Linear Algebra and its Applications

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1 Linear Algebra and its Applications 430 (2009) Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa Computing tight upper bounds on the algebraic connectivity of certain graphs Oscar Rojo, Department of Mathematics, Universidad Católica del Norte, Casilla 280, Antofagasta, Chile A R T I C L E I N F O A B S T R A C T Article history: Received 22 July 2007 Accepted 22 August 2008 Available online 30 September 2008 Submitted by RA Brualdi AMS classification: 5C50 5A48 05C05 Keywords: Laplacian matrix Algebraic connectivity Bethe trees Generalized Bethe trees A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree Let B be a generalized Bethe tree The algebraic connectivity of: the generalized Bethe tree B, a tree obtained from the union of B andatreet isomorphic to a subtree of B such that the root vertex of T is the root vertex of B, a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices, a graph obtained from the cycle C r by attaching B,byitsroot,to each vertex of the cycle, and a tree obtained from the path P r by attaching B,byitsroot,toeach vertex of the path, is the smallest eigenvalue of a special type of symmetric tridiagonal matrices In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs 2008 Elsevier Inc All rights reserved Introduction Let G be a simple undirected graph on n vertices The Laplacian matrix of G is the n n matrix L(G) = D(G) A(G) where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of vertex degrees L(G) is a real symmetric matrix From this fact and Geršgorin s theorem, it follows Work supported by Project Fondecyt , Chile Fax: address: orojo@ucncl Part of this research was conducted while the author was a visitor at the Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile /$ - see front matter 2008 Elsevier Inc All rights reserved doi:006/jlaa

2 O Rojo / Linear Algebra and its Applications 430 (2009) that its eigenvalues are nonnegative real numbers Moreover, since its rows sum to 0, 0 is the smallest eigenvalue of L(G) Let 0 = λ (G) λ 2 (G) λ n (G) be the Laplacian eigenvalues of G Fiedler [] proved that G is a connected graph if and only if 0 is a simple eigenvalue of L(G) That is, if and only if λ 2 (G) >0 This eigenvalue is called the algebraic connectivity of G and denoted by a(g) According to Mohar [2], a(g) is probably the most important information contained in the spectrum of a graph In [3], a survey on a(g) is given, in which old and new results are presented, some applications are described and a complete set of references is included In a tree, any vertex can be choosen as the root vertex The level of a vertex on a tree is one more than its distance from the root vertex A Bethe tree B k (d) [4] is a rooted unweighted tree of k levels in which the root has degree d, the vertices at the level j (2 j k ) have degree (d + ) and the vertices at the level k (the pendant vertices) have degree If d = 2 then B k (2) is a balanced binary tree of k levels In [5], Molitierno et al obtain quite tight upper and lower bounds on the algebraic connectivity of B k (2) These bounds are a(b k (2)) (2 k 2k + 3) 2k 2 2 k and (2 k 2k + 2) 2k 2(2k 2 k ) 2 k + 2(2 k ) 3 2 ( ) 2 cos π 2k a(b k (2)) In [6] we obtain quite tight upper and lower bounds on a(b k (d)) for d > 2 They are (d ) a(b k (d)) 2 d k (2k 2)d + (2k ) (2k 2)(d ) d k and ( d k (2k )(d ) ) a(b k (d)) d + (d )(2k ) + (d ) 2 d k 2 + (d+) 2 π d cos 2k A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree Throughout this paper B k (d) denotes a generalized Bethe tree of k levels (k > ) where d = (, d 2, d 3,, d k, d k ) is the vertex degree vector in which d k j+ is the degree of the vertices at the level j ( j k) Thus, d k is degree of the root and d = is the degree of the pendant vertices For j =, 2, 3,, k, the numbers n k j+ denote the number of vertices at the level j Then, n k = and n is the number of pendant vertices Throughout this paper, Ω is the following set of indices: Ω ={j : j k and n j > n j+ } We introduce the following notation As usual, let P r and C r be the path and the cycle on r vertices, respectively Let v(b k (d), T) be the tree obtained from the union of B k (d) andatreetisomorphic to a subtree of B k (d) such that the root vertex of T is the root vertex of B k (d) Let v(b, B 2,, B r ) be the tree obtained from the union of the generalized Bethe trees B k (d ), B k2 (d 2 ),, B kr (d r ) joined at their respective root vertices Let G {G 2 } be the graph obtained from the graph G by attaching the same rooted graph G 2,byits root, to each vertex of G The algebraic connectivity of :ageneralized Bethe tree B k (d), atreev(b k (d), T), atreev(b, B 2,, B r ),agraphc r {B k (d)}, andatreep r {B k (d)}, is the smallest eigenvalue of a special type

3 534 O Rojo / Linear Algebra and its Applications 430 (2009) of symmetric tridiagonal matrices In this paper, we first derive a numerical procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices Finally we apply this procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs Denote by λ (A) λ 2 (A) λ m (A) the eigenvalues of an m m matrix A with only real eigenvalues and by σ(a) the set of eigenvalues of a matrix A 2 The basic procedure For the graphs mentioned above, as we will see later, the algebraic connectivity is the smallest eigenvalue of a symmetric tridiagonal matrix having the form a2 a2 a 2 a3 A(a, m, α) = a3 a 3 () am am a m + α of order m m, with a j > forj = 2, 3,, m and α> Therefore we search for a procedure to compute a tight upper bound on λ (A(a, m, α)) Lemma (a) Let A j denote the j j leading principal submatrix of A(a, m, α) Then det A j = for j =, 2,, m and det A(a, m, α) = + α (b) If α>, then A(a, m, α) is a positive definite matrix Proof (a) Clearly det A = Let 2 j m By application of the Gaussian elimination procedure, without row interchanges, we can obtain det A j = and deta(a, m, α) = + α (b) It follows from (a) and from Sylvester s Theorem which states that a real symmetric m m matrix A is a positive definite matrix if and only if det A j > 0, j =, 2,, m, where A j is the j j leading principal submatrix of A Let a 2 a3 a3 a S(a, m, α) = 3 am am a m + α We recall the following lemmas Lemma 2 [7] If A is an m m symmetric tridiagonal matrix with nonzero codiagonal entries then the eigenvalues of any (m ) (m ) principal submatrix strictly interlace the eigenvalues of A Lemma 3 [6] LetAbeanm m matrix with only positive eigenvalues and B be an (m ) (m ) matrix whose eigenvalues interlace the eigenvalues of A Then λ (A) trace(a ) trace(b ) Corollary If A = A(a, m, α) and S = S(a, m, α) then λ (A) trace(a ) trace(s ) (2)

4 O Rojo / Linear Algebra and its Applications 430 (2009) Proof From Lemma, A is a positive definite matrix From Lemma 2, the eigenvalues of S strictly interlace the eigenvalues of A Finally, we apply Lemma 3 to obtain (2) For brevity, let A = A(a, m, α) and S = S(a, m, α) Thus, in order to get an upper bound for the smallest eigenvalue of A, we need to compute the difference trace(a ) trace(s ) We recall the notion of the adjoint of a matrix B = (b i,j ) The cofactor of an entry b i,j is the number B i,j = ( ) i+j D i,j, where D i,j is the determinant of the (n ) (n ) matrix obtained from B by omitting the ith row and jth column of B The adjoint matrix of B is defined to be the n n matrix whose (i, j) entry is B j,i It is well known that if B is an invertible, then B = det B adjb Therefore trace(b ) = det B trace(adjb) For j = 2,, m, let a j a j+ a j+ a j+ τ j = det am am a m am am a m + α Lemma 4 (a) The terms τ j can be computed in the order τ m, τ m 2,, τ 2 from τ m+ =, τ m = a m + α and, for j =, 3,, m 2, (b) τ m j = a m j τ m j+ ( a m j+ ) τ m j+2 (3) trace(a ) = + + α m τ j (4) j=2 Proof (a) Let j m 2 We have a m j a m j+ a m j+ a m j+ τ m j = det am am a m am am a m + α We expand about the first row of τ m j Clearly the cofactor for a m j is τ m j+ The cofactor for the entry a m j+ is a m j+ a m j+2 0 a det m j+2 am 0 am a m + α = a m j+ τ m j+2

5 536 O Rojo / Linear Algebra and its Applications 430 (2009) Therefore τ m j = a m j τ m j+ (a m j+ )τ m j+2 (b) Since det A = + α, it follows that trace(a ) = trace(adja) (5) + α The diagonal entries of adja are (adja), = τ 2, (adja) 2,2 = τ 3, (adja) 3,3 = (det A 2 )τ 4 = τ 4, (adja) 4,4 = (det A 3 )τ 5 = τ 5, (adja) m,m = (det A m 2 )τ m = τ m, (adja) m,m = det A m = Hence m trace(adja) = + τ j j=2 From (5) and (6) we obtain (4) (6) It remains to compute trace(s ) We define σ =, σ 2 = a 2, a 2 a3 a3 a 3 σ j = det a j a j a j for j = 3, 4,, m, and σ m = det S Lemma 5 (a) The terms σ j can be computed in the order σ 3, σ 4,, σ m from σ =, σ 2 = a 2, σ j = a j σ j (a j )σ j 2 for j = 3, 4,, m, and (b) σ m = (a m + α)σ m (a m )σ m 2 trace(s ) = m σ σ j τ j+2 m j= (7)

6 O Rojo / Linear Algebra and its Applications 430 (2009) Proof (a) Expanding about the last row of σ j one can obtain the result in a way that is similar to that in the proof of (3) (b) Since σ m = det S, it follows that trace(s ) = σ m trace(adj S) The diagonal entries of adj S are (adjs), = τ 3 = σ τ 3, (adjs) 2,2 = σ 2 τ 4, (adjs) 3,3 = σ 3 τ 5, (adjs) 4,4 = σ 4 τ 6, (adjs) m,m = σ m = σ m τ m+ (8) Hence m trace(adjs) = σ j τ j+2 j= Finally (8) and (9) imply (7) (9) We have derived the following procedure Algorithm For computing an upper bound on the smallest eigenvalue of the matrix A(a, m, α)defined in () For j =, 2, 3,, m 2, compute τ m j = a m j τ m j+ (a m j+ )τ m j+2 with τ m+ =, τ m = a m + α ( 2 Compute trace(a ) = + ) m +α j=2 τ j 3 For j = 3, 4,, m, compute σ j = a j σ j (a j )σ j 2 with σ =, σ 2 = a 2, and σ m = (a m + α)σ m (a m )σ m 2 4 Compute trace(s ) = σ m m j= σ jτ j+2 5 Compute the upper bound trace(a ) trace(s ) 3 Computing upper bounds on the algebraic connectivity In this section, based on our previous papers, we first describe the eigenvalues of the graphs mentioned in the Introduction The corresponding algebraic connectivity is the smallest eigenvalue of a symmetric tridiagonal matrix A(a, m, α) of the form specified in () Finally, we apply the Algorithm to A(a, m, α) to compute a tight upper bound on its smallest eigenvalue 3 The tree B k (d) Consider the tree B k (d) For j =, 2,, k, let M j be the j j leading principal submatrix of the k k symmetric tridiagonal matrix

7 538 O Rojo / Linear Algebra and its Applications 430 (2009) d2 d2 d 2 d3 M = d3 d 3 dk dk d k dk dk d k From Lemma, it follows that M j is a positive definite matrix and det M j =, for j =,, k Moreover, Lemma with a m = d k + and α = yields det M = 0 In [8], we prove that : Theorem The set of eigenvalues of L(B k (d)) is σ(l(b k (d))) = j Ω σ(m j ) σ(m) Theorem 2 If d k > then smallest eigenvalue of the (k ) (k ) symmetric tridiagonal matrix d2 d2 d 2 d3 M k = d3 d 3 dk dk d k is the algebraic connectivity of the generalized Bethe tree B k (d) Hence, if d k >, an upper bound on a(b k (d)) can be obtained by applying the Algorithm to A(a, m, α) = M k with m = k, a j = d j where 2 j k, and α = 0 Proof Suppose d k > Then n k = d k > = n k Hence k Ω and thus, from Theorem, each eigenvalue of M k is a Laplacian eigenvalue Moreover, from Theorem, each eigenvalue of the M is also a Laplacian eigenvalue From Lemma 2, it follows that the eigenvalues of M j strictly interlace the eigenvalues of M j+ ( j k 2) and the eigenvalues of M k strictly interlace the eigenvalues of M Finally, since 0 is the smallest eigenvalue of M, we conclude that the smallest eigenvalue of M k is the algebraic connectivity of B k (d) Example Consider the tree B 6 (d) with d = (, 3, 4, 3, 3, 5) Since d 6 = 5 >, we may apply Algorithm to compute an upper bound on a(b 6 (d)) We have m = 6 = 5, a 2 = d 2 = 3, a 3 = d 3 = 4, a 4 = d 4 = 3, a 5 = d 5 = 3 and α = 0 To 6 decimal places, a(b 6 (d)) = and the upper bound obtained by the Algorithm is Corollary 2 If d k > then the generalized Bethe treesb k (, d 2, d 3,, d k, d k ) andb k (, d 2, d 3,, d k,2) have the same algebraic connectivity Proof It is an immediate consequence of Theorem 2 32 The tree v(b(d), T) We recall the following result Lemma 6 [9, Corollary 42] Let v be a pendant vertex of the graph G Let G be the graph obtained from G by removing v and its edge Then the eigenvalues of L(G) interlace the eigenvalues of L( G)

8 O Rojo / Linear Algebra and its Applications 430 (2009) From Lemma 6, it follows the following corollary Corollary 3 The algebraic connectivity of a graph does not increase if a pendant vertex and its edge are added to the graph Corollary 4 Let T be a subtree of the tree T Then a( T) a(t) Proof Since T is a subtree of a tree T, we can construct T from T by successively adding in pendants vertices and edges Applying Corollary 3, we obtain the result We are ready to find the algebraic connectivity of v(b(d), T) Theorem 3 If d k > then a(v(b k (d), T)) = a(b k (d)) (0) Hence, if d k >, an upper bound on a(v(b k (d), T)) can be obtained by applying the Algorithm with m = k, a j = d j (2 j k ) and α = 0 Proof We choose as the root vertex for v(b k (d), T), the common vertex root of B k (d) and T Let d k be the degree of the root vertex of v(b k (d), T) Clearly d k > d k Let d = (, d2, d 3,, d k, d k ) Since B k (d) is a subtree of v(b k (d), T) and v(b k (d), T) is a subtree of B k ( d), it follows from Corollary 4 that a(b k ( d)) a(v(b k (d), T)) a(b k (d)) () We now use Corollary 2 to see that a(b k ( d)) = a(b k (, d 2, d 3,, d k,2)) = a(b k (d)) (2) Thus (2) combined with () gives (0) Example 2 Consider the tree This tree is obtained from the union of B 4 (, 4, 3, 2) and the tree T isomorphic to a subtree of B 4 (, 4, 3, 2) such that the root vertex of T is the root vertex of B 4 (, 4, 3, 2) Hence the algebraic connectivity of v(b 4 (, 4, 3, 2), T) is the smallest eigenvalue of

9 540 O Rojo / Linear Algebra and its Applications 430 (2009) To four decimal places the algebraic connectivity is The upper bound obtained by the Algorithm is The tree v(b, B 2,, B r ) For i =, 2,, r and j =, 2, 3,, k i,letd i,ki j+ and n i,ki j+ be the degree of the vertices and the number of them at the level j of the generalized Bethe tree B i = B ki (d i ), d i = (, d i,2, d i,3,, d i,ki ) For i =, 2,, r, let d i,2 d i,2 d i,2 T i,ki = d i,ki 2 d i,ki 2 d i,ki 2 d i,ki d i,ki d i,ki In [0], we study the case r = 2 We recall that a weighted graph G is a graph in which each edge e has a positive weight w(e) If w(e) = for all edge e then G is an unweighted graph In [], we derive the Laplacian spectrum of v(b, B 2,, B r ) assuming that, for i =, 2,, r, the edges of B i joining the vertices at the level j with the vertices at the level (j + ) have weight w i,ki j for j =, 2,, k i For unweighted trees B i, which is the case in this paper, we have Theorem 4 [, Theorem 4] (a) If d i,ki > for i =, 2,, r then smallest eigenvalue of T,k T 2,k2 T r,kr is the algebraic connectivity of v(b, B 2,, B r ) (b)if, in addition, there exists B s such that each B i is isomorphic to a subtree of B s then the smallest eigenvalue of T s,ks is the algebraic connectivity of v(b, B 2,, B r ) Hence, if d i,ki > for i =, 2,, r and each B i is isomorphic to a subtree of B s then an upper bound on the algebraic connectivity of v(b, B 2,, B r ) can be obtained applying the Algorithm to A(a, m, α) = T s,ks That is, to m = k s, a j = d s,j (2 j k s ) and α = 0 Clearly, B i = B ki (d i ) is isomorphic to a subtree of B s = B ks (d s ) if and only if k i k s and d i,j d s,j for j =, 2,, k i Example 3 Let B = B k (d ), B 2 = B k2 (d 2 ) and B 3 = B k3 (d 3 ) where d = (, 2, 2, 2, 3), d 2 = (, 2, 3, 3, 2) and d 3 = (, 2, 3, 4, 3, 4) ThenB andb 2 are both isomorphic to subtrees ofb 3 Hence the algebraic connectivity of v(b, B 2, B 3 ) is the smallest eigenvalue of the matrix 2 2 T 3,5 = To 6 decimal places, the exact algebraic connectivity is The Algorithm, applied to T 3,5 gives the upper bound

10 O Rojo / Linear Algebra and its Applications 430 (2009) Graph C r {B k (d)} The unicyclic graph C r (B k (d)), r > 2, is obtained from the cycle C r by attaching B k (d),byitsroot, to each vertex of C r Letr = 2s or r = 2s + For l = 0,,, s,lets k,l be the k k symmetric tridiagonal matrix given below: d2 d2 d 2 S k,l = dk dk d k dk dk d k cos 2πl r For j =, 2, 3,, k, let S j be the j j leading principal submatrix of S k,0 Lemma with a m = d k + and α = yields det S k,0 = 0 Theorem 5 [2, Theorem 2] Let r = 2s orr= 2s + The set of eigenvalues of the Laplacian matrix of C r {B k (d)} is σ(l(c r {B k (d)})) = j Ω σ(s j ) s l=0 σ(s k,l) Theorem 6 [2, Theorem 3(f)] The smallest eigenvalue of d2 d2 d 2 d3 S k, = d3 dk dk dk d k cos 2π r is the algebraic connectivity of C r {B k (d)} Hence, an upper bound on the algebraic connectivity of C r {B k (d)} can be obtained applying Algorithm to A(a, m, α) = S k, That is, to m = k, a j = d j (2 j k ), a k = d k + and α = 2 cos 2π r Example 4 Consider the graph C 5 (B 6 (d)) with d = (, 3, 4, 3, 5, 4) To 6 decimal places, the algebraic connectivity is The Algorithm, with a 2 = 3, a 3 = 4, a 4 = 3, a 5 = 5, a 6 = 5 and α = 2 cos 2π 5, gives the upper bound The tree P r {B k (d)} Lemma 7 [3] The eigenvalues of a Hermitian matrix do not decrease if a positive semidefinite matrix is added to it The tree P r {B k (d)} is obtained from P r by attaching the same generalized Bethe B k (d),byitsroot, to each vertex of P r For l =,, r, let d2 d2 d 2 d3 d3 d 3 T k,l = dk dk d k dk dk d k cos πl r

11 542 O Rojo / Linear Algebra and its Applications 430 (2009) Clearly T k,j is a matrix of order k k For j =, 2,, k, let d2 d2 d 2 d3 T j = d3 d 3 d j d j d j It is also clear that T j is a matrix of order j j In [4], we study the case r = 2 We prove σ(l(p 2 {B k (d)})) = j Ω σ(t j ) σ(t k, ) σ(t k,2 ) For r > 2, the Laplacian eigenvalues of P r {B k (d)} can be found in a similar way to that used for the graph C r {B k (d)} [2] They are given in the following theorem Theorem 7 For r 2, the set of eigenvalues of the Laplacian matrix of P r {B k (d)} is σ(l(p r {B k (d)})) = j Ω σ(t j ) r l= σ(t k,l) Theorem 8 For r 2, the smallest eigenvalue of the k k real symmetric tridiagonal matrix d2 d2 d 2 T k,r = dk dk d k dk dk d k cos (r )π r is the algebraic connectivity of P r (B k (d)) Hence, an upper bound on the algebraic connectivity of P r (B k (d)) can be obtained applying the Algorithm to A(a, m, α) = T k,r That is, to m = k, a j = d j (2 j k ), a k = d k + and α = + 2 cos (r )π r Proof For j =, 2,, k 2, the eigenvalues of T j strictly interlace the eigenvalues of T j+ and, for l =, 2,, r, the eigenvalues of T k strictly interlace the eigenvalues of T k,l From Theorem 7, each eigenvalue of T k,l is a Laplacian eigenvalue Moreover, Lemma with a m = d k + and α = yields det T k,r = 0 Therefore 0 = λ (T k,r )<λ (T k,l ) for l =, 2,, r and thus the algebraic connectivity of P r {B k (d)} is min{λ (T k,l ) : l =, 2,, r } We may write T k,l = T k,l+ + E, where 0 0 E = cos π(l+) r + 2 cos πl r We claim that E is a positive semidefinite matrix In fact cos π(l + ) r + cos πl r = 2 sin π(2l + ) 2r sin π 2r > 0

12 O Rojo / Linear Algebra and its Applications 430 (2009) for l =, 2,, r Hence, from Lemma 7, λ (T k,l+ ) λ (T k,l ) for l =, 2,, r 2 It follows that min{λ (T k,l ) : l =, 2,, r } =λ (T k,r ) Example 5 Consider the tree P 3 (B 6 (d)) with d = (, 3, 4, 3, 3, 5) Then m = k = 6, a 2 = 3, a 3 = 4, a 4 = 3, a 5 = 3, a 6 = 5 + = 6 and α = + 2 cos 2π = 0 To 6 decimal 3 places, the exact algebraic connectivity is The upper bound obtained by the Algorithm is Acknowledgement The author wishes to thank the referee for the valuable comments which led to an improved version of the paper References [] M Fiedler, Algebraic connectivity of graphs, Czechoslovak Math J 23 (973) [2] B Mohar, The Laplacian spectral of graphs, in: Y Alavi, G Chartrand, OR Oellermann, AJ Schwenk (Eds), Graph Theory Combinatorics and Applications, vol 2, 99, pp [3] NMM de Abreu, Old and new results on algebraic connectivity of graphs, Linear Algebra Appl 423 (2007) [4] OJ Heilmann, EH Lieb, Theory of monomer dimer systems, Commun Math Phys 25 (972) [5] JJ Molitierno, M Neumann, BS Shader, Tight bounds on the algebraic connectivity of a balanced binary tree, Electron J Linear Algebra 6 (March) (2000) 62 7 [6] O Rojo, L Medina, Tight bounds on the algebraic connectivity of Bethe trees, Linear Algebra Appl 48 (2006) [7] GH Golub, CF Van Loan, Matrix Computations, second ed, Johns Hopkins University Press, Baltimore, 989 [8] O Rojo, R Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl 403 (2005) 97 7 [9] R Grone, R Merris, VS Sunder, The Laplacian spectrum of a graph, SIAM Matrix Anal Appl (2) (990) [0] O Rojo, On the spectra of certain rooted trees, Linear Algebra Appl 44 (2006) [] O Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra Appl 428 (2008) [2] O Rojo, The spectra of a graph obtained from copies of a generalized Bethe tree, Linear Algebra Appl 420 (2007) [3] RA Horn, CR Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 99 [4] O Rojo, The spectra of some trees and bounds for the largest eigenvalue of any tree, Linear Algebra Appl 44 (2006) 99 27