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WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin Investigation on spectrum of the adjacency matrix and Laplacian matrix of graph Gl SHUHUA YIN Computer Science and Information Technology College Zhejiang Wanli University Ningbo 315100, PEOPLE’S REPUBLIC OF CHINA Abstract: Let Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl , and dk−j+1 be the degree of vertices in the level j. In this paper we study the spectrum of the adjacency matrix A(Gl ) and the Laplacian matrix L(Gl ) for all positive integer l, and give some results about the spectrum of the adjacency matrix A(Gl ) and the Laplacian matrix L(Gl ). By using these results, an upper bound for the largest eigenvalue of the adjacency matrix A(Gl ) is obtained: q λ1 (A(Gl )) < max{ max { dj − 1 + q dj+1 − 1}, 2≤j≤k−2 q dk−1 − 1 + p dk − l + 1, p dk − l + 1 + l − 1}, and an upper bound for the largest eigenvalue of the Laplacian matrix L(Gl ) is also obtained: µ1 (L(Gl )) < max n p max { 2≤j≤k−2 dj − 1 + dj + p dj+1 − 1}, p dk−1 − 1 + dk−1 + p dk − l + 1, p o dk − l + 1 + dk + 1 . Key–Words: Adjacency matrix, Laplacian matrix, complete graph, spectrum 1 Introduction For example, the complete graph Kn has n vertices, and each distinct pair are adjacent. Thus, the graph K4 has adjacency matrix Let G be a simple undirected graph with vertex set V = {v1 , v2 , ...vn }, which n = |V |. Let A(G) be a (0, 1)-adjacency matrix of G. Since A(G) is a real symmetric matrix, all of its eigenvalues are real. Without loss of generality, that they are ordered in nonincreasing order, i.e., λ1 (G) ≥ λ2 (G) ≥ ... ≥ λn (G), A(K4 ) = 1 1 1 0 , About the spectrum and the spectral radius of graphs, a great deal of investigation is carried out [1][2][3]. Specially, to the special graphs, for example [4] studied the spectral radius of bicyclic graphs with n vertices and diameter d, [5] studied the spectral radius of trees with fixed diameter. In [6] V. Nikiforov proved that if G is a graph of order n ≥ 2, maximum degree ∆, and girth at least 5, then √ λ1 (G) ≤ min{∆, n − 1}, and their muliplicities are m(λ1 ), m(λ2 ), ..., m(λs ), then we shall write ISSN: 1109-2777 1 1 0 1 and an easy calculation shows that that spectrum of K4 is: ! 3 −1 SpecA(K4 ) = 1 3 λ1 (G) > λ2 (G) > ... > λs (G), SpecA(G) = 1 0 1 1 (1) and call them the spectrum of G, The largest eigenvalue λ1 (G) is called the spectral radius of G. If the destine eigenvalues of A(G) are λ1 (G) λ2 (G) ... λs (G) m(λ1 ) m(λ2 ) ... m(λs ) 0 1 1 1 ! . 362 Issue 4, Volume 7, April 2008 WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin 2 where λ1 (G) is the largest eigenvalue of the adjacency matrix of G. In [7] X.D.Zhang proved Preliminaries Let T be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Kl be a complete graph on l vertices. Let Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl . Similar to the definition of tree’s level, we agree that the complete graph Kl is at level 1, and that Gl has k levels. Thus the vertices in the level k have degree 1. For j = 1, 2, 3, ..., k, let nk−j+1 and dk−j+1 be the number of vertices and the degree of them in the level j. Observe that nk = l is the number of vertices in level 1 and n1 the number of vertices in level k(the number of pendant vertices). Then, p 2∆ − 1 − 2 ∆(∆ − 1) λ1 (G) < ∆ − , n(n − 1)∆ where G be a simple connected non-regular graph of order n and ∆ be the maximum degree of G. Let d(vi ) denote the degree of vi ∈ V , i = 1, 2, ..., n, and let D(G) = diag(d(v1 ), d(v2 ), ..., d(vn )) be the diagonal matrix of vertex degrees. The Laplacian matrix of G is L(G) = D(G) − A(G) . Clearly, L(G) is a real symmetric matrix. From this fact and Gers̆gorin’s Theorem, it follows that its eigenvalues are nonnegative real numbers. Therefore, the eigenvalues of L(G), which are call the Laplacian eigenvalue of G , can be denote by nk−1 = (dk − l + 1)nk , nk−j = (dk−j+1 − 1)nk−j+1 , j = 2, 3, ..., k − 1. Observe that dk is the degree of vertices of the complete graph Kl in Gl , d1 is the degree of the vertices in the level k, nk = l . The total number of vertices in the graph Gl is µ1 (G) ≥ µ2 (G) ≥ ... ≥ µn (G) = 0. We call µ1 (G) the Laplacian spectral radius of G. If the destine eigenvalues of L(G) are µ1 (G) > µ2 (G) > ... > µs (G), n= and their muliplicities are Example 2.1 Follow (Fig.1) is an example of a such graph G4 for k = 3, n1 = 24, n2 = 8, n3 = 4, d1 = 1, d2 = 4, d3 = 5. 21r 20 22 r 19 23r rB r 18r 17 r BB 16 B r @ 24a r @B D B then we shall write µ1 (G) µ2 (G) ... µs (G) m(µ1 ) m(µ2 ) ... m(µs ) ! . r B aa aBr @ For the Laplacian eigenvalues and the Laplacian spectral radius of simple graphs, there are many good results. In [8], some of the many results known for Laplacian matrices are given. Fiedler [9] proved that G is a connected graph if and only if the second smallest eigenvalue of L(G) is positive. This eigenvalue is call the algebraic connectivity of G, denoted by α(G). In [10] Li and Pan proved the following result: Let G be a simple connected graph with n vertices and m edges. Denote by ∆, δ the largest and smallest degrees of vertices in G. Then µ1 (G) ≤ q J r 36Je e e % r 33 % % 25r r 26 1 r B 2 r 3 r B r Br 4 r 6 5 v u n 1 u 1 2 X t µ1 (G) ≤ dn + + (dn − ) + di (di − dn ). e e er A34 A AAr r 27Lr 28 Abb 12 r A b11 L A r r r r L 10 9 7 8 Fig.1 graph G4 i=1 In general, using the labels n, n − 1..., 1, in this order, our labeling for the vertices of Gl is: where d1 ≥ d2 ≥ ... ≥ dn is the degree sequence of G. ISSN: 1109-2777 % e% %e In [11], Shu,Hong and Wenren proved a sharp upper bound as follows: 2 D Dr 15 r 30 14 !r ! r ! r r % 35 29 13 % 31 32JJ 2∆2 + 4m − 2δ(n − 1) + 2∆(δ − 1) 2 nj + l. j=1 m(µ1 ), m(µ2 ), ..., m(µs ), SpecL(G) = k−1 X 363 Issue 4, Volume 7, April 2008 WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin (1) First, we label the vertices of Kl with clockwise direction. (2) For one of vertices of level j(j = 1, 2, ..., k − 1), the bigger its labeling is , then the vertex of level j + 1 adjacent to it should be labeled first. (3) Label from level 1 to level k in turn. [12], [13] studied the spectrum of the adjacency matrix A(Gl ) and the eigenvalues of Laplacian matrix L(Gl ) for case l = 1 and l = 2 respectively. In this paper we will study the spectrum of the adjacency matrix A(Gl ) and the eigenvalues of Laplacian matrix L(Gl ) for all positive integer l. We introduce the following notations: (1) 0 is the all zeros matrix, the order of 0 will be clear from the context in which it is used. (2)Im is the identity matrix of order m × m. nj (3) mj = nj+1 , for j = 1, 2, ..., k − 1. (4) em is the all ones column vetor of dimension m. For j = 1, 2, ..., k − 1, Cj is the block diagonal matrix In general, our labeling yields to 0 CT 1 A(Gl ) = emj Cj = .. . emj C2 .. . .. . C2T .. . .. . .. . .. .. where Bl = . . Ck−1 T Ck−1 Bl , 0 1 1 ··· 1 1 0 1 ··· 1 . .. .. .. . . .. . . . . . 1 1 1 ··· 0 L(Gl ) = d1 In1 −C1T −C1 d2 In2 emj C1 0 −C2T −C2 .. . .. . .. . .. . .. . .. . −Ck−1 Ul dk−1 Ink−1 T −Ck−1 , where with nj+1 diagonal blocks. Thus, the order of Cj is nj × nj+1 . For example we use these notation with the graph G4 in Fig.1 m1 = nn12 = 3, m2 = nn23 = 2, then Ul = dk Ink − Bl = dk −1 −1 · · · −1 −1 dk −1 · · · −1 .. .. .. .. .. . . . . . . −1 −1 −1 · · · dk C1 = diag{e3 , e3 , e3 , e3 , e3 , e3 , e3 , e3 }, Apply the Gaussian elimination procedure we obtained the following lemma: Lemma 2.1 Let M= C2 = diag{e2 , e2 , e2 , e2 }, The adjacency matrix A(G4 ) in Fig.1 become 0 A(G4 ) = C1T 0 where B4 = 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 C1 0 C2T α1 In1 C1T 0 C2 , B4 and d1 I24 −C1 L(G4 ) = −C1T d2 I8 0 −C2T C1 α2 In2 C2T U4 = d3 I4 − B4 = ISSN: 1109-2777 C2 .. . .. . .. . .. . .. . .. , . αk−1 Ink−1 T Ck−1 Ck−1 αk Ink + Bl let β1 = α1 0 −C2 , U4 and βj = αj − where 5 −1 −1 −1 −1 5 −1 −1 −1 −1 5 −1 −1 −1 −1 5 nj−1 1 , j = 2, 3, ..., k, βj−1 6= 0. nj βj−1 If βj 6= 0 for all j = 1, 2, ..., k − 1, then . detM 364 n k−1 = β1n1 β2n2 ...βk−1 ×(βk + l − 1)(βk − 1)l−1 . Issue 4, Volume 7, April 2008 (2) WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin Hence, Proof. Apply the Gaussian elimination procedure, without row interchanges, to M to obtain the block upper triangular matrix β1 I n 1 n k−1 det(βk Ink − Bl ). detM0 = β1n1 β2n2 ...βk−1 C1 β2 I n 2 Since C2 β3 I n 3 .. .. . . .. . βk−1 Ink−1 Ck−1 βk Ink + Bl . det(λI − Bl ) = (λ − l + 1)(λ + 1)l−1 , so det(βk Ink − Bl ) = (βk − l + 1)(βk + 1)l−1 . Hence, Then n k−1 det(βk Ink + Bl ). detM = β1n1 β2n2 ...βk−1 n k−1 (βk − l + 1)(βk + 1)l−1 . detM0 = β1n1 β2n2 ...βk−1 Since det(λI − Bl ) = (λ − l + 1)(λ + 1)l−1 , Thus, (3) is proved. # so 3 det(βk Ink + Bl ) = (−1)l det(−βk Ink − Bl ) = (βk + l − 1)(βk − 1)l−1 . Then In this section we will apply Lemma3.1, Lemma 3.2 and Lemma 3.3 to study the spectrum of the adjacency matrix and the Laplacian matrix of Gl Lemma 3.1[14] Let H be k ×k symmetric tridiagonal matrix: n k−1 detM = β1n1 β2n2 ...βk−1 (βk + l − 1)(βk − 1)l−1 . Thus, (2) is proved. # Lemma 2.2 Let M0 = α1 In1 −C1T −C1 α2 In2 The spectrum of the adjacency matrix and the Laplacian matrix of Gl −C2 .. . .. . −C2T .. .. .. . .. . . . αk−1 Ink−1 T −Ck−1 −Ck−1 αk Ink − Bl a1 b1 b 1 a2 , H= b2 b2 .. . .. . nj−1 1 , j = 2, 3, ..., k, βj−1 6= 0. nj βj−1 detM0 = ×(βk − l + 1)(βk + 1)l−1 . (3) Proof. Apply the Gaussian elimination procedure, without row interchanges, to M0 to obtain the block upper triangular matrix β1 I n 1 −C1 β2 I n 2 −C2 β3 I n 3 ISSN: 1109-2777 .. . .. . .. . βk−1 Ink−1 −Ck−1 βk Ink − Bl .. . . ak−1 bk−1 bk−1 ak Q0 (λ) = 1, Q1 (λ) = λ − a1 , Qj (λ) = (λ − aj )Qj−1 (λ) − b2j−1 Qj−2 (λ), (j = 2, 3, ..., k). If βj 6= 0 for all j = 1, 2, ..., k − 1, then nk−1 β1n1 β2n2 ...βk−1 . and Qj (λ)(j = 0, 1, 2, ..., k) be the characteristic polynomials of the j × j leading principal submatrix of matrix H. Then and βj = αj − . .. .. let β1 = α1 .. , . 365 (4) Lemma 3.2[14] Let H and Qj (λ)(j = 0, 1, 2, ..., k) (j) be matrix as in Lemma 3.1, then all roots λi (i = 1, 2, ..., j) of Qj (j = 0, 1, 2, ..., k) are real and simple: (j) (j) (j) λ1 > λ2 > ... > λj , and the roots of Qj−1 and Qj , respectively, separate each other strictly: (j) (j−1) λ1 > λ1 (j) (j−1) > λ2 > λ2 (j−1) (j) > ... > λj−1 > λj . Issue 4, Volume 7, April 2008 WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin n1 1 n2 β1 n1 1 = λ− n2 S1 (λ) λS1 (λ) − nn21 S0 (λ) = S1 (λ) S2 (λ) = 6= 0. S1 (λ) Lemma 3.3[15] Let A, B be n × n Hermitian matrix. Assume that B is positive semidefite and that the eigenvalues of A and A+B are arranged in decreasing order as in (1). Then β2 = λ − λi (A) ≤ λi (A + B) for all i = 1, 2, ..., n. 3.1 The spectrum of the adjacency matrix of Gl Let φ = {1, 2, ..., k − 1} Similarly, for j = 3, 4, ..., k − 1, k and Ω = {j ∈ φ : nj > nj+1 }. βj Observe that nk−j = (dk−j+1 − 1)nk−j+1 , j = 2, 3, ..., k − 1 and nk−1 = (dk − l + 1)nk . Observe also that if j ∈ φ − Ω then nj = nj+1 and Cj is the identity matrix of order nj . Theorem 3.1.1 Let S0 (λ) = 1, S1 (λ) = λ, Sj (λ) = 2, 3, ..., k, nj−1 nj Sj−2 (λ), for λSj−1 (λ) − j = Thus Sk− (λ) = (λ + 1)Sk−1 (λ) − nk−1 Sk−2 (λ) l βk + 1 = and = Sk+ (λ) nk−1 = (λ − l + 1)Sk−1 (λ) − Sk−2 (λ). l = Then (i) If Sj (λ) 6= 0, for j = 1, 2, ..., k − 1, then det(λI − A(Gl )) = (Sk− (λ))l−1 Sk+ (λ) Q n −n × j∈Ω Sj j j+1 (λ). = βk − l + 1 = = Proof. Suppose Sj (λ) 6= 0 for all j = 1, 2, ..., k − 1. We apply Lemma 2.2 to M0 = λI − A(Gl ). M0 = λI − A(Gl ) = λIn1 −C T 1 = = −C1 λIn2 −C2T −C2 .. . .. . .. .. .. . . .. . . λInk−1 −Ck−1 T −Ck−1 λInk − Bl . Sk (λ) −l+1 Sk−1 (λ) Sk (λ) − (l − 1)Sk−1 (λ) Sk−1 (λ) n (λ − l + 1)Sk−1 (λ) − k−1 l Sk−2 (λ) Sk−1 (λ) + Sk (λ) . Sk−1 (λ) Therefore, from Lemma 2.2, det(λI − A(Gl )) n k−1 S2n2 (λ) Sk−1 (λ) Sk+ (λ) (Sk− (λ))l−1 ... n k−1 l−1 S1n2 (λ) Sk−2 (λ) Sk−1 (λ) Sk−1 (λ) = S1n1 (λ) = k−1 S1n1 −n2 (λ)S2n2 −n3 (λ)...Sk−1 n −nk (λ) ×Sk+ (λ)(Sk− (λ))l−1 We have = β1 = λ = S1 (λ) 6= 0, ISSN: 1109-2777 Sk (λ) +1 Sk−1 (λ) Sk (λ) + Sk−1 (λ) Sk−1 (λ) n (λ + 1)Sk−1 (λ) − k−1 l Sk−2 (λ) Sk−1 (λ) − Sk (λ) , Sk−1 (λ) (5) (ii)The spectrum of A(Gl ) is σ(A(Gl )) = (∪j∈Ω {λ : Sj (λ) = 0}) ∪ {λ : Sk− (λ) = 0} ∪ {λ : Sk+ (λ) = 0}. nj−1 1 nj βj−1 nj−1 Sj−2 (λ) = λ− nj Sj−1 (λ) n λSj−1 (λ) − nj−1 Sj−2 (λ) j = Sj−1 (λ) Sj (λ) 6= 0. = Sj−1 (λ) = λ− (Sk− (λ))l−1 Sk+ (λ) Y n −nj+1 Sj j (λ). j∈Ω 366 Issue 4, Volume 7, April 2008 WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin Thus (i) is proved. Similar to the proof in [13], we can get (ii) by (i) . # Let Rk+ and Rk− be the k × k symmetric tridiagonal matrices Rk+ = √ 0 d2 − 1 √ d2 − 1 0 √ p d3 − 1 dk−1 − 1 √ d3 − 1 .. . 0 √ dk − l + 1 simply. Finally, we use (5) and Theorem 3.1.2 to obtain(ii). # .. . √ dk − l + 1 l−1 and Rk− = √ 0 d2 − 1 √ d2 − 1 0 √ p d3 − 1 dk−1 − 1 √ d3 − 1 .. . 0 √ dk − l + 1 .. . √ dk − l + 1 −1 . Observe that det(λI − Rk− ) = Sk− (λ), det(λI − Rk+ ) = Sk+ (λ). Proof. We apply Lemma 3.1, in our case, a1 = a2 = ... = ak−1 = 0, ak = l − 1(or ak = −1) and nk−1 p = dk − l + 1, nk bj = q nj = dj+1 − 1 nj+1 for j = 1, 2, 3, ..., k − 2. For these values, the recursion formula (4) gives the polynomials Sj (λ), j = 0, 1, 2, ..., k − 1, Sk+ (λ) and Sk− (λ). This completes the proof. # Theorem 3.1.3 Let Rj , j = 1, 2, ..., k−1, Rk+ and Rk− as above. then (i)σ(A(Gl )) = (∪j∈Ω σ(Rj )) ∪ σ(Rk+ ) ∪ σ(Rk− ). (ii) The multiplicity of each eigenvalue of the matrix Rj , as an eigenvalue of A(Gl ), is at least nj −nj+1 for j ∈ Ω, 1 for the eigenvalues of Rk+ and l − 1 for the eigenvalues of Rk− . Proof. (i) is an immediate consequence of Theorem 3.1.1 and Theorem 3.1.2. From Lemma 3.2 that the eigenvalues of Rj , j = 1, 2, ..., k − 1, Rk+ and Rk− are ISSN: 1109-2777 22 21 20 r r r 24 r A JJ A 19 r 26 25r rHH r ! A! r r 38 J H 17 37 27 LL @ r r @ 18 16 r Z ZL r r @ L ZLr 39 44 L r 36L L L L Lr 43 r45 r 49 T @ T @ r T @ 28 r 15 T @ T @ r 35 Tr @r # 48 50AZ # # r A ZZ r r 14 # Z 40 A # 13 A Z# A # ZZ Zr 47 46 Ar# r A r 30 A 29 A A A 42 Ar r 41 12 r @ ,B 31 ,, B @ @ "r B " D @r 34 B " r" D 32 Br r 33 1 e D A r D r ee r r r A r r 2 r Ar 4 r 3 7 10 6 det(λI − Rj ) = Sj (λ), j = 1, 2, ..., k − 1, s we can get the eigenvalues of Rk+ are greater or equal to the eigenvalues of Rk− . Now (a3 ) follow from this fact and Lemma3.2. # 23 Theorem 3.1.2 For j = 1, 2, 3, ..., k − 1, let Rj be the j × j leading principal submatrix Rk+ . Then bk−1 = Rk+ = Rk− + diag{0, 0, ..., 0, l}, Example 3.1.1 Rk+ = Rk− + diag{0, 0, ..., 0, l}. r Theorem 3.1.4 Let A(Gl ) be the adjacency matrix of Gl . Then (a1 ) σ(Rj−1 ) ∩ σ(Rj ) = φ for j = 2, 3, ..., k − 1. (a2 ) σ(Rk−1 ) ∩ σ(Rk+ ) = φ and σ(Rk−1 ) ∩ σ(Rk− ) = φ. (a3 ) The largest eigenvalue of Rk+ is the largest eigenvalue of A(Gl ) and the largest eigenvalue of Rk− is the second largest eigenvalue of A(Gl ). Proof. (a1 ) and (a2 ) follow from Lemma 3.2. By Lemma 3.3 and 5 8 Fig.2 graph G5 367 9 Issue 4, Volume 7, April 2008 11 WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin For the graph G5 in Fig.2 for l = 5, k = 4, n1 = 30, n2 = 10, n3 = n4 = 5, d1 = 1, d2 = 4, d3 = 3, d4 = 5. R1 = 0, R2 = √0 3 and Pk− (µ) = (µ−(dk −l +1))Pk−1 (µ)− √ ! 3 0 nk−1 Pk−2 (µ). l Then (i) If Pj (µ) 6= 0, for all j = 1, 2, ..., k − 1, then √ 0 3 √ √ R3 = 3 √0 2 0 2 0 √ 0 3 √0 0 √ 2 0 3 √0 R4+ = 0 2 0 1 0 0 1 4 √ 0 3 0 0 √ √ 2 0 3 √0 − R4 = 0 2 0 1 0 0 1 −1 det(µI − L(Gl )) = (Pk+ (µ))l−1 Pk− (µ) Q n −n × j∈Ω Pj j j+1 (µ). (ii)The spectrum of L(Gl ) is σ(L(Gl )) = (∪j∈Ω {µ : Pj (µ) = 0}) ∪ {µ : Pk− (µ) = 0} ∪ {µ : Pk+ (µ) = 0}. Proof. Suppose Pj (µ) 6= 0 for all j = 1, 2, ..., k − 1. We apply Lemma 2.1 to M = µI − L(Gl ), we denote µ − dj = xj , (j = 1, 2, ..., k), then M = µI − L(Gl ) = x1 In1 C1T and Ω = {1, 2}. By Theorem 3.1.3, the eigenvalues of A(G5 ) in Fig.2 are the eigenvalues of R1 , R2 , R4+ and R4− , they are R1 : 0 R2 : −1.7320 1.7320 R4− : −2.4142 −1.3028 0.4142 2.3028 R4+ : −2.2696 −0.1444 2.1444 4.2696 C1 x2 In2 C2T C2 .. . .. . .. . .. . .. . .. . xk−1 Ink−1 T Ck−1 Ck−1 xk Ink + Bl We have β1 = µ − d1 = µ − 1 = P1 (µ) 6= 0, The spectral radius of G5 in Fig.2 is λ1 (A(G5 )) = 4.2696. SpecA(G5 ) = 4.2696 2.3028 2.1444 1.7320 1 4 1 5 n1 1 n2 β1 n1 1 = µ − d2 − n2 P1 (µ) (µ − d2 )P1 (µ) − nn12 P0 (µ) = P1 (µ) P2 (µ) = 6= 0. P1 (µ) β2 = (µ − d2 ) − 0.4142 0 −0.1444 −1.3028 4 20 1 4 −1.7320 −2.2696 −2.4142 5 1 4 ! . Similarly, for j = 3, 4, ..., k − 1, k 3.2 The spectrum of the Laplacian matrix of Gl βj Theorem 3.2.1 Let P0 (µ) = 1, P1 (µ) = µ − 1, Pj (µ) = (µ − dj )Pj−1 (µ) − 2, 3, ..., k, nj−1 nj Pj−2 (µ), for Pk+ (µ) = (µ − (dk + 1))Pk−1 (µ) − ISSN: 1109-2777 (6) j= nk−1 Pk−2 (µ) l 368 nj−1 1 nj βj−1 nj−1 Pj−2 (µ) = µ − dj − nj Pj−1 (µ) n (µ − dj )Pj−1 (µ) − nj−1 Pj−2 (µ) j = Pj−1 (µ) Pj (µ) = 6= 0. Pj−1 (µ) = (µ − dj ) − Issue 4, Volume 7, April 2008 . WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin Theorem 3.2.2 For j = 1, 2, 3, ..., k − 1, let Wj be the j × j leading principal submatrix Wk+ . Then Thus Pk (µ) −1 Pk−1 (µ) Pk (µ) − Pk−1 (µ) Pk−1 (µ) (µ − dk − 1)Pk−1 (µ) − Pk−1 (µ) + Pk (µ) , Pk−1 (µ) βk − 1 = = = = βk + l − 1 = = det(µI − Wj ) = Pj (µ), j = 1, 2, ..., k − 1, det(µI − Wk− ) = Pk− (µ), det(µI − Wk+ ) = Pk+ (µ). nk−1 l Pk−2 (µ) Proof. Similar to the proof of Theorem 3.1.2, in this case, a1 = 1, aj = dj forj = 2, 3, ..., k − 1, ak = dk + 1 (or ak = dk − l + 1) and r bk−1 = Pk (µ) +l−1 Pk−1 (µ) Pk (µ) + (l − 1)Pk−1 (µ) Pk−1 (µ) s = (µ − dk + l − 1)Pk−1 (µ) − Pk−1 (µ) = Pk− (µ) . Pk−1 (µ) nk−1 Pk−2 (µ) l det(µI − L(Gl )) n k−1 P2n2 (µ) Pk−1 (µ) Pk− (µ) (Pk+ (µ))l−1 ... nk−1 n2 l−1 P1 (µ) Pk−2 (µ) Pk−1 (µ) Pk−1 (µ) P1n1 (µ) = k−1 P1n1 −n2 (µ)P2n2 −n3 (µ)...Pk−1 n −nk (µ) ×Pk− (µ)(Pk+ (µ))l−1 = (Pk+ (µ))l−1 Pk− (µ) Y nj −nj+1 Pj (µ). j∈Ω Thus (i) is proved. Similar to the proof Theorem 3.1.1, we can get (ii) by (i) . # Let Wk+ and Wk− be the k × k symmetric tridiagonal matrices Wk+ = √ 1 d2 − 1 √ d2 − 1 d2 √ p d3 − 1 dk−1 − 1 √ d3 − 1 .. . d k−1 √ dk − l + 1 .. . √ dk − l + 1 dk + 1 and Wk− = √ 1 d2 − 1 √ d2 − 1 d2 √ p d3 − 1 dk−1 − 1 √ d3 − 1 .. . √ dk−1 dk − l + 1 .. . √ dk − l + 1 dk − l + 1 . = Wk− ISSN: 1109-2777 q nj = dj+1 − 1 nj+1 for j = 1, 2, 3, ..., k − 2. Theorem 3.2.4 Let L(Gl ) be the Laplacian matrix of Gl . Then (b1 ) σ(Wj−1 ) ∩ σ(Wj ) = φ for j = 2, 3, ..., k − 1. (b2 ) σ(Wk−1 ) ∩ σ(Wk+ ) = φ and σ(Wk−1 ) ∩ σ(Wk− ) = φ. (b3 ) det Wj = 1 for j = 1, 2, ..., k−1, det Wk− = 0 and det Wk+ = l. (b4 ) The largest eigenvalue of Wk+ is the largest eigenvalue of L(Gl ) . (b5 ) The smallest eigenvalue of Wk+ is the algebraic connectivity Gl . Proof. (b1 ) and (b2 ) follow from Lemma 3.2. Now we apply the Gaussian elimination procedure, without row interchanges, to reduce Wj to the upper triangular matrix Wj ⇒ Observe that Wk+ bj = For these values, recursion formula (4) gives the polynomials Pj (µ), j = 0, 1, 2, ..., k − 1, Pk+ (µ) and Pk− (µ). This completes the proof. # Similar to the proof of Theorem 3.1.3, we can get: Theorem 3.2.3 Let Wj , j = 1, 2, ..., k − 1, Wk+ and Wk− as above. then (i)σ(L(Gl )) = (∪j∈Ω σ(Wj ))∪σ(Wk+ )∪σ(Wk− ). (ii) The multiplicity of each eigenvalue of the matrix Wj , as an eigenvalue of L(Gl ), is at least nj −nj+1 for j ∈ Ω, 1 for the eigenvalues of Wk− and l − 1 for the eigenvalues of Wk+ . Therefore, from Lemma 2.1, = nk−1 p = dk − l + 1, nk + diag{0, 0, ..., 0, l}. 369 1 √ d2 − 1 1 √ d3 − 1 1 .. . p dj−1 − 1 1 p dj − 1 1 Issue 4, Volume 7, April 2008 . WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin The same procedure applied Wk+ and Wk− gives the triangular matrices Wk+ ⇒ 1 √ d2 − 1 1 √ d3 − 1 .. . The spectral radius of G5 in Fig.2 is µ1 (L(G5 )) = 6.4244. SpecL(G5 ) = 6.4244 5.5287 5.2586 4.7912 4 1 4 5 . p dk−1 − 1 √ 1 dk − l + 1 .. 2.8327 2.2511 1 0.6385 1 4 20 1 l 0.2088 0.0658 0 5 4 1 and Wk− ⇒ 1 √ d2 − 1 1 √ d3 − 1 .. . . 4 . . p dk−1 − 1 √ 1 dk − l + 1 .. 0 respectively. Thus (b3 ) is proved and 0 is the smallest eigenvalue of Wk− . Since An upper bound for the largest eigenvalue of the adjacency matrix A(Gl ) and the Laplacian matrix L(Gl ) In this section we will give an upper bound for the largest eigenvalue of the adjacency matrix and the Laplaican matrix of graph Gl . Lemma 4.1[3] Let B = (bij ) be a nonnegative n × n matrix and λ is the largest eigenvalue of matrix B. Denote the ith row sum of B by si (B). Then Wk+ = Wk− + diag{0, 0, ..., 0, l}, by Lemma 3.3, the eigenvalues of Wk+ are greater or equal to the eigenvalues of Wk− . Now (b4 ) and (b5 ) follow from this fact and Lemma3.2. # min si (B) ≤ λ ≤ max si (B), 1≤i≤n Example 3.2.1 For the graph G5 in Fig.2 √ ! 1 3 W1 = 1, W2 = √ 3 4 √ 1 3 √0 √ W3 = 3 √4 2 0 2 3 √ 1 3 √0 0 √ 2 0 3 √4 W4+ = 0 2 3 1 0 0 1 6 √ 0 3 √0 0 √ 2 0 3 √0 W4− = 0 2 0 1 0 0 1 1 1≤i≤n the left equality holds if and only if the right equality also holds. 4.1 An upper bound for the largest eigenvalue of the adjacency matrix A(Gl ) Theorem 4.1.1 Let Gl (l is a positive integer) be the graph as above and that Gl has k levels, dk−j+1 be the degree of vertices in the level j, then λ1 (A(Gl )) < max n p p max2≤j≤k−2 { dj − 1 + dj+1 − 1}, o p √ √ dk−1 − 1 + dk − l + 1, dk − l + 1 + l − 1 . Proof. Let Rk+ = √ 0 d2 − 1 and Ω = {1, 2}. By Theorem 3.2.2, the eigenvalues of L(G5 ) in Fig.2 are the eigenvalues of W1 , W2 , W4+ and W4− , they are √ d2 − 1 0 √ p d3 − 1 dk−1 − 1 √ d3 − 1 .. . 0 √ dk − l + 1 .. . √ dk − l + 1 l−1 , we now apply Lemma 4.1 to conclude W1 : 1 W2 : 0.2088 4.7912 W4− : 0 0.6385 2.8327 5.5287 + W4 : 0.0658 2.2511 5.2586 6.4244 ISSN: 1109-2777 ! + λ n1 (Rk ) < max p p max2≤j≤k−2 { dj − 1 + dj+1 − 1}, o p √ √ dk−1 − 1 + dk − l + 1, dk − l + 1 + l − 1 . 370 Issue 4, Volume 7, April 2008 WSEAS TRANSACTIONS on SYSTEMS Shuhua Yin From Theorem 3.3.4, we easily have λ n1 (A(Gl )) < max p p max2≤j≤k−2 { dj − 1 + dj+1 − 1}, o p √ √ dk−1 − 1 + dk − l + 1, dk − l + 1 + l − 1 .# 4.2 An upper bound for the largest eigenvalue of the Laplacian matrix L(Gl ) In this section we give an upper bound for the largest eigenvalue of the Laplacian matrix L(Gl ). Theorem 4.2.1 Let Gl (l is a positive integer) be the graph as above and that Gl has k levels, dk−j+1 be the degree of vertices in the level j, then λ n1 (A(Gl )) < max p p max2≤j≤k−2 { dj − 1 + dj+1 − 1}, o p √ √ dk−1 − 1 + dk − l + 1, dk − l + 1 + l − 1 . µ n1 (L(Gl )) < max p p max2≤j≤k−2 { dj − 1 + dj + dj+1 − 1}, p √ dk−1 − 1 + dk−1 + o dk − l + 1, √ dk − l + 1 + dk + 1 . Proof. Let Wk+ = √ 1 d2 − 1 √ d2 − 1 d2 √ p d3 − 1 dk−1 − 1 √ d3 − 1 .. . d k−1 √ dk − l + 1 .. . √ dk − l + 1 dk + 1 we now apply Lemma 4.1 to conclude µ1 (Wk+ ) < max n p for j ∈ Ω, 1 for the eigenvalues of Rk+ and l − 1 for the eigenvalues of Rk− . (I3 )The largest eigenvalue of Rk+ is the largest eigenvalue of A(Gl ) and the largest eigenvalue of Rk− is the second largest eigenvalue of A(Gl ). It is very convenient with conclusions (I1 )(I2 )(I3 ) to calculate the spectrum of the adjacency matrix A(Gl ). In section 4.1, according to the results(I1 )(I2 )(I3 ) and Lemma 4.1, an upper bound for the largest eigenvalue of the adjacency matrix A(Gl ) is obtained: p max2≤j≤k−2 { dj − 1 + dj + dj+1 − 1}, p √ dk−1 − 1 + dk−1 + o dk − l + 1, √ dk − l + 1 + dk + 1 . (II). Let Wj , j = 1, 2, ..., k − 1, Wk+ and Wk− as in section 3.2. We found that: (II1 ) σ(L(Gl )) = (∪j∈Ω σ(Wj )) ∪ σ(Wk+ ) ∪ σ(Wk− ). (II2 ) The multiplicity of each eigenvalue of the matrix Wj , as an eigenvalue of L(Gl ), is at least nj − nj+1 for j ∈ Ω, 1 for the eigenvalues of Wk− and l − 1 for the eigenvalues of Wk+ . (II3 ) The largest eigenvalue of Wk+ is the largest eigenvalue of L(Gl ). (II4 ) The smallest eigenvalue of Wk+ is the algebraic connectivity Gl . In section 4.2, according to the results(II1 )(II2 )(II3 ) and Lemma 4.1, an upper bound for the largest eigenvalue of the Laplacian matrix L(Gl ) is obtained: µ1 (L(Gl )) < max n p From Theorem 3.2.4, we easily have µ n1 (L(Gl )) < max p p max2≤j≤k−2 { dj − 1 + dj + dj+1 − 1}, p √ dk−1 − 1 + dk−1 + o dk − l + 1, √ dk − l + 1 + dk + 1 .# 5 References: [1] Dragan Stevanocic‘, Bounding the langest eigenvalue of trees in terms of the largest vertex degree, Linear Algebra and its Applications 360, 2003, pp. 35–42. [2] AiMei Yu, Mei Lu and Feng Tian, On the spectral radius of graphs, Linear Algebra and its Applications 387, 2004, pp. 41–49. [3] M.N.Ellingham, X.Zha, The spectral radius of graph on surfaces, Journal of Combinatorial Theory, Ser B 78, 2000, pp. 45–46. [4] ShuGuang Guo, On the spectral radius of bicyclic graphs with n vertices and diameter d, Linear Algebra and its Applications 422, 2007, pp. 119–132. Conclusion We studied the spectrum of the adjacency matrix A(Gl ) and the spectrum of the Laplacian matrix L(Gl ) for all positive integer l with an effective way. (I). Let Rj , j = 1, 2, ..., k − 1, Rk+ and Rk− as in section 3. We found that: (I1 )σ(A(Gl )) = (∪j∈Ω σ(Rj )) ∪ σ(Rk+ ) ∪ σ(Rk− ). (I2 ) The multiplicity of each eigenvalue of the matrix Rj , as an eigenvalue of A(Gl ), is at least nj −nj+1 ISSN: 1109-2777 p max2≤j≤k−2 { dj − 1 + dj + dj+1 − 1}, p √ dk−1 − 1 + dk−1 + o dk − l + 1, √ dk − l + 1 + dk + 1 . 371 Issue 4, Volume 7, April 2008 WSEAS TRANSACTIONS on SYSTEMS [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] Shuhua Yin J.M. Guo, J.Y. Shao, On the spectral radius of trees with fixed diameter, Linear Algebra and its Applications 413(1), 2006, pp. 131–147. Vladimir Nikiforov, Boundson graph eigenvalues I, Linear Algebra and its Applications 420, 2007, pp. 667–671. X.D.Zhang, Eigenvectors and eigenvalues ofnon-regular graphs, LinearAlgebra andits Applications 409, 2005, pp. 79–86. R.Merris, Laplacian matrices of graphs: a survey, Linear Algebra and its Applications, 197198, 1994, pp. 143–176. M.Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. , 23, 1973, pp. 298–305. J.S.Li, Y.L.Pan, de Cane’s inequality and bounds on the largest Laplacian eigenvalue of a graph, Linear Algebra and its Applications, 328, 2001, pp. 153–160. J.L.Shu, Y.Hong, K.WenRen, A sharp bound on the largest eigenvalue of the Laplacian matrix of a graph, Linear Algebra and its Applications, 347, 2002, pp. 123–129. Oscar Rojo, Ricardo Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra and its Applications 403, 2005, pp. 97–117. Oscar Rojo,The spectra of some trees and bounds for the lagest eigenvalue of any tree, Linear Algebra and its Applications 414, 2006, pp. 199–217. J.Stoer, R.Bulirsich, Introduction to Numerical Analysis (Second Edition), Springer-Verlag 2004. R.A.Horn, C.R.Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1991. ISSN: 1109-2777 372 Issue 4, Volume 7, April 2008