainingunsolved case for ¯λ k (n) is **the** case n ≡ 0(**modk**), 7 k [n/2]. For this case,we write n = kt (t 2) and let¯Γ k,t ={T ∈ Γ kt | λ k (T ) = ¯λ k (kt)}.The **trees** in ¯Γ k,t are called **the** extremal **trees**.To be clear, we give **the** same definitions as those in [1] below.Definition 1. Let X k,t be **the** subset **of** **trees** in Γ kt which consists **of** k disjointstars S 1 ,...,S k **of** **the** order t(S 1∼ = S2 ∼ = ···∼ = Sk ∼ = K1,t−1 ) toge**the**r **with** ano**the**rk − 1 edges e 1 ,e 2 ,...,e k−1 such that **the** two end vertices **of** each edge e i (i =1, 2,...,k− 1) are noncentral vertices **of** different stars. We call S 1 ,...,S k **the** stars**of** this tree T ∈ X k,t , call **the** edges e 1 ,...,e k−1 **the** nonstar edges **of** T, and call **the**o**the**r edges **the** star edges **of** T.Definition 2. We define **the** condensed tree ̂T **of** T as V(̂T)= (S 1 ,S 2 ,...,S k ),and**the**re is an edge [S i ,S j ] (i /= j) in ̂T if and only if **the**re exists some nonstar edge **of**T **with** one end in S i and **the** o**the**r end in S j .Definition 3. Define X ′ k,tas **the** subset **of** X k,t which consists **of** those **trees** T inX k,t such that for any star S i **of** T, **the**re is only one vertex in S i incident to somenonstar edges **of** T.A considerable necessary condition for extremal **trees** obtained in [1] is that if T ∈¯Γ k,t (k 2,t 5),**the**nT ∈ X k,t and Δ(̂T) 3, where Δ(̂T)is **the** maximal degree**of** **the** condensed tree ̂T . In this paper, we establish a fur**the**r necessary condition forextremal **trees**.

180 Y. Zou et al. / Linear Algebra and its Applications 320 (2000) 173–182and from (2.15)( )√t − 1 + λ 2 f 2cosπk+1 √ t − 1 + λ 2 (f 2 ). (3.9)Combining (3.6)–(3.8), we obtain (3.5). □Remark 1. We have also verified that (3.5) holds for k = 7, 8, 9. So, for k 7andt 4, if we denote byP k,t ={T ∈ X k,t | ̂T = P k },J k,t ={T ∈ X k,t | ̂T = J k },andL k,t ={T ∈ X k,t | ̂T = L k },**the**n, from **the** previous results and Theorem 8, it suffices to find **the** extremal **trees**in P k,t ∪ J k,t ∪ L k,t .4. Some fur**the**r discussionsIn this section, we establish some fur**the**r results about **the** left problem **of** finding**the** extremal **trees** in J k,t .Lemma 9. Let P k ,J k k 4 as in Fig. 2. We havei.e.λ i (J k ) = 2cos(2i − 1)π, i = 1, 2,...,k, (4.1)2k − 2λ i (J k ) = λ 2i−1 (P 2k−3 ). (4.2)Pro**of**. From Fig. 2, we can write( ) A αA(J k ) =α T ,0where A is **the** adjacent matrix **of** P k−1 , α = (0, 1, 0,...,0) T .Let J k (λ) = det(λI − A(J k )). Then, we have **the** recursive relation as follows:J k (λ) = λJ k−1 (λ) − J k−2 (λ),since x 1,2 = λ ± √ λ 2 − 4/2 are **the** two roots **of** x 2 − λx + 1 = 0, we haveJ k (λ) = c 1 x1 k + c 2x2 k . (4.3)**On** **the** o**the**r hand,J 4 (λ) = λ 3 − 3λ 2 = c 1 x1 4 + c 2x2 4 , (4.4)

Y. Zou et al. / Linear Algebra and its Applications 320 (2000) 173–182 181J 5 (λ) = λ 5 − 4λ 3 + 2λ = c 1 x1 5 + c 2x2 5 . (4.5)Combining (4.4) and (4.5), we havec 1 = J 5(λ) − J 4 (λ)x 2x 5 1 − x4 1 x 2From (4.3) and (4.6), we haveJ k (λ) = J 5(λ) − J 4 (λ)x 2x k−4x 1 − x 2Let(2i − 1)πλ i = 2cos2k − 2 .Then **the** direct computations giveJ k (λ i ) = 0, i = 1, 2,...,k.Thus we obtain (4.1). Noticingλ i (P k ) = 2coswe have (4.2).□iπk + 1 ,, c 2 = J 5(λ) − J 4 (λ)x 1x2 5 − x4 2 x . (4.6)11+ J 5(λ) − J 4 (λ)x 1x 2 − x 1x k−42.Theorem 10. If **the**re is no extremal tree in J k,t , **the**n **the**re is no extremal tree inJ s,t for k + 1 s 2k − 2.Pro**of**. By Lemma 9, we haveλ 1 (J k ) = λ 1 (P 2k−3 ), λ 1 (J s )>λ 1 (J k ),andλ 1 (P s )>λ 1 (P k ), k + 1 s 2k − 2.So, from (2.15), we have√t − 1 + λ2 (f λ1 (J k ))

182 Y. Zou et al. / Linear Algebra and its Applications 320 (2000) 173–182References[1] Jia-yu Shao, **On** **the** **largest** **kth** **eigenvalues** **of** **trees**, Linear Algebra Appl. 221 (1995) 131–157.[2] D.M. Cretkvoic, M. Doob, H. Sachs, Spectra **of** Graphs, Academic, New York, 1980 (Appendix,Table 2).