Computing the Laplacian spectra of some graphs - Semantic Scholar
PAM preprint. Collection: Cadernos de Matem´ atica - S´erie de Investiga¸c˜ ao Institution: University of Aveiro (submitted May 9, 2010)
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Computing the Laplacian spectra of some graphs Domingos M. Cardoso(1) Universidade de Aveiro, Portugal [email protected] Enide Andrade Martins(1) Universidade de Aveiro, Portugal [email protected] Mar´ıa Robbiano(2) Universidad Cat´olica del Norte, Chile [email protected] Vilmar Trevisan(3) Instituto de Matem´atica, Universidade Federal do Rio Grande do Sul Porto Alegre, RS, Brazil [email protected] May 5, 2010 Abstract In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain an upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained by copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of regular graph or a path.
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1
Introduction
Along this paper we deal with simple graphs, just called graphs. An (n, m)−graph G is a graph with n vertices and m edges. The vertex set of G is denoted by V (G) and its edge set is denote by E(G). Furthermore, D(G) denotes the diagonal matrix of vertex degrees and A(G) denotes the (0, 1)−adjacency matrix. The eigenvalues of A(G) are simple called the eigenvalues of G. The set of these eigenvalues, λ1 , λ2 , . . . , λn (together with their multiplicities) form the spectrum of the graph G, σ(G). The matrix L(G) = D(G) − A(G) is the Laplacian matrix of G. The eigenvalues of L(G), µ1 , µ2 , . . . , µn form the Laplacian spectrum of G. The computation of the Laplacian spectrum of G is, in general, a hard problem. However, there some classes of graphs for which its Laplacian spectrum is characterized, as it is the case of complete graphs, complete bipartite graphs, cycles and paths. There has been some significant work on the characterization of the Laplacian spectra of trees, which are connected acyclic graphs. However, very few classes of graphs containing cycles have their spectra explicitly computed. In this paper the spectra of a family of graphs is characterized in terms of the eigenvalues of symmetric tridiagonal matrices, from which the Laplacian spectrum can be computed. These graphs contain many cycles and are obtained by copies of generalized Bethe trees, connecting the vertices at same level by edges and identifying its roots with the vertices of a regular graph or a path. As an application, upper and lower bounds on the Laplacianenergy-like (cf. [12]) are obtained and some inequalities, regarding Laplacian eigenvalues of particular cases of graphs obtained by the above graph operations, are deduced.
2
Notation and basic definitions
We describe now the graphs considered in this note. The level of a vertex on a tree is one unit more than its distance from the root vertex. For k ≥ 2, a generalized Bethe tree, Bk , of k levels [16] is a rooted tree in which vertices at same level have equal degrees. For j = 1, . . . , k, we denote by dk−j+1 and by nk−j+1 the degree of the vertices at level j and their number, respectively. Thus, d1 = 1 is the degree of the vertices at the level k and dk is the degree of the root vertex.
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In this paper we deal with generalized Bethe trees (herein called modified generalized Bethe trees) modified according to the following definition. Definition 1 Fixing i such that 1 ≤ i ≤ k − 1 and dk−i+1 ≥ 3. A modified generalized Bethe tree with parameters k and i is the graph Bk [i], obtained from the generalized Bethe tree Bk connecting the vertices at level i + 1 by paths such such that two vertices at level i + 1 are in the same path if both are connected at the same vertex in the level i. We recall that in Bk at the levels i, i + 1 there are nk−i+1 , nk−i vertices, respectively, and such implies that in Bk [i] at level i + 1 there are nk−i+1 paths each one with pi = nk−i /nk−i+1 vertices. As an example, the modified generalized Bethe tree B4 [2] is depicted in the Figure 1. 31
28t
19 t B t B t
1
2
HH H
20 t
B t B t
3
4
t PPP PP P
29 t
HHH 21 22 24 HH HH 23 t t t t B B B B t Bt t B t t B t t Bt
5
67 8
9 10
PP P
25
t B t B t
PP 30t P HH H
11 12 13 14
26 t
B t B t
HH t27 B t Bt
15 16
17 18
Figure 1: The graph B4 [2] · T1 (0) There are several deep works on the characterization of the spectra of some graph constructions with generalized Bethe trees. In 2006, Rojo (cf. [17]) found the adjacency and Laplacian spectrum of graphs which are obtained connecting by an edge the roots of two copy of a generalized Bethe tree. Later, in 2007 [16] the author characterizes the adjacency and Laplacian spectrum of graphs obtained by identifying the roots of r copies of generalized Bethe trees with the vertices of a r-cycle. In [14], the previous results are extended, considering copies of generalized Bethe trees and attaching their roots to the vertices of an arbitrary connected graph. Recently, Rojo and Medina in [15] characterized the Laplacian and adjacency eigenvalues of the trees obtained attaching the roots of k Bethe trees to a k-vertex path in a such way that the root of the i-th Bethe tree is identified with the i-th vertex of the path.
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Figure 2: The graph 3B4 [2] · T3 (2)
In this paper we characterize the Laplacian spectrum of graphs obtained by the following operations on r copies of modified generalized Bethe trees Bk [i]. Each of the r copies of the obtained graphs Bk [i] is attached to a distinct vertex of a r-vertex connected d-regular graph Tr (d) (producing the graph rBk [i] · Tr (d)) or to a distinct vertex of a r-vertex path Pr (producing the graph rBk [i] · Pr ). Notice that, in particular, Bk [i] · T1 (0) corresponds to Bk [i]. Further examples are given in Figure 1 with the graph B4 [2] · T1 (0), in Figure 2 with the graph 3B4 [2] · T3 (2) and in Figure 3 with the graph 3B4 [2] · P3 . Considering a generalized Bethe tree, Bk , the labeling of the vertices of Bk [i] can be obtained accordingly with the following procedures: 1. The vertices of each copy of Bk [i] are labeled from the bottom to the root vertex and, in each level, from the left to the right. This labeling is illustrated in the Figure 1. 2. At the level i + 1 in Bk [i], the vertices have degrees dk−i + 1 or dk−i + 2, where dk−i is the degree of the vertices of the (i + 1)-th level of Bk . In the case of the graphs rBk [i] · Tr (d) and rBk [i] · Pr the labeling of the vertices can be obtained as follows:
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Figure 3: The graph 3B4 [2] · P3
1. We label the vertices of the first level, from the left to the right. 2. The vertices of the second level are labeled using rn1 +1, rn1 +2, . . . , rn1 + rn2 . 3. The labeling of the vertices of the higher levels can be done in the same way. 4. Finally, the last vertex (which belongs to the regular graph or to the path) is labeled using n. For illustrative purpose, consider the graphs 3B4 [2]·T3 (2) and 3B4 [2]·P3 depicted in Figures 2 and 3, respectively. Since, the main results of the next sections are based in matricial techniques related with the Kronecker product, let us define this product and review some of its properties. For the matrices A = (aij ) ∈ Rr×s and B = (bij ) ∈ Rp×q , the Kronecker product is denoted A ⊗ B ∈ Rrp×sq . Several properties of the Kronecker product may be found, for instance, in [11]. From now on, 0 denotes the all zero matrix with appropriate order, Im is the identity matrix of order m and em denotes the all ones m− dimensional vector. The Laplacian matrix of the q-vertex path Pq is denoted Lq , and then 1 −1 −1 2 −1 . . . . . . Lq = . . . . −1 2 −1 −1 1
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Considering a fixed i, 1 ≤ i ≤ k − 1 such that dk−i+1 ≥ 3, and taking into account that nk−i , pi = nk−i+1 let us define mj = dj+1 − 1 =
nj = pk−j , for j = 1, . . . , k − 2. nj+1
mk−1 = dk = nk−1 .
(1) (2)
Then, we are able to deal with the matrices Jpi = Ink−i+1 ⊗ Lpi ,
(3)
of order nk−i and also with the matrices Hpi = Ir ⊗ Jpi = Ir ⊗ Ink−i+1 ⊗ Lpi , In the sequel, Cj ( 1 ≤ j ≤ k − 1) is the block diagonal matrix defined by Cj = Inj+1 ⊗ emj ,
(4)
with nj+1 diagonal blocks equals to emj . Thus, Cj is an nj × nj+1 matrix, for j = 1, . . . , k − 1, where Ck−1 = enk−1 . Let Kj = Ir ⊗ Cj ,
(5)
the matrix of order rnj , where each of its r diagonal blocks is equal to Cj , that is, since Cj is composed by nj+1 diagonal blocks equal to emj , in total the matrix Kj has rnj+1 diagonal blocks emj . As direct consequence of the properties of the Kronecker product, we have the following equalities: T KjT Kj = mj Irnj+1 , for j = 1, . . . , k − 2, Kk−1 Kk−1 = nk−1 Ir = dk Ir , T T Kj cIrnj − Hpk−j = cKj for j = 1, . . . , k − 1,
where c is an arbitrary polynomial.
(6) (7)
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Example 2 Let us consider a generalized Bethe tree Bk and the graph Tr (d). Then, in the particular case of k = 4, in B4 we have the following pairs (di , ni ). (d1 , n1 ) = (1, 12), (d2 , n2 ) = (3, 6), (d3 , n3 ) = (4, 2), (d4 , n4 ) = (2, 1). In B4 [2], at level 3, we have n3 = 2 paths with p2 = n2 /n3 = 6/2 = 3 vertices and also m1 = 2, m2 = 3 and m3 = 2. Therefore, J3 = Jp2 = I2 ⊗ L3 . Furthermore, Hp2 = I3 ⊗ Jp2 = I3 ⊗ I2 ⊗ L3 = I6 ⊗ L3 and the matrices defined in (4) are C1 = I6 ⊗ e2 , C2 = I2 ⊗ e3 and C3 = e2 . Taking into account the above referred labeling for the vertices, the Laplacian matrices L(3B4 [2] · T3 (2)) and L(3B4 [2] · P3 ) become: I36 −K1 0 0 −K1T 3I18 + Hp −K2 0 , 2 T 0 −K2 4I6 −K3 0 0 −K3T W where in the case of the matrix L(3B4 [2] · T3 (2)), W = (2 + 2)I3 − A(T3 (2)), and in the case of the matrix L(3B4 [2] · P3 ), W = 2I3 + L3 . In the general case, the Laplacian matrices L(rBk [i]·Tr (d)) and L(rBk [i]· Pr ) become: d1 Irn1 −K1 −K1T d2 Irn2 . . . .. .. . . −Kk−(i+1) (8) dk−i Irnk−i + Hpi −Kk−i . . T .. .. −K −K k−1 k−i .. T . −Kk−1 W where in the case of the matrix L(rBk [i] · Tr (d)), W = (dk + d)Ir − A(Tr (d)) and in the case of the matrix L(rBk [i] · Pr ), W = dk Ir + Lr .
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Lemma 3 Let M be the following block tridiagonal matrix α1 Irn1 K1 K1T α2 Irn2 K2 ... ... Kk−(i+1) T Kk−i Kk−(i+1) αk−i Irnk−i − Hpi , . T . . αk−i+1 Irnk−i+1 Kk−i . .. Kk−1 αk−1 Irnk−1 .. T . Kk−1 αk Ir + L where L is a r × r symmetric matrix. If β1 = α1 6= 0 1 6= 0, βj = αj − mj−1 βj−1
2 ≤ j ≤ k − 1,
1 βk = αk − dk βk−1 6= 0,
then r
Y n k−1 det(M ) = βj j j=1 j6=k−i
pi Y l=1
!rnk−i+1 (βk−i − νl )
r Y
! (βk + δj ) ,
(9)
j=1
where δ1 , . . . , δr are the eigenvalues of L and ν1 , . . . , νpi are the eigenvalues of Lpi . Proof. For this proof, we take into account the equalities (6) and (7). Then, performing elementary operation over the line blocks of M , the matrices KjT , for j = 1, . . . , k − 1 are eliminated using the blocks βj Irnj , with βj 6= 0, T excluding the matrix Kk−i which is eliminated according to the item (c). Therefore, the matrix M is transformed according to the following items: (a) For j = 1, . . . , k − (i + 2), each diagonal block αj+1 Irnj+1 is replaced by αj+1 Irnj+1 − βj−1 KjT Kj = αj+1 Irnj+1 − mj βj−1 Irnj+1 = βj+1 Irnj+1 . (b) For j = k −(i+1), the block αk−i Irnk−i −Hpi is replaced by βk−i Irnk−i − Hpi .
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(c) For j = k − i the block βk−i Irnk−i − Hpi is multiplied on left by −1 T T −βk−i Kk−i and added to the block Kk−i . Then this block is eliminated taking into account (7). Additionally, taking into account (6), the corresponding diagonal block is replaced by −1 T αk−i+1 Irnk−i+1 − βk−i Kk−i Kk−i = βk−i+1 Irnk−i+1 .
(d) For j = k − (i − 1), . . . , k − 1, the block Kj is eliminated by the block βj Irnj and the corresponding diagonal block is replaced by βj+1 Irnj+1 . Observe that after the last operation of item (d), αk is replaced by αk − −1 dk βk−1 = βk . Hence Eq. (9) follows.
3
The spectrum of the Laplacian matrix
The technics used in the proof of the next results are similar to those used in [19, Th. 5]. In what follows, γ1 , . . . , γr are the eigenvalues of Tr (d) and , for j = 1, . . . , r, are the eigenvalues of Lr . Furthermore, µj = 2 + 2 cos πj r considering Φ = {0, 1, 2, . . . , k − i − 1, k − i + 1, . . . , k − 1}, then Ω = {j ∈ Φ − {0} : nj > nj+1 } ∪ {0} .
(10)
Along this section we need also to define the following polynomials (where mj is as (1)): Q0 (λ) Q1 (λ) Qj (λ) Qkj (λ) Lkj (λ)
= = = = =
1, (11) λ − d1 (12) (λ − dj )Qj−1 (λ) − mj−1 Qj−2 (λ) , j = 2, . . . , k − 1, (13) (λ − dk − d + γj ) Qk−1 (λ) − mk−1 Qk−2 (λ) , j = 1, . . . , r,(14) (λ − dk − µj ) Qk−1 (λ) − mk−1 Qk−2 (λ) , j = 1, . . . , r. (15)
Now we are able to introduce the main result of this section. Theorem 4 If H is a graph belonging to the set {rBk [i] · Tr (d), rBk [i] · Pr }, with i ∈ {1, . . . , k − 1}, then pi −1
det(λI − L(H)) =
Y t∈Ω
rn −rn Qt t t+1
Y l=1
(Qk−i − νl Qk−i−1 )rnk−i+1 Ψ(H),
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Q Q where Ψ(H) = rj=1 Qkj (λ) when H = rBk [i]·Tr (d) and Ψ(H) = rj=1 Lkj (λ) when H = rBk [i] · Pr . Furthermore, pi −1
σ(L(H)) =
[
{λ : Qt (λ) = 0} ∪
t∈Ω
[
! {λ : Qk−i (λ) − νl Qk−i−1 (λ) = 0}
∪X
l=1
Sr where X = j=1 {λ : Qkj (λ) = 0} when H = rBk [i] · Tr (d), and X = Sr j=1 {λ : Lkj (λ) = 0} when H = rBk [i] · Pr . Proof. For H ∈ {rBk [i] · Tr (d), rBk [i] · Pr }, taking into account the expression of L (H) in (8) and applying Lemma 3 to the matrix M = λI − L (H), we obtain αj = λ − dj , j = 1, . . . , k − 1, λ − d − dk , if H = rBk [i] · Tr (d); αk = λ − dk , if H = rBk [i] · Pr . Let βj be as in Lemma 3 and suppose that λ ∈ R is such that Qj (λ) 6= 0, when j = 1, . . . , k − 1. For the sake of simplicity, let us denote Qj (λ) and Qkj (λ) by Qj and Qkj , respectively. Then β1 = λ − 1 =
Q1 Q0
6= 0,
1 = βj = (λ − dj ) − mj−1 βj−1
= and, for i = 1, . . . , r, ( βk + δi = ( = ( =
Qj Qj−1
(λ−d3 )Qj−1 −mj−1 Qj−2 Qj−1
6= 0, for j = 2, . . . , k − 1,
1 λ − d − dk + γi − dk βk−1 , if H = rBk [i] · Tr (d); 1 λ − dk + µi − dk βk−1 , if H = rBk [i] · Pr (λ−d−dk +γj )Qk−1 −dk Qk−2 , Qk−1 (λ−dk +µi )Qk−1 −dk Qk−2 , Qk−1 Qki , Qk−1 Lki , Qk−1
if H = rBk [i] · Tr (d); if H = rBk [i] · Pr
if H = rBk [i] · Tr (d); if H = rBk [i] · Pr .
Therefore, using (9), it follows
(16)
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r
det M
Y nj k−1 = βj j=1 j6=k−i
= =
=
n Q1 1 n1 Q0
n Q2 2 n Q1 2
pi Y
!rnk−i+1 (βk−i − νl )
nk−i−1
···
nk−i−1 Qk−i−2
(βk + δj )
j=1
l=1 Qk−i−1
r Y
r
pi Y
Qk−i ( Qk−i−1
!rnk−i+1 − νl )
nk−1
nk−i+1
Qk−i+1
nk−i+1 Qk−i
l=1 r(nk−i−1 −nk−i) r(n1 −n2 ) Q1 · · · Qk−i−1 nk−i+1 pY i −1 nk−1 r Qk−1 Qk−i+1 rnk−i+1 rnk−i+1 (Qk−i − νl Qk−i−1 ) Qk−i nk−i+1 · · · nk−1 Qk−i Qk−2 l=1 r(nk−1 −nk ) r(nk−i−1 −nk−i ) r(nk−i+1 −nk−i+2 ) r(n −n ) · · · Qk−1 Q1 1 2 · · · Qk−i−1 Qk−i+1 pY i −1 (Qk−i − νl Qk−i−1 )rnk−i+1 Ψ(H). l=1
...
Qk−1
r
nk−1 Qk−2
Ψ(H) Qrk−1
Ψ(H) Qrk−1
Q Q where Ψ(rBk [i] · Tr (d)) = rj=1 Qkj and Ψ(rBk [i] · Pr ) = rj=1 Lkj . Notice that the above expression is obtained taking into account that (according to its definition) νpi = 0. Consider now λ0 ∈ R such that Qt (λ0 ) = 0 for some t = 1, . . . , k −1. Since the zeros of any polynomial are isolated there exists a neighborhood V (λ0 ) of λ0 such that Qj (λ) 6= 0, for all λ ∈ V (λ0 )−{λ0 } and for all j = 1, . . . , k −1. Hence, the equality (17) follows, for all λ ∈ V (λ0 ) − {λ0 } . By continuity, taking the limit as λ tends to λ0 , we obtain the desired expression. Corollary 5 If H ∈ {rBk [k − 1] · Tr (d), rBk [k − 1] · Pr }, with k ≥ 2, then for all 1 ≤ l ≤ pk−1 − 1, µl = 3 + 2 cos lπ is an eigenvalue with multiplicity pi at least rn2 of H. Proof. Taking into at level k ≥ 2 there are n2 ≥ 1 paths with pk−1 = nn12 > 1 vertices, then by Theorem 4, we obtain {λ : Q1 (λ) − νl Q0 (λ) = 0} ⊆ σ(H). Hence since Q0 (λ) = 1, using (12), the result follows. Definition 6 For t = 1, . . . , k − 1, let Tt (H) be the t × t leading principal
(17)
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submatrix of the k × k symmetric tridiagonal matrix √ d d − 1 1 2 √ d2 − 1 d2 .. . Tkj (H) = p . . .. .. d − 1 k−1 p √ dk−1 − 1 d√k−1 dk dk dk + θj d − γj , if H = rBk [k − 1] · Tr (d); where, for j = 1, . . . , r, θj = 2 + 2 cos πj , if H = rBk [k − 1] · Pr . r and for l = 1, . . . , pi −1, let consider the dk−i ×dk−i symmetric tridiagonal matrix √ d2 − 1 d1 .. √ . d2 d2 − 1 Sl = . p . . .. .. d − 1 k−i p dk−i − 1 dk−i + νl Remark 7 We remark that σ(L(Tr (d))) = {d − γj : 1 ≤ j ≤ r}. Lemma 8 If Tt , t = 1, . . . , k − 1, and Sl , l = 1, . . . , pi − 1 are the leading principal submatrices of Tkj (H) and the dk−i × dk−i tridiagonal symmetric matrices referred in Definition 6, respectively. Then det(λI − Tt ) = Qt (λ) , t = 1, . . . , k − 1,
(18)
det (λI − Tkj (H)) = Qkj (λ)
(19)
det(λI − Sl ) = Qk−i (λ) − νl Qk−i−1 (λ).
(20)
for j = 1, . . . , r and for l = 1, . . . , pi − 1
Proof. Formulas in (18) and in (19) have already been proved in [19]. To obtain (20) , we see that if i = k − 1 the result follows by Corollary 5. Now we suppose that i < k − 1. By (13) we see that Qk−i − νl Qk−i−1 = (λ − dk−i − νl )Qk−i−1 − mk−i−1 Qk−i−2 = (λ − dk−i − νl )Qk−i−1 − (dk−i − 1) Qk−i−2 , which, by [20] implies the result.
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Lemma 9 For t = 1, . . . , k − 1 , l = 1, . . . , pi − 1 and j = 1, . . . , r the zeros of the polynomials Qt , Qk−i − νl Qk−i−1 and Qkj (or Lkj ), respectively are real and simple. Proof. In view of Lemma 8 and as the matrices in Definition 6 are symmetric and tridiagonal with nonzero codiagonal entries it follows that its eigenvalues are simple (cf. [4]). The next theorem gives a complete characterization of the eigenvalues of the Laplacian matrix in the case when H = rBk [i] · Tr (d) or H = rBk [i] · Pr . i hS S S r pi −1 σ(T (H)) . σ(S ))∪ σ (T ) Theorem 10 (a) σ (H) = ∪( kj l t j=1 l=1 t∈Ω−{0} (b) The multiplicity of each eigenvalue of the matrix Tt as eigenvalue of L(H) is at least rnt − rnt+1 for t ∈ Ω and the multiplicity of each eigenvalue of the matrices Tkj (H) for j = 1, . . . , r is at least 1. (c) Each eigenvalue of Sl , for all 1 ≤ l ≤ pi − 1, is an eigenvalue of L(H). (d) Each eigenvalue of Sl is an eigenvalue of L (H) with multiplicity at least rnk−i+1 . Proof. (a), (b) and (d) are consequences of Theorem 4, Lemma 9 and Lemma 8. The assertion in (c) is due to nk−i+1 ≥ 1. Next we present an illustrative example for the case H = 3B4 [2] · T3 (2) . Example 11 In the graph 3B4 [2] · T3 (2) let Φ = {0, 1, 3} and Ω = {0, 1, 3} .
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By Remark 7 we have {d − γj : 1 ≤ j ≤ 3} = {0, 3, 3} . Thus the matrices Definition 6 correspond to, √ √ 2 0 0 1 2 0 0 1 √ √ √ √ √1 2 3 3 √0 2 √3 3 √0 2 √ , T42 = , T43 = T41 = 0 3 √4 2 0 3 √4 2 0 0 0 2 5 0 0 2 5 0 and
S1 =
√ √ 2 1 2 √1 √ , S2 = . 2 3+3 2 3+1
in √
2 √0 0 3 √0 , √3 3 √4 2 0 2 2
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To four decimal places, the eigenvalues and its multiplicities are given by: T41 : T42 : T43 : T1 : T3 : S1 : S2 :
0.0496 0.0496 0 1 0.0746 0.6277 0.4384
2.1456 4.3968 6.4080 2.1456 4.3968 6.4080 1.2087 3 5.7913 2.4481 5.4774 6.3723 4.5616
1 1 1 rn1 − rn2 = 18 rn3 − rn4 = 3 rn3 = 6 rn3 = 6.
The referred graph operations with modified generalized Bethe trees can be generalized in the following way: for a fixed 1 ≤ s ≤ k − 1, let ˆı = (i1 , i2 , . . . , is ) with 2 ≤ i1 < i2 < · · · < is ≤ k −1. We construct an unweighed graph Bk [ˆı] obtained from a generalized Bethe tree Bk and the union of paths at the level iq + 1 such that two vertices are in the same path if and only if both are connected at the same vertex at level iq , with 1 ≤ q ≤ s. A more general graph operation rBk [ˆı] · Tr (d) obtained from r copies of Bk [ˆı] and a regular connected graph Tr (d) (or a path Pr ) with r vertices, each one of degree d, can be considered by identifying the root of each copy of Bk [ˆı] with each vertex of Tr (d) (Pr ). The previous obtained results can be extended to the graphs produced by these more general graph operations, using techniques very similar to the ones above presented.
4
Some applications
In this section the singular values of the Laplacian matrices (and then the eigenvalues) of the graphs rBk [i] · Tr (d) and H = rBk [i] · Pr are compared, when Pr is isomorphic to an Hamiltonian path of Tr (d) (if there is). First, it is convenient to introduce the following notation (c.f. [1, 2]). A > 0 means that A is positive semidefinite and if A and B are hermitian matrices, then A > B if A − B > 0. For a complex m × n matrix C we recall that its nonzero singular values correspond to the nonzero eigenvalues of the positive 1/2 semidefinite matrix |C| = C T C , (see, e. g. [11]). The singular values of a matrix C are enumerated as s1 (C) ≥ · · · ≥ sn (C). Now, we are able to remind the following results (see [2, 1]). Lemma 12 [2] Let X and Y be positive semidefite n × n hermitian matrices, with eigenvalues λ1 (X) ≥ · · · ≥ λn (X) and λ1 (Y ) ≥ · · · ≥ λn (Y ), respectively. If X > Y , then λj (X) ≥ λj (Y ) for j = 1, . . . , n.
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Theorem 13 Consider a modified generalized Bethe tree Bk [i]. If a d-regular graph of order r, Tr (d), has an Hamilton path isomorphic to Pr , then λj (L(rBk [i] · Tr (d))) ≥ λj (L(rBk [i] · Pr )), for j = 1, . . . , n. Proof. The Laplacian matrices of the graphs rBk [i] · Tr (d) and rBk [i] · Pr are positive semidefinite. Taking into account the expression (8) for these matrices, then 0 0 L(rBk [i] · Tr (d))) − L(rBk [i] · Pr ) = . 0 L(Tr (d)) − L(Pr ) Therefore, since L(Tr (d)) − L(Pr ) = L(Tr (d) \ Pr ), where Tr (d) \ Pr is the graph obtained from Tr (d) after deleting the edges corresponding to Pr , it follows that L(rBk [i] · Tr (d))) − L(rBk [i] · Pr ) > 0 ⇔ L(rBk [i] · Tr (d))) > L(rBk [i] · Pr ). Applying Lemma 12, the conclusion follows. The Laplacian-energy-like of an (n, m)−graph G (denoted by LEL [G]), introduced by Liu and Liu [12] is defined as LEL [G] =
n X √
µj .
(22)
j=1
where µj , for j = 1, . . . , n, are the eigenvalues of the Laplacian matrix of G. For a m × n complex matrix C, we define the Laplacian-energy-like of the matrix C by Xq λj (|C|). LEL(C) = j
As a consequence of the previous definitions the Laplacian-energy-like of any graph G is given by LEL [G] = LEL(L(G)). The next theorem obtains bounds for the Laplacian-energy-like for the graphs studied in this paper.
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Theorem 14 Let G be an (n, m)−graph such that det(L(G) − λIn ) = P1 (λ)t1 P2 (λ)t2 · · · Ps (λ)ts ,
(23)
where Pj (λ) = det(Mj − λIj ), Mj are square matrices of order j, with tj > 0, for j = 1, . . . , s. Then s X p q LEL [G] ≤ tj j trace(Mj ) (24) j=1
Proof. From the decomposition in (23) it is clear that LEL [G] = =
s X j=1 s X j=1
tj LEL (Mj ) X √ µ
tj
µ∈σ(Mj )
Thus, via Cauchy Schwarz inequality 1/2 s X X √ 2 p LEL [G] ≤ tj j ( µ) j=1
=
s X
µ∈σ(Mj )
p q tj j trace (Mj ),
j=1
and the inequality holds. We fix i, 2 ≤ i ≤ k − 2. Now we consider the connected graph H ∈ {Bk [i] · Tr (d) , Bk [i] · Pr }. We shall take n and m as the number of vertices and of edges, respectively, of H. Applying Theorem 4, Lemma 8 and Theorem 14 we obtain the following result. Corollary 15 For 1 ≤ j ≤ r, 1 ≤ u ≤ k − 1 and 1 ≤ l ≤ pi − 1. Let Tkj (H), Tu and Sl defined according to the Definition 6. Then LEL (H) ≤ X√ u∈Ω
u(rnu −rnu+1 )
pi −1 r Xp X p √ q p traceTu +rnk−i+1 dk−i traceSl + k traceTkj (H). l=1
j=1
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Finally, as immediate consequence of Theorem 13, we have the following corollary. Corollary 16 Under the conditions of Theorem 13, LEL(rBk [i] · Tr (d)) ≥ LEL(rBk [i] · Pr ). Acknowledgements: (1) Research supported by the Center for Research and Development in Mathematics and Applications from the Funda¸c˜ao para a Ciˆencia e a Tecnologia, cofinanced by the European Community Fund FEDER/POCI 2010. (2) Research supported by Proyecto Mecesup 2 UCN 0605, Chile y FondecytIC Project 11090211, Chile. (3) Research partially supported by CNPq - Grant 309531/2009-8.
References [1] R. Bhatia, F. Kitaneh, The singular valuies of A + B and A + iB. Linear Algebra Appl. 431 (20019): 1502-1508. [2] R. Bhatia, Matrix Analysis. Springer, 1997. [3] D. Cvetkovi´c, M. Doob, H. Sachs. Spectra of Graphs - Theory and Applications, Deutcher Verlag der Wissenschaften, Academic Press, BerlinNew York (1980), second ed. (1980), third ed. Verlag, Heidelberg Leipzig (1995). [4] G. H. Golub, C. F. Van Loan. Matrix Computations, second ed., Johns Hopkins University Press, Baltimore, 1989. [5] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978): 1-22. [6] I. Gutman, The energy of a Graph: Old and New Results, Algebraic Combinatorics and Applications (2001): 196-211. Springer-Verlag. Berlin. [7] I. Gutman, S. Zare Firoozabadi, J. A. de la Pe˜ na, J. Rada, On the energy of regular graphs, MATCH Commun. Math. Comput. Chem. 57 (2007): 435-442.
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[8] O. J. Heilmann, E. H. Lieb. Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972): 190-232. [9] R. A. Horn, C. R. Johnson. Matrix Analysis. Cambridge University Press (1991). Cambridge. [10] G. Indulal, A. Vijayakumar, A note on energy of some graphs, MATCH Commun. Math. Comput. Chem. 59 (2008): 269-274. [11] P. Lancaster, M.Tismenetsky. The theory of matrices. Academic Press. INC. [12] J. Liu, B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 v. 2 (2008): 355-372. [13] V. Nikiforov, The energy of graphs and matrices, Journal of Mathematical Analysis and Applications 326, Issue 2 (2007): 1472-1475. [14] O. Rojo. Spectra of copies of a generalized Bethe tree attached to any graph. Linear Algebra and its Appl. 431 (2009): 862-882. [15] O. Rojo, L. Medina. Spectra of generalized Bethe trees attached to a path. Linear Algebra and its Appl. 430 (2009): 483-503. [16] O. Rojo. The spectra of a graph obtained from copies of a generalized Bethe tree. Linear Algebra and its Appl. 420, I. 2-3 15 (2007): 490-507. [17] O. Rojo. The spectra of some trees and bounds for the largest eigenvalues of any tree. Linear Algebra Appl. 414 (2006): 199-217. [18] O. Rojo and R. Soto. The spectra of the adjacency matrix and Laplacian matrix for some balanced trees. Linear Algebra Appl. 403 (2005): 97117. [19] O. Rojo and M. Robbiano. On the spectra of some weighted rooted tree and applications, Linear Algebra Appl. 420 (2007): 310-328. [20] L. N. Trefethen. D. Bau III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics, 1997. [21] R. S. Varga. Matrix Iterative Analysis. Prentice-Hall, Inc., 1965.
A principal inten¸c˜ ao desta s´erie de publica¸c˜oes, Cadernos de Matem´atica, ´e de divulgar trabalho original t˜ao depressa quanto poss´ıvel. Como tal, os artigos publicados n˜ao sofrem a revis˜ao usual na maior parte das revistas. Os autores, apenas, s˜ ao respons´aveis pelo conte´ udo, interpreta¸c˜ao dos dados e opini˜oes expressas nos artigos. Todos os contactos respeitantes aos artigos devem ser endere¸cados aos autores. The primary intent of this publication, Cadernos de Matem´atica, is to share original work as quickly as possible. Therefore, articles which appear are not reviewed as is the usual practice with most journals. The authors alone are responsible for the content, interpretation of data, and opinions expressed in the articles. All communications concerning the articles should be addressed to the authors.
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Computing the Laplacian spectra of some graphs Domingos M. Cardoso(1) Universidade de Aveiro, Portugal [email protected] Enide Andrade Martins(1) Universidade de Aveiro, Portugal [email protected] Mar´ıa Robbiano(2) Universidad Cat´olica del Norte, Chile [email protected] Vilmar Trevisan(3) Instituto de Matem´atica, Universidade Federal do Rio Grande do Sul Porto Alegre, RS, Brazil [email protected] May 5, 2010 Abstract In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain an upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained by copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of regular graph or a path.
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1
Introduction
Along this paper we deal with simple graphs, just called graphs. An (n, m)−graph G is a graph with n vertices and m edges. The vertex set of G is denoted by V (G) and its edge set is denote by E(G). Furthermore, D(G) denotes the diagonal matrix of vertex degrees and A(G) denotes the (0, 1)−adjacency matrix. The eigenvalues of A(G) are simple called the eigenvalues of G. The set of these eigenvalues, λ1 , λ2 , . . . , λn (together with their multiplicities) form the spectrum of the graph G, σ(G). The matrix L(G) = D(G) − A(G) is the Laplacian matrix of G. The eigenvalues of L(G), µ1 , µ2 , . . . , µn form the Laplacian spectrum of G. The computation of the Laplacian spectrum of G is, in general, a hard problem. However, there some classes of graphs for which its Laplacian spectrum is characterized, as it is the case of complete graphs, complete bipartite graphs, cycles and paths. There has been some significant work on the characterization of the Laplacian spectra of trees, which are connected acyclic graphs. However, very few classes of graphs containing cycles have their spectra explicitly computed. In this paper the spectra of a family of graphs is characterized in terms of the eigenvalues of symmetric tridiagonal matrices, from which the Laplacian spectrum can be computed. These graphs contain many cycles and are obtained by copies of generalized Bethe trees, connecting the vertices at same level by edges and identifying its roots with the vertices of a regular graph or a path. As an application, upper and lower bounds on the Laplacianenergy-like (cf. [12]) are obtained and some inequalities, regarding Laplacian eigenvalues of particular cases of graphs obtained by the above graph operations, are deduced.
2
Notation and basic definitions
We describe now the graphs considered in this note. The level of a vertex on a tree is one unit more than its distance from the root vertex. For k ≥ 2, a generalized Bethe tree, Bk , of k levels [16] is a rooted tree in which vertices at same level have equal degrees. For j = 1, . . . , k, we denote by dk−j+1 and by nk−j+1 the degree of the vertices at level j and their number, respectively. Thus, d1 = 1 is the degree of the vertices at the level k and dk is the degree of the root vertex.
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In this paper we deal with generalized Bethe trees (herein called modified generalized Bethe trees) modified according to the following definition. Definition 1 Fixing i such that 1 ≤ i ≤ k − 1 and dk−i+1 ≥ 3. A modified generalized Bethe tree with parameters k and i is the graph Bk [i], obtained from the generalized Bethe tree Bk connecting the vertices at level i + 1 by paths such such that two vertices at level i + 1 are in the same path if both are connected at the same vertex in the level i. We recall that in Bk at the levels i, i + 1 there are nk−i+1 , nk−i vertices, respectively, and such implies that in Bk [i] at level i + 1 there are nk−i+1 paths each one with pi = nk−i /nk−i+1 vertices. As an example, the modified generalized Bethe tree B4 [2] is depicted in the Figure 1. 31
28t
19 t B t B t
1
2
HH H
20 t
B t B t
3
4
t PPP PP P
29 t
HHH 21 22 24 HH HH 23 t t t t B B B B t Bt t B t t B t t Bt
5
67 8
9 10
PP P
25
t B t B t
PP 30t P HH H
11 12 13 14
26 t
B t B t
HH t27 B t Bt
15 16
17 18
Figure 1: The graph B4 [2] · T1 (0) There are several deep works on the characterization of the spectra of some graph constructions with generalized Bethe trees. In 2006, Rojo (cf. [17]) found the adjacency and Laplacian spectrum of graphs which are obtained connecting by an edge the roots of two copy of a generalized Bethe tree. Later, in 2007 [16] the author characterizes the adjacency and Laplacian spectrum of graphs obtained by identifying the roots of r copies of generalized Bethe trees with the vertices of a r-cycle. In [14], the previous results are extended, considering copies of generalized Bethe trees and attaching their roots to the vertices of an arbitrary connected graph. Recently, Rojo and Medina in [15] characterized the Laplacian and adjacency eigenvalues of the trees obtained attaching the roots of k Bethe trees to a k-vertex path in a such way that the root of the i-th Bethe tree is identified with the i-th vertex of the path.
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Figure 2: The graph 3B4 [2] · T3 (2)
In this paper we characterize the Laplacian spectrum of graphs obtained by the following operations on r copies of modified generalized Bethe trees Bk [i]. Each of the r copies of the obtained graphs Bk [i] is attached to a distinct vertex of a r-vertex connected d-regular graph Tr (d) (producing the graph rBk [i] · Tr (d)) or to a distinct vertex of a r-vertex path Pr (producing the graph rBk [i] · Pr ). Notice that, in particular, Bk [i] · T1 (0) corresponds to Bk [i]. Further examples are given in Figure 1 with the graph B4 [2] · T1 (0), in Figure 2 with the graph 3B4 [2] · T3 (2) and in Figure 3 with the graph 3B4 [2] · P3 . Considering a generalized Bethe tree, Bk , the labeling of the vertices of Bk [i] can be obtained accordingly with the following procedures: 1. The vertices of each copy of Bk [i] are labeled from the bottom to the root vertex and, in each level, from the left to the right. This labeling is illustrated in the Figure 1. 2. At the level i + 1 in Bk [i], the vertices have degrees dk−i + 1 or dk−i + 2, where dk−i is the degree of the vertices of the (i + 1)-th level of Bk . In the case of the graphs rBk [i] · Tr (d) and rBk [i] · Pr the labeling of the vertices can be obtained as follows:
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Figure 3: The graph 3B4 [2] · P3
1. We label the vertices of the first level, from the left to the right. 2. The vertices of the second level are labeled using rn1 +1, rn1 +2, . . . , rn1 + rn2 . 3. The labeling of the vertices of the higher levels can be done in the same way. 4. Finally, the last vertex (which belongs to the regular graph or to the path) is labeled using n. For illustrative purpose, consider the graphs 3B4 [2]·T3 (2) and 3B4 [2]·P3 depicted in Figures 2 and 3, respectively. Since, the main results of the next sections are based in matricial techniques related with the Kronecker product, let us define this product and review some of its properties. For the matrices A = (aij ) ∈ Rr×s and B = (bij ) ∈ Rp×q , the Kronecker product is denoted A ⊗ B ∈ Rrp×sq . Several properties of the Kronecker product may be found, for instance, in [11]. From now on, 0 denotes the all zero matrix with appropriate order, Im is the identity matrix of order m and em denotes the all ones m− dimensional vector. The Laplacian matrix of the q-vertex path Pq is denoted Lq , and then 1 −1 −1 2 −1 . . . . . . Lq = . . . . −1 2 −1 −1 1
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Considering a fixed i, 1 ≤ i ≤ k − 1 such that dk−i+1 ≥ 3, and taking into account that nk−i , pi = nk−i+1 let us define mj = dj+1 − 1 =
nj = pk−j , for j = 1, . . . , k − 2. nj+1
mk−1 = dk = nk−1 .
(1) (2)
Then, we are able to deal with the matrices Jpi = Ink−i+1 ⊗ Lpi ,
(3)
of order nk−i and also with the matrices Hpi = Ir ⊗ Jpi = Ir ⊗ Ink−i+1 ⊗ Lpi , In the sequel, Cj ( 1 ≤ j ≤ k − 1) is the block diagonal matrix defined by Cj = Inj+1 ⊗ emj ,
(4)
with nj+1 diagonal blocks equals to emj . Thus, Cj is an nj × nj+1 matrix, for j = 1, . . . , k − 1, where Ck−1 = enk−1 . Let Kj = Ir ⊗ Cj ,
(5)
the matrix of order rnj , where each of its r diagonal blocks is equal to Cj , that is, since Cj is composed by nj+1 diagonal blocks equal to emj , in total the matrix Kj has rnj+1 diagonal blocks emj . As direct consequence of the properties of the Kronecker product, we have the following equalities: T KjT Kj = mj Irnj+1 , for j = 1, . . . , k − 2, Kk−1 Kk−1 = nk−1 Ir = dk Ir , T T Kj cIrnj − Hpk−j = cKj for j = 1, . . . , k − 1,
where c is an arbitrary polynomial.
(6) (7)
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Example 2 Let us consider a generalized Bethe tree Bk and the graph Tr (d). Then, in the particular case of k = 4, in B4 we have the following pairs (di , ni ). (d1 , n1 ) = (1, 12), (d2 , n2 ) = (3, 6), (d3 , n3 ) = (4, 2), (d4 , n4 ) = (2, 1). In B4 [2], at level 3, we have n3 = 2 paths with p2 = n2 /n3 = 6/2 = 3 vertices and also m1 = 2, m2 = 3 and m3 = 2. Therefore, J3 = Jp2 = I2 ⊗ L3 . Furthermore, Hp2 = I3 ⊗ Jp2 = I3 ⊗ I2 ⊗ L3 = I6 ⊗ L3 and the matrices defined in (4) are C1 = I6 ⊗ e2 , C2 = I2 ⊗ e3 and C3 = e2 . Taking into account the above referred labeling for the vertices, the Laplacian matrices L(3B4 [2] · T3 (2)) and L(3B4 [2] · P3 ) become: I36 −K1 0 0 −K1T 3I18 + Hp −K2 0 , 2 T 0 −K2 4I6 −K3 0 0 −K3T W where in the case of the matrix L(3B4 [2] · T3 (2)), W = (2 + 2)I3 − A(T3 (2)), and in the case of the matrix L(3B4 [2] · P3 ), W = 2I3 + L3 . In the general case, the Laplacian matrices L(rBk [i]·Tr (d)) and L(rBk [i]· Pr ) become: d1 Irn1 −K1 −K1T d2 Irn2 . . . .. .. . . −Kk−(i+1) (8) dk−i Irnk−i + Hpi −Kk−i . . T .. .. −K −K k−1 k−i .. T . −Kk−1 W where in the case of the matrix L(rBk [i] · Tr (d)), W = (dk + d)Ir − A(Tr (d)) and in the case of the matrix L(rBk [i] · Pr ), W = dk Ir + Lr .
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Lemma 3 Let M be the following block tridiagonal matrix α1 Irn1 K1 K1T α2 Irn2 K2 ... ... Kk−(i+1) T Kk−i Kk−(i+1) αk−i Irnk−i − Hpi , . T . . αk−i+1 Irnk−i+1 Kk−i . .. Kk−1 αk−1 Irnk−1 .. T . Kk−1 αk Ir + L where L is a r × r symmetric matrix. If β1 = α1 6= 0 1 6= 0, βj = αj − mj−1 βj−1
2 ≤ j ≤ k − 1,
1 βk = αk − dk βk−1 6= 0,
then r
Y n k−1 det(M ) = βj j j=1 j6=k−i
pi Y l=1
!rnk−i+1 (βk−i − νl )
r Y
! (βk + δj ) ,
(9)
j=1
where δ1 , . . . , δr are the eigenvalues of L and ν1 , . . . , νpi are the eigenvalues of Lpi . Proof. For this proof, we take into account the equalities (6) and (7). Then, performing elementary operation over the line blocks of M , the matrices KjT , for j = 1, . . . , k − 1 are eliminated using the blocks βj Irnj , with βj 6= 0, T excluding the matrix Kk−i which is eliminated according to the item (c). Therefore, the matrix M is transformed according to the following items: (a) For j = 1, . . . , k − (i + 2), each diagonal block αj+1 Irnj+1 is replaced by αj+1 Irnj+1 − βj−1 KjT Kj = αj+1 Irnj+1 − mj βj−1 Irnj+1 = βj+1 Irnj+1 . (b) For j = k −(i+1), the block αk−i Irnk−i −Hpi is replaced by βk−i Irnk−i − Hpi .
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(c) For j = k − i the block βk−i Irnk−i − Hpi is multiplied on left by −1 T T −βk−i Kk−i and added to the block Kk−i . Then this block is eliminated taking into account (7). Additionally, taking into account (6), the corresponding diagonal block is replaced by −1 T αk−i+1 Irnk−i+1 − βk−i Kk−i Kk−i = βk−i+1 Irnk−i+1 .
(d) For j = k − (i − 1), . . . , k − 1, the block Kj is eliminated by the block βj Irnj and the corresponding diagonal block is replaced by βj+1 Irnj+1 . Observe that after the last operation of item (d), αk is replaced by αk − −1 dk βk−1 = βk . Hence Eq. (9) follows.
3
The spectrum of the Laplacian matrix
The technics used in the proof of the next results are similar to those used in [19, Th. 5]. In what follows, γ1 , . . . , γr are the eigenvalues of Tr (d) and , for j = 1, . . . , r, are the eigenvalues of Lr . Furthermore, µj = 2 + 2 cos πj r considering Φ = {0, 1, 2, . . . , k − i − 1, k − i + 1, . . . , k − 1}, then Ω = {j ∈ Φ − {0} : nj > nj+1 } ∪ {0} .
(10)
Along this section we need also to define the following polynomials (where mj is as (1)): Q0 (λ) Q1 (λ) Qj (λ) Qkj (λ) Lkj (λ)
= = = = =
1, (11) λ − d1 (12) (λ − dj )Qj−1 (λ) − mj−1 Qj−2 (λ) , j = 2, . . . , k − 1, (13) (λ − dk − d + γj ) Qk−1 (λ) − mk−1 Qk−2 (λ) , j = 1, . . . , r,(14) (λ − dk − µj ) Qk−1 (λ) − mk−1 Qk−2 (λ) , j = 1, . . . , r. (15)
Now we are able to introduce the main result of this section. Theorem 4 If H is a graph belonging to the set {rBk [i] · Tr (d), rBk [i] · Pr }, with i ∈ {1, . . . , k − 1}, then pi −1
det(λI − L(H)) =
Y t∈Ω
rn −rn Qt t t+1
Y l=1
(Qk−i − νl Qk−i−1 )rnk−i+1 Ψ(H),
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Q Q where Ψ(H) = rj=1 Qkj (λ) when H = rBk [i]·Tr (d) and Ψ(H) = rj=1 Lkj (λ) when H = rBk [i] · Pr . Furthermore, pi −1
σ(L(H)) =
[
{λ : Qt (λ) = 0} ∪
t∈Ω
[
! {λ : Qk−i (λ) − νl Qk−i−1 (λ) = 0}
∪X
l=1
Sr where X = j=1 {λ : Qkj (λ) = 0} when H = rBk [i] · Tr (d), and X = Sr j=1 {λ : Lkj (λ) = 0} when H = rBk [i] · Pr . Proof. For H ∈ {rBk [i] · Tr (d), rBk [i] · Pr }, taking into account the expression of L (H) in (8) and applying Lemma 3 to the matrix M = λI − L (H), we obtain αj = λ − dj , j = 1, . . . , k − 1, λ − d − dk , if H = rBk [i] · Tr (d); αk = λ − dk , if H = rBk [i] · Pr . Let βj be as in Lemma 3 and suppose that λ ∈ R is such that Qj (λ) 6= 0, when j = 1, . . . , k − 1. For the sake of simplicity, let us denote Qj (λ) and Qkj (λ) by Qj and Qkj , respectively. Then β1 = λ − 1 =
Q1 Q0
6= 0,
1 = βj = (λ − dj ) − mj−1 βj−1
= and, for i = 1, . . . , r, ( βk + δi = ( = ( =
Qj Qj−1
(λ−d3 )Qj−1 −mj−1 Qj−2 Qj−1
6= 0, for j = 2, . . . , k − 1,
1 λ − d − dk + γi − dk βk−1 , if H = rBk [i] · Tr (d); 1 λ − dk + µi − dk βk−1 , if H = rBk [i] · Pr (λ−d−dk +γj )Qk−1 −dk Qk−2 , Qk−1 (λ−dk +µi )Qk−1 −dk Qk−2 , Qk−1 Qki , Qk−1 Lki , Qk−1
if H = rBk [i] · Tr (d); if H = rBk [i] · Pr
if H = rBk [i] · Tr (d); if H = rBk [i] · Pr .
Therefore, using (9), it follows
(16)
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r
det M
Y nj k−1 = βj j=1 j6=k−i
= =
=
n Q1 1 n1 Q0
n Q2 2 n Q1 2
pi Y
!rnk−i+1 (βk−i − νl )
nk−i−1
···
nk−i−1 Qk−i−2
(βk + δj )
j=1
l=1 Qk−i−1
r Y
r
pi Y
Qk−i ( Qk−i−1
!rnk−i+1 − νl )
nk−1
nk−i+1
Qk−i+1
nk−i+1 Qk−i
l=1 r(nk−i−1 −nk−i) r(n1 −n2 ) Q1 · · · Qk−i−1 nk−i+1 pY i −1 nk−1 r Qk−1 Qk−i+1 rnk−i+1 rnk−i+1 (Qk−i − νl Qk−i−1 ) Qk−i nk−i+1 · · · nk−1 Qk−i Qk−2 l=1 r(nk−1 −nk ) r(nk−i−1 −nk−i ) r(nk−i+1 −nk−i+2 ) r(n −n ) · · · Qk−1 Q1 1 2 · · · Qk−i−1 Qk−i+1 pY i −1 (Qk−i − νl Qk−i−1 )rnk−i+1 Ψ(H). l=1
...
Qk−1
r
nk−1 Qk−2
Ψ(H) Qrk−1
Ψ(H) Qrk−1
Q Q where Ψ(rBk [i] · Tr (d)) = rj=1 Qkj and Ψ(rBk [i] · Pr ) = rj=1 Lkj . Notice that the above expression is obtained taking into account that (according to its definition) νpi = 0. Consider now λ0 ∈ R such that Qt (λ0 ) = 0 for some t = 1, . . . , k −1. Since the zeros of any polynomial are isolated there exists a neighborhood V (λ0 ) of λ0 such that Qj (λ) 6= 0, for all λ ∈ V (λ0 )−{λ0 } and for all j = 1, . . . , k −1. Hence, the equality (17) follows, for all λ ∈ V (λ0 ) − {λ0 } . By continuity, taking the limit as λ tends to λ0 , we obtain the desired expression. Corollary 5 If H ∈ {rBk [k − 1] · Tr (d), rBk [k − 1] · Pr }, with k ≥ 2, then for all 1 ≤ l ≤ pk−1 − 1, µl = 3 + 2 cos lπ is an eigenvalue with multiplicity pi at least rn2 of H. Proof. Taking into at level k ≥ 2 there are n2 ≥ 1 paths with pk−1 = nn12 > 1 vertices, then by Theorem 4, we obtain {λ : Q1 (λ) − νl Q0 (λ) = 0} ⊆ σ(H). Hence since Q0 (λ) = 1, using (12), the result follows. Definition 6 For t = 1, . . . , k − 1, let Tt (H) be the t × t leading principal
(17)
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submatrix of the k × k symmetric tridiagonal matrix √ d d − 1 1 2 √ d2 − 1 d2 .. . Tkj (H) = p . . .. .. d − 1 k−1 p √ dk−1 − 1 d√k−1 dk dk dk + θj d − γj , if H = rBk [k − 1] · Tr (d); where, for j = 1, . . . , r, θj = 2 + 2 cos πj , if H = rBk [k − 1] · Pr . r and for l = 1, . . . , pi −1, let consider the dk−i ×dk−i symmetric tridiagonal matrix √ d2 − 1 d1 .. √ . d2 d2 − 1 Sl = . p . . .. .. d − 1 k−i p dk−i − 1 dk−i + νl Remark 7 We remark that σ(L(Tr (d))) = {d − γj : 1 ≤ j ≤ r}. Lemma 8 If Tt , t = 1, . . . , k − 1, and Sl , l = 1, . . . , pi − 1 are the leading principal submatrices of Tkj (H) and the dk−i × dk−i tridiagonal symmetric matrices referred in Definition 6, respectively. Then det(λI − Tt ) = Qt (λ) , t = 1, . . . , k − 1,
(18)
det (λI − Tkj (H)) = Qkj (λ)
(19)
det(λI − Sl ) = Qk−i (λ) − νl Qk−i−1 (λ).
(20)
for j = 1, . . . , r and for l = 1, . . . , pi − 1
Proof. Formulas in (18) and in (19) have already been proved in [19]. To obtain (20) , we see that if i = k − 1 the result follows by Corollary 5. Now we suppose that i < k − 1. By (13) we see that Qk−i − νl Qk−i−1 = (λ − dk−i − νl )Qk−i−1 − mk−i−1 Qk−i−2 = (λ − dk−i − νl )Qk−i−1 − (dk−i − 1) Qk−i−2 , which, by [20] implies the result.
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Lemma 9 For t = 1, . . . , k − 1 , l = 1, . . . , pi − 1 and j = 1, . . . , r the zeros of the polynomials Qt , Qk−i − νl Qk−i−1 and Qkj (or Lkj ), respectively are real and simple. Proof. In view of Lemma 8 and as the matrices in Definition 6 are symmetric and tridiagonal with nonzero codiagonal entries it follows that its eigenvalues are simple (cf. [4]). The next theorem gives a complete characterization of the eigenvalues of the Laplacian matrix in the case when H = rBk [i] · Tr (d) or H = rBk [i] · Pr . i hS S S r pi −1 σ(T (H)) . σ(S ))∪ σ (T ) Theorem 10 (a) σ (H) = ∪( kj l t j=1 l=1 t∈Ω−{0} (b) The multiplicity of each eigenvalue of the matrix Tt as eigenvalue of L(H) is at least rnt − rnt+1 for t ∈ Ω and the multiplicity of each eigenvalue of the matrices Tkj (H) for j = 1, . . . , r is at least 1. (c) Each eigenvalue of Sl , for all 1 ≤ l ≤ pi − 1, is an eigenvalue of L(H). (d) Each eigenvalue of Sl is an eigenvalue of L (H) with multiplicity at least rnk−i+1 . Proof. (a), (b) and (d) are consequences of Theorem 4, Lemma 9 and Lemma 8. The assertion in (c) is due to nk−i+1 ≥ 1. Next we present an illustrative example for the case H = 3B4 [2] · T3 (2) . Example 11 In the graph 3B4 [2] · T3 (2) let Φ = {0, 1, 3} and Ω = {0, 1, 3} .
(21)
By Remark 7 we have {d − γj : 1 ≤ j ≤ 3} = {0, 3, 3} . Thus the matrices Definition 6 correspond to, √ √ 2 0 0 1 2 0 0 1 √ √ √ √ √1 2 3 3 √0 2 √3 3 √0 2 √ , T42 = , T43 = T41 = 0 3 √4 2 0 3 √4 2 0 0 0 2 5 0 0 2 5 0 and
S1 =
√ √ 2 1 2 √1 √ , S2 = . 2 3+3 2 3+1
in √
2 √0 0 3 √0 , √3 3 √4 2 0 2 2
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To four decimal places, the eigenvalues and its multiplicities are given by: T41 : T42 : T43 : T1 : T3 : S1 : S2 :
0.0496 0.0496 0 1 0.0746 0.6277 0.4384
2.1456 4.3968 6.4080 2.1456 4.3968 6.4080 1.2087 3 5.7913 2.4481 5.4774 6.3723 4.5616
1 1 1 rn1 − rn2 = 18 rn3 − rn4 = 3 rn3 = 6 rn3 = 6.
The referred graph operations with modified generalized Bethe trees can be generalized in the following way: for a fixed 1 ≤ s ≤ k − 1, let ˆı = (i1 , i2 , . . . , is ) with 2 ≤ i1 < i2 < · · · < is ≤ k −1. We construct an unweighed graph Bk [ˆı] obtained from a generalized Bethe tree Bk and the union of paths at the level iq + 1 such that two vertices are in the same path if and only if both are connected at the same vertex at level iq , with 1 ≤ q ≤ s. A more general graph operation rBk [ˆı] · Tr (d) obtained from r copies of Bk [ˆı] and a regular connected graph Tr (d) (or a path Pr ) with r vertices, each one of degree d, can be considered by identifying the root of each copy of Bk [ˆı] with each vertex of Tr (d) (Pr ). The previous obtained results can be extended to the graphs produced by these more general graph operations, using techniques very similar to the ones above presented.
4
Some applications
In this section the singular values of the Laplacian matrices (and then the eigenvalues) of the graphs rBk [i] · Tr (d) and H = rBk [i] · Pr are compared, when Pr is isomorphic to an Hamiltonian path of Tr (d) (if there is). First, it is convenient to introduce the following notation (c.f. [1, 2]). A > 0 means that A is positive semidefinite and if A and B are hermitian matrices, then A > B if A − B > 0. For a complex m × n matrix C we recall that its nonzero singular values correspond to the nonzero eigenvalues of the positive 1/2 semidefinite matrix |C| = C T C , (see, e. g. [11]). The singular values of a matrix C are enumerated as s1 (C) ≥ · · · ≥ sn (C). Now, we are able to remind the following results (see [2, 1]). Lemma 12 [2] Let X and Y be positive semidefite n × n hermitian matrices, with eigenvalues λ1 (X) ≥ · · · ≥ λn (X) and λ1 (Y ) ≥ · · · ≥ λn (Y ), respectively. If X > Y , then λj (X) ≥ λj (Y ) for j = 1, . . . , n.
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Theorem 13 Consider a modified generalized Bethe tree Bk [i]. If a d-regular graph of order r, Tr (d), has an Hamilton path isomorphic to Pr , then λj (L(rBk [i] · Tr (d))) ≥ λj (L(rBk [i] · Pr )), for j = 1, . . . , n. Proof. The Laplacian matrices of the graphs rBk [i] · Tr (d) and rBk [i] · Pr are positive semidefinite. Taking into account the expression (8) for these matrices, then 0 0 L(rBk [i] · Tr (d))) − L(rBk [i] · Pr ) = . 0 L(Tr (d)) − L(Pr ) Therefore, since L(Tr (d)) − L(Pr ) = L(Tr (d) \ Pr ), where Tr (d) \ Pr is the graph obtained from Tr (d) after deleting the edges corresponding to Pr , it follows that L(rBk [i] · Tr (d))) − L(rBk [i] · Pr ) > 0 ⇔ L(rBk [i] · Tr (d))) > L(rBk [i] · Pr ). Applying Lemma 12, the conclusion follows. The Laplacian-energy-like of an (n, m)−graph G (denoted by LEL [G]), introduced by Liu and Liu [12] is defined as LEL [G] =
n X √
µj .
(22)
j=1
where µj , for j = 1, . . . , n, are the eigenvalues of the Laplacian matrix of G. For a m × n complex matrix C, we define the Laplacian-energy-like of the matrix C by Xq λj (|C|). LEL(C) = j
As a consequence of the previous definitions the Laplacian-energy-like of any graph G is given by LEL [G] = LEL(L(G)). The next theorem obtains bounds for the Laplacian-energy-like for the graphs studied in this paper.
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Theorem 14 Let G be an (n, m)−graph such that det(L(G) − λIn ) = P1 (λ)t1 P2 (λ)t2 · · · Ps (λ)ts ,
(23)
where Pj (λ) = det(Mj − λIj ), Mj are square matrices of order j, with tj > 0, for j = 1, . . . , s. Then s X p q LEL [G] ≤ tj j trace(Mj ) (24) j=1
Proof. From the decomposition in (23) it is clear that LEL [G] = =
s X j=1 s X j=1
tj LEL (Mj ) X √ µ
tj
µ∈σ(Mj )
Thus, via Cauchy Schwarz inequality 1/2 s X X √ 2 p LEL [G] ≤ tj j ( µ) j=1
=
s X
µ∈σ(Mj )
p q tj j trace (Mj ),
j=1
and the inequality holds. We fix i, 2 ≤ i ≤ k − 2. Now we consider the connected graph H ∈ {Bk [i] · Tr (d) , Bk [i] · Pr }. We shall take n and m as the number of vertices and of edges, respectively, of H. Applying Theorem 4, Lemma 8 and Theorem 14 we obtain the following result. Corollary 15 For 1 ≤ j ≤ r, 1 ≤ u ≤ k − 1 and 1 ≤ l ≤ pi − 1. Let Tkj (H), Tu and Sl defined according to the Definition 6. Then LEL (H) ≤ X√ u∈Ω
u(rnu −rnu+1 )
pi −1 r Xp X p √ q p traceTu +rnk−i+1 dk−i traceSl + k traceTkj (H). l=1
j=1
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Finally, as immediate consequence of Theorem 13, we have the following corollary. Corollary 16 Under the conditions of Theorem 13, LEL(rBk [i] · Tr (d)) ≥ LEL(rBk [i] · Pr ). Acknowledgements: (1) Research supported by the Center for Research and Development in Mathematics and Applications from the Funda¸c˜ao para a Ciˆencia e a Tecnologia, cofinanced by the European Community Fund FEDER/POCI 2010. (2) Research supported by Proyecto Mecesup 2 UCN 0605, Chile y FondecytIC Project 11090211, Chile. (3) Research partially supported by CNPq - Grant 309531/2009-8.
References [1] R. Bhatia, F. Kitaneh, The singular valuies of A + B and A + iB. Linear Algebra Appl. 431 (20019): 1502-1508. [2] R. Bhatia, Matrix Analysis. Springer, 1997. [3] D. Cvetkovi´c, M. Doob, H. Sachs. Spectra of Graphs - Theory and Applications, Deutcher Verlag der Wissenschaften, Academic Press, BerlinNew York (1980), second ed. (1980), third ed. Verlag, Heidelberg Leipzig (1995). [4] G. H. Golub, C. F. Van Loan. Matrix Computations, second ed., Johns Hopkins University Press, Baltimore, 1989. [5] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978): 1-22. [6] I. Gutman, The energy of a Graph: Old and New Results, Algebraic Combinatorics and Applications (2001): 196-211. Springer-Verlag. Berlin. [7] I. Gutman, S. Zare Firoozabadi, J. A. de la Pe˜ na, J. Rada, On the energy of regular graphs, MATCH Commun. Math. Comput. Chem. 57 (2007): 435-442.
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[8] O. J. Heilmann, E. H. Lieb. Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972): 190-232. [9] R. A. Horn, C. R. Johnson. Matrix Analysis. Cambridge University Press (1991). Cambridge. [10] G. Indulal, A. Vijayakumar, A note on energy of some graphs, MATCH Commun. Math. Comput. Chem. 59 (2008): 269-274. [11] P. Lancaster, M.Tismenetsky. The theory of matrices. Academic Press. INC. [12] J. Liu, B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 v. 2 (2008): 355-372. [13] V. Nikiforov, The energy of graphs and matrices, Journal of Mathematical Analysis and Applications 326, Issue 2 (2007): 1472-1475. [14] O. Rojo. Spectra of copies of a generalized Bethe tree attached to any graph. Linear Algebra and its Appl. 431 (2009): 862-882. [15] O. Rojo, L. Medina. Spectra of generalized Bethe trees attached to a path. Linear Algebra and its Appl. 430 (2009): 483-503. [16] O. Rojo. The spectra of a graph obtained from copies of a generalized Bethe tree. Linear Algebra and its Appl. 420, I. 2-3 15 (2007): 490-507. [17] O. Rojo. The spectra of some trees and bounds for the largest eigenvalues of any tree. Linear Algebra Appl. 414 (2006): 199-217. [18] O. Rojo and R. Soto. The spectra of the adjacency matrix and Laplacian matrix for some balanced trees. Linear Algebra Appl. 403 (2005): 97117. [19] O. Rojo and M. Robbiano. On the spectra of some weighted rooted tree and applications, Linear Algebra Appl. 420 (2007): 310-328. [20] L. N. Trefethen. D. Bau III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics, 1997. [21] R. S. Varga. Matrix Iterative Analysis. Prentice-Hall, Inc., 1965.
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