Available online at www.sciencedirect.com
Linear Algebra and its Applications 430 (2009) 483–503
www.elsevier.com/locate/laa
Spectra of generalized Bethe trees attached to a path聻
Oscar Rojo ∗ , Luis Medina
Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile
Received 29 May 2008; accepted 12 August 2008
Available online 21 September 2008
Submitted by R.A. Brualdi
Abstract
A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the
same degree. Let Pm be a path of m vertices. Let {Bi : 1 i m} be a set of generalized Bethe trees.
Let Pm {Bi : 1 i m} be the tree obtained from Pm and the trees B1 , B2 , . . . , Bm by identifying the
root vertex of Bi with the ith vertex of Pm . We give a complete characterization of the eigenvalues of the
Laplacian and adjacency matrices of Pm {Bi : 1 i m}. In particular, we characterize their spectral radii
and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply
the results to the case B1 = B2 = · · · = Bm .
© 2008 Elsevier Inc. All rights reserved.
AMS classification: 5C50; 15A48
Keywords: Tree; Bethe tree; Generalized Bethe tree; Laplacian matrix; Adjacency matrix; Algebraic connectivity
1. 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 and D(G) is the diagonal matrix of
vertex degrees. It is easy to see that L(G) is singular and positive semidefinite. Fiedler [1] proved
that G is a connected graph if and only if the second smallest eigenvalue of L(G) is positive. This
eigenvalue is called the algebraic connectivity of G and it is denoted by a(G).
聻
∗
Work supported by Project Fondecyt 1070537 and Project Mecesup UCN0202, Chile.
Corresponding author.
E-mail address:
[email protected] (O. Rojo).
0024-3795/$ - see front matter ( 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.laa.2008.08.009
484
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
A bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such
that every edge connects a vertex in U to one in V . A tree is a connected acyclic graph. Every
tree is a bipartite graph.
Let B(G) = D(G) + A(G).
Lemma 1 [3]. If G is a bipartite graph then B(G) and L(G) are unitarily similar.
A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root
have the same degree. Throughout this paper {Bi : 1 i m} is a set of generalized Bethe
trees.
In [5], we give a complete characterization of the spectra of the Laplacian and adjacency
matrices of a tree obtained from the union of the trees Bi joined at their respective root vertices.
Let Pm be a path of m vertices. In this paper, we study the tree Pm {Bi : 1 i m} obtained
from Pm and B1 , B2 , . . . , Bm by identifying the root vertex of Bi with the ith vertex of Pm . For
brevity, we write Pm {Bi } instead of Pm {Bi : 1 i m}. We obtain a complete characterization
of the eigenvalues of the Laplacian and adjacency matrices of Pm {Bi } together with results
concerning their multiplicities. In particular, we characterize the spectral radii and the algebraic
conectivity.
From Lemma 1, it follows that L(Pm {Bi }) and B(Pm {Bi }) have the same eigenvalues. Then
we consider B(Pm {Bi }) instead of L(Pm {Bi }).
For j = 1, 2, 3, . . . , ki , let di,ki −j +1 and ni,ki −j +1 be the degree of the vertices and the number
of them at the level j of Bi . Observe that ni,ki = 1 and ni,ki −1 = di,ki . We have
ni,ki −j = (di,ki −j +1 − 1)ni,ki −j +1 , 2 j ki − 1
(1)
m ki −1
The total number of vertices in Pm {Bi } is n = i=1 j =1 ni,j + m.
We introduce the following additional notation.
|A| is the determinant of A.
0 and I are the all zeros matrix and the identity matrix of the appropiate order, respectively.
Ir is the identity matrix of order r × r.
er is the all ones column vector of dimension r.
n
For 1 i m and 1 j ki − 2, mi,j = ni,ji,j+1 and Ci,j is the block diagonal matrix defined
by
Ci,j = diag{emi,j , emi,j , . . . , emi,j }
with ni,j +1 diagonal. The order of Ci,j is ni,j × ni,j +1 .
−2
For 1 i m, let si = kji=1
ni,j and Ei be the matrix of order si × m defined by
1 if q = i and si + 1 p si + ni,ki −1 ,
Ei (p, q) =
0 elsewhere.
(2)
(3)
We label the vertices of Pm {Bi } as follows:
−1
1. Using the labels 1, 2, . . . , jk1=1
n1,j , we label the vertices of B1 from the bottom to level
2 and, at each level, from the left to the right.
−1
−1
−1
2. Using the labels jk1=1
n1,j + 1, . . . , jk1=1
n1,j + kj2=1
n2,j , we label the vertices of
B2 from the bottom to level 2 and, at each level, from the left to the right.
3. We continue labelling the vertices of B3 , B4 , . . . , Bm , in this order, from the bottom to
level 2 and, at each level, from the left to the right.
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
485
4. Finally, using the labels n − m + 1, n − m + 2, . . . , n, we label the vertices of Pm from
the left to the right.
Example 1. This example illustrates our vertex labelling and the notation introduced above.
35
9
5
10
6
1
36
7
2 3
37
17
18
19
11
12 13
14 15
28
26
8
4
38
33
34
27
29
16
20 21 22 23 24
30
31
32
25
This is a tree obtained from the path P4 and four generalized Bethe trees by identifying the
roots of them with the vertices of P4 . We have k1 = 4, k2 = 3, k3 = 4, k4 = 3 and
n1,1 = 4, d1,1 = 1, n1,2 = 4, d1,2 = 2, n1,3 = 2, d1,3 = 3
n2,1 = 6, d2,1 = 1, n2,2 = 3, d2,2 = 3
n3,1 = 6, d3,1 = 1, n3,2 = 2, d3,2 = 4, n3,3 = 1, d3,3 = 3
n4,1 = 4, d4,1 = 1, n4,2 = 2, d4,2 = 3.
The degree of the vertices 35, 36, 37 and 38 is d1,4 + 1 = 3, d2,3 + 2 = 5, d3,4 + 2 = 3 and
d4,3 + 1 = 3, respectively. The total number of vertices is n = 38. The matrices defined in (2) are
C1,1 = diag{e1 , e1 , e1 , e1 } = I4 ,
C1,2 = diag{e2 , e2 }
C2,1 = diag{e2 , e2 , e2 }
C3,1 = diag{e3 , e3 },
C3,2 = e2
C4,1 = diag{e2 , e2 }.
The nonzero entries of the matrices Ei defined in (3) are
E1 (9, 1) = E1 (10, 1) = 1,
E3 (9, 3) = 1,
E2 (7, 2) = E2 (8, 2) = E2 (9, 2) = 1
E4 (5, 4) = E4 (6, 4) = 1.
For the above mentioned labelling, the adjacency matrix A(Pm {Bi }) and the matrix B(Pm {Bi })
become
⎡
⎤
A1
0
···
0
E1
⎢
⎥
..
⎢
⎥
.
E2 ⎥
A2
⎢0
⎢
⎥
.
.. ⎥
..
..
A(Pm {Bi }) = ⎢
(4)
⎢ ..
.
.
0
. ⎥
⎥
⎢
⎥
⎢
0
Am
Em ⎦
⎣0
E1T
and
E2T
···
T
Em
Am+1
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
⎡
0
B1
⎢
⎢
⎢0
⎢
.
B(Pm (Bi )) = ⎢
⎢ ..
⎢
⎢
⎣0
E1T
B2
..
.
..
.
0
E2T
···
⎤
0
E1
0
Bm
E2
..
.
Em
T
Em
Bm+1
···
..
.
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎦
(5)
where, for i = 1, 2, . . . , m, the matrices Ai and Bi are the following block tridiagonal
matrices:
⎡
0
⎢C T
⎢ i,1
⎢
⎢
Ai = ⎢
⎢
⎢
⎢
⎣
⎡
0
T
Ci,2
Ci,2
..
.
..
.
..
..
.
.
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
Ci,ki −2 ⎦
0
T
Ci,k
i −2
Ini,1
⎢C T
⎢ i,1
⎢
⎢
Bi = ⎢
⎢
⎢
⎢
⎣
⎤
Ci,1
Ci,1
di,2 Ini,2
T
Ci,2
Ci,2
..
.
..
.
..
..
.
.
T
Ci,k
i −2
Ci,ki −2
di,ki −1 Ii,nki −1
(6)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(7)
and
⎡
Am+1
0
⎢1
⎢
⎢
⎢
=⎢
⎢
⎢
⎢
⎣
⎡
Bm+1
1
0
1
1
..
.
..
.
..
..
.
d1,k1 + 1
⎢
1
⎢
⎢
⎢
=⎢
⎢
⎢
⎢
⎣
both of order m × m.
⎤
.
1
1
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
1⎦
0
d2,k2 + 2
1
(8)
1
..
.
..
.
⎤
..
.
dm−1,km−1 + 2
1
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
1
⎦
dm,km + 1
(9)
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
487
2. Preliminaries
Lemma 2. Let
⎡
X1
⎢
⎢ 0
⎢
.
X=⎢
⎢ ..
⎢
⎣ 0
−E1T
0
X2
..
.
−E2T
···
..
.
..
.
0
···
0
0
Xm
T
−Em
−E1
⎤
⎥
−E2 ⎥
⎥
.. ⎥ ,
. ⎥
⎥
−Em ⎦
Xm+1
where, for i = 1, 2, . . . , m, Xi is the block tridiagonal matrix
⎡
⎤
−Ci,1
αi,1 Ini,1
T
⎢ −Ci,1
⎥
αi,2 Ini ,2 −Ci,2
⎢
⎥
⎢
⎥
..
..
T
⎥
.
.
−C
Xi = ⎢
i,2
⎢
⎥
⎢
⎥
.
.
..
..
⎣
−Ci,ki −2 ⎦
T
−Ci,k
αi,ki −1 Ii,nki −1
i −2
and
⎡
Xm+1
α1
⎢−1
⎢
⎢
=⎢
⎢
⎢
⎣
−1
α2
−1
−1
..
.
..
..
..
.
⎤
.
.
−1
For i = 1, 2, . . . , m, let
⎥
⎥
⎥
⎥.
⎥
⎥
−1⎦
αm
βi,1 = αi,1 ,
ni,j −1 1
,
ni,j βi,j −1
1
.
βi = αi − ni,ki −1
βi,ki −1
j = 2, 3, . . . , ki − 1,
βi,j = αi,j −
If βi,j =
/ 0 for all i = 1, 2, . . . , m and all j = 1, 2, . . . , ki − 1 then
⎞
⎛
|X| = ⎝
where
Ym+1
m ki −1
i=1 j =1
⎡
β1
⎢−1
⎢
⎢
=⎢
⎢
⎢
⎣
n
βi,ji,j ⎠ |Ym+1 |,
−1
β2
−1
−1
..
.
..
..
..
.
(10)
⎤
.
.
−1
⎥
⎥
⎥
⎥.
⎥
⎥
−1⎦
βm
(11)
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Proof. Suppose β1,j =
/ 0 for all j = 1, 2, . . . , k1 − 1. After some steps of the Gaussian elimination procedure, without row interchanges, we reduce X to the intermediate matrix given
below:
⎤
⎡
0
···
0
−E1
R1
⎥
⎢
..
⎢0
.
−E2 ⎥
X2
⎥
⎢
.. ⎥ ,
⎢ ..
..
..
⎢ .
.
.
. ⎥
0
⎥
⎢
⎣0
0
Xm
−Em ⎦
(1)
0
−E2T · · · −E2T Xm+1
where R1 is the upper triangular matrix
⎡
β1,1 In1,1
C1,1
⎢
..
⎢
.
β1,2 In1,2
⎢
R1 = ⎢
.
..
⎣
⎤
C1,k1 −2
β1,k1 −1 In1,k1 −1
(1)
and Xm+1 is the tridiagonal matrix
⎡
β1 −1
⎢−1 α2 −1
⎢
⎢
..
..
(1)
.
.
−1
Xm+1 = ⎢
⎢
⎢
.
.
..
..
⎣
−1
⎥
⎥
⎥
⎥
⎦
⎤
⎥
⎥
⎥
⎥.
⎥
⎥
−1⎦
αm
Suppose, in addition, βi,j =
/ 0 for all i = 2, 3, . . . , m and all j = 1, 2, . . . , ki − 1. We continue
the Gaussian elimination procedure to finally obtain the upper triangular matrix
⎡
⎤
R1
0
···
0
−E1
⎢
⎥
..
⎢0
.
−E2 ⎥
R2
⎢
⎥
⎢ .
.. ⎥
..
..
⎢ ..
⎥,
.
.
0
.
⎢
⎥
⎣0
0
Rm −Em ⎦
0
0
···
0
Ym+1
where, for i = 2, 3, . . . , m,
⎡
βi,1 Ini,1
−Ci,1
⎢
⎢
βi,2 Ini,2
Ri = ⎢
⎢
⎣
..
.
..
.
⎤
−Ci,ki −2
βi,ki −1 Ini,ki −1
and Ym+1 is as in (11). Thus, (10) is proved.
⎥
⎥
⎥
⎥
⎦
the submatrix obtained from A by deleting its last row and its
From now on, we denote by A
last column and, for i = 1, 2, . . . , m − 1, by Fi the matrix of order ki × ki+1 whose are 0 except
for the entry Fi (ki , ki+1 ) = 1.
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Lemma 3. Let
⎡
T1
⎢−F T
⎢ 2
⎢
Mm = ⎢
⎢
⎣
−F1
T2
−F2T
−F2
..
.
···
⎤
.
Tm−1
T
−Fm−1
where Ti is a matrix of order ki × ki . Let
|T1 | |T1 |
|T2 | |T2 | |T2 |
..
..
.
.
Dm =
|T3 |
..
. |Tm−1 |
|T
m|
Then
..
489
⎥
⎥
⎥
⎥,
⎥
−Fm−1 ⎦
Tm
.
|T
|
m−1
|Tm |
|M2 | = |D2 |
(12)
and, for all m,
|Mm | = |Dm |.
(13)
Proof. By induction on m. Clearly |M1 | = |D1 |. We may write
⎡
⎤
p
0
0
T1
⎢qT T1 (k1 , k1 ) 0
T1
−F1
−1 ⎥
⎥,
M2 =
=⎢
T
⎣
⎦
−F1
T2
0
0
T2
r
T
0
−1
s
T2 (k2 , k2 )
where p, q, r, s are clear from the context and 0 is the all zeros vector of the appropiate order. We
know that the determinant function is linear transformation on each column (row). We apply this
property on the last column of M2 to obtain
T1
T1
p
0
0
p
0
0
T
T
q
q
T1 (k1 , k1 ) 0
0
T1 (k1 , k1 ) 0 −1
+
|M2 | =
0
T2
r
0
T2
0
0
0
T
T
0
−1
s
T2 (k2 , k2 )
0
−1
s
0
T1
p
0
0
2
T
2k1 +k2
k
2
= |T1 ||T2 | − (−1)
0
T2 = |T1 ||T2 | − (−1) |T1 |
0
−1 sT
0 −1 sT
= |T1 ||T2 | + (−1)k2 (−1)k2 +1 |T1 ||T2 | = |T1 ||T2 | − |T1 ||T2 |.
Thus (12) is proved. Observe that (12) is (13) for m = 2. Let m 2. Suppose that (13) is true for
all integer less or equal to m. We have
−Km
Mm
Mm+1 =
,
T
−Km
Tm+1
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
where Km is the matrix of order (k1 + · · · + km ) × km+1 whose entries are zeros except for the
entry
Km (k1 + · · · + km , km+1 ) = 1.
We apply (12) to Mm+1 obtaining
|Mm |
|M
m|
= |Mm ||Tm+1 | − |T
|Mm+1 | =
m+1 ||M
m|
|Tm+1 | |Tm+1 |
= |Dm ||Tm+1 | − |T
m+1 ||M
m |.
Since
Mm−1
0
M
m =
0
T
m
we have |Mm | = |Mm−1 ||T
m |. Hence
|Mm+1 | = |Dm ||Tm+1 | − |T
m+1 ||Mm−1 ||T
m |.
On the other hand
|T1 |
|T |
|Dm+1 | = 2
|T1 |
..
.
..
.
..
.
|Tm |
|T
m+1 |
(14)
|T
|
m
|Tm+1 |
Dm−1
0
= |Tm+1 ||Dm | − |T
m+1 |
∗
|Tm |
= |Tm+1 ||Dm | − |T
m+1 ||Dm−1 ||T
m |.
From the hypothesis of induction, we have |Dm−1 | = |Mm−1 |. Therefore
|Dm+1 | = |Tm+1 ||Dm | − |T
m+1 ||Mm−1 ||T
m |.
From (14) and (15), we obtain (13) for all m 2.
(15)
3. The spectrum of the Laplacian matrix
Definition 1. For i = 1, 2, . . . , m, let
Pi,0 (λ) = 1, Pi,1 (λ) = λ − di,1
and for j = 2, 3, . . . , ki − 1, let
Pi,j (λ) = (λ − di,j )Pi,j −1 (λ) −
ni,j −1
Pi,j −2 (λ).
ni,j
Moreover, let
P1 (λ) = (λ − d1,k1 − 1)P1,k1 −1 (λ) − n1,k1 −1 P1,k1 −2 (λ),
Pi (λ) = (λ − di,ki − 2)Pi,ki −1 (λ) − ni,ki −1 Pi,ki −2 (λ)
for i = 2, 3, . . . , m − 1, and
Pm (λ) = (λ − dm,km − 1)Pm,km −1 (λ) − nm,km −1 P1,km −2 (λ).
(16)
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
491
For i = 1, 2, . . . , m, let
i = {j : 1 j ni,ki −1 : ni,j > ni,j +1 }.
Theorem 1. We have
(a)
m
|λI − L(Pm {Bi })| = P (λ)
n
Pi,ji,j
−ni,j +1
(17)
(λ)
i=1 j ∈i
where
P1 (λ)
P
(λ)
P (λ) = 2,k2 −1
P1,k1 −1 (λ)
..
.
..
.
..
.
..
.
Pm,km −1 (λ)
(b) The set of eigenvalues of L(Pm {Bi }) is
Pm−1,km−1 −1 (λ)
Pm (λ)
σ (L(Pm {Bi })) = (∪m
i=1 ∪j ∈i {λ : Pi,j (λ) = 0}) ∪ {λ : P (λ) = 0}.
Proof. (a) We consider B(Pm {Bi }) instead of L(Pm {Bi }). We know that B(Pm {Bi }) is given
by (5), (7) and (9). We apply Lemma 2 to the matrix X = λI − B(Pm {Bi }). For this matrix
αi,j = λ − di,j (1 i m, 1 j ki − 1)
α1 = λ − d1,k1
αi = λ − di,ki − 2 (2 i m − 1)
αm = λ − dm,km − 1.
Let βi,j , βi be as in Lemma 2. We first suppose that λ ∈ R is such that Pi,j (λ) =
/ 0 for all i =
1, 2, . . . , m and for all j = 1, 2, . . . , ki − 1. For brevity, we write Pi,j (λ) = Pi,j and P (λ) = P .
We have
Pi,1
βi,1 = λ − di,1 =
=
/ 0,
Pi,0
ni,1 Pi,0
ni,1 1
= (λ − di,2 ) −
,
βi,2 = (λ − di,2 ) −
ni,2 βi,1
ni,2 Pi,1
=
(λ − di,2 )Pi,1 −
ni,1
ni,2 Pi,0
Pi,1
=
Pi,2
=
/ 0,
Pi,1
..
.
βi,ki −1 = (λ − di,ki −1 ) −
=
ni,ki −2 Pi,ki −3
nki −2 1
= (λ − di,ki −1 ) −
nki −1 βi,ki −2
ni,ki −1 Pi,ki −2
(λ − di,ki −1 )Pi,ki −2 −
Pi,ki −2
nki −2
ni,ki −1 Pi,ki −3
=
Pi,ki −1
=
/ 0.
Pi,ki −2
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Moreover
β1 = (λ − d1,k1 − 1) − n1,k1 −1
1
β1,k1 −1
= (λ − d1,k1 − 1) − n1,k1 −1
P1,k1 −2
P1,k1 −1
=
P1
(λ − d1,k1 − 1)P1,k1 −1 − n1,k1 −1 P1,k1 −2
=
,
P1,k1 −1
P1,k1 −1
βm = (λ − dm,km − 1) − nm,km −1
=
βm,k1 −1
= (λ − dm,km − 1) − nm,km −1
Pm,km −2
Pm,km −1
(λ − dm,km − 1)Pm,km −1 − nm,km −1 Pm,km −2
Pm
=
Pm,km −1
Pm,km −1
and for i = 2, . . . , m − 1
βi = (λ − di,ki − 2) − ni,ki −1
=
1
Pi,ki −2
1
= (λ − di,ki − 2) − ni,ki −1
βi,ki −1
Pi,ki −1
(λ − di,ki − 2)Pi,ki −1 − ni,ki −1 Pi,ki −2
Pi
=
.
Pi,ki −1
Pi,ki −1
From (10)
⎛
det(λI − B(Pm {Bi })) = ⎝
=
=
m ki −1
i=1 j =1
n
m
⎞
n
βi,ji,j ⎠ |Cm+1 |
n
n
n
n
Pi,1i,1 Pi,2i,2 Pi,3i,3
n
i=1
m
Pi,0i,1 Pi,1i,2 Pi,2i,3
···
n −n
n −n
Pi,1i,1 i,2 Pi,2i,2 i,3
i=1
ni,k
−2
ni,k
−1
ni,k
−2
ni,k
−1
i
i
Pi,ki −2
Pi,ki −1
i
i
Pi,ki −3
Pi,ki −2
|Cm+1 |
ni,ki −2 −ni,ki −1 ni,ki −1
· · · Pi,ki −2
Pi,ki −1
where
P
1
P1,k1 −1
−1
|Cm+1 | =
−1
P2
P2,k2 −1
−1
1
i=1 Pi,ki −1
= m
−1
..
.
..
.
P1
P2,k −1
2
..
.
..
.
−1
P1,k1 −1
P2
..
.
−1
Pm
P
m,km −1
..
..
.
.
Pm,km −1
.
Pm−1,km−1 −1
Pm
|Cm+1 |
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Hence
m ni,ki −1
|λI − L(Pm {Bi })| = P (λ)
= P (λ)
n
Pi,ji,j
i=1 j =1
m
n
Pi,ji,j
−ni,j +1
−ni,j +1
(λ).
i=1 j ∈i
/ 0, for all i = 1, 2, . . . , m and for all j =
Thus (17) is proved for all λ ∈ R such that Pi,j (λ) =
1, 2, . . . , ki − 1. Now, we consider λ0 ∈ R such that Pl,s (λ0 ) = 0 for some 1 l m and for
1 s kl − 1. Since the zeros of any nonzero polynomial are isolated, there exists a neighborhood N(λ0 ) of λ0 such that Pi,j (λ) =
/ 0 for all λ ∈ N (λ0 ) − {λ0 } and for all i = 1, 2, . . . , m and
for all j = 1, 2, . . . , ki − 1. Hence
m
|λI − B(Pm {Bi })| = P (λ)
n
Pi,ji,j
−ni,j +1
(λ)
i=1 j ∈i
for all λ ∈ N (λ0 ) − {λ0 }. By continuity, taking the limit as λ tends to λ0 we obtain
m
|λ0 I − B(Pm {Bi })| = P (λ0 )
n
Pi,ji,j
−ni,j +1
(λ0 ).
i=1 j ∈i
Therefore (17) holds for all λ ∈ R.
(b) It is an immediate consequence of part (a).
Definition 2. For i = 1, 2, 3, . . . , m, let
⎡
di,2 − 1
d
i,1
⎢ di,2 − 1
di,2
⎢
⎢
..
Ti = ⎢
.
⎢
⎣
di,k −1 − 1
i
⎤
di,ki −1 − 1
di,ki −1
di,ki
di,ki
di,ki + c
⎥
⎥
⎥
⎥
⎥
⎦
of order ki × ki , where c = 2 for i = 2, 3, . . . , m − 1 and c = 1 for i = 1 and i = m. Moreover, for i = 1, 2, . . . , m and for j = 1, 2, 3, . . . , ki − 1, let Ti,j be the j × j leading principal
submatrix of Ti .
Lemma 4. For i = 1, 2, . . . , m and for j = 1, 2, . . . , ki − 1, we have
|λI − Ti,j | = Pi,j (λ).
(18)
Moreover,
|λI − Ti | = Pi (λ)
(19)
for i = 2, 3, . . . , m − 1, and
|λI − T1 | = P1 (λ),
|λI − Tm | = Pm (λ).
(20)
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Proof. It is well known [7, page 229] that the characteristic polynomials, Qj , of the j × j leading
principal submatrix of the k × k symmetric tridiagonal matrix
⎡
c1
⎢b1
⎢
⎢
⎢
⎢
⎢
⎣
b1
c2
..
.
b2
..
.
..
..
.
⎤
⎥
⎥
⎥
⎥
⎥
⎥
bk−1 ⎦
ck
.
ck−1
bk−1
satisfy the three-term recursion formula
Qj (λ) = (λ − cj )Qj −1 (λ) − bj2−1 Qj −2 (λ)
(21)
with
Q0 (λ) = 1 and Q1 (λ) = λ − c1 .
We recall that, for i = 1, 2, . . . , m and for j = 1, 2, . . . , ki , the polynomials
Pi,j are defined
n
i,j
by the recursion formula (16). Let 1 i m be fixed. From (1), ni,j +1 = di,j +1 − 1 for
For the matrix Ti,ki −1 , we have cj = di,j for j = 1, 2, . . . , ki − 1 and
j = 1, 2, . . . , ki − 2.
n
bj = di,j +1 − 1 = ni,ji,j+1 for j = 1, 2, . . . , ki − 2. Replacing in the recursion formula (21),
we get the polynomials Pi,j , j = 1, 2, . . . , ki − 1. Thus (18) is proved. The proof for (19) and
(20) are similar.
Lemma 5. Let r =
⎡
T1
⎢ T
⎢F
1
G=⎢
⎢
⎣
m
i=1 ki .
Let G be the symmetric matrix of order r × r defined by
F1
T2
..
.
..
.
..
.
T
Fm−1
⎤
⎥
⎥
⎥.
⎥
Fm−1 ⎦
Tm
Then
|λI − G| = P (λ).
Proof. We apply Lemma 3 to λI − G to obtain that
|λI − T1 |
|λI
− T2 |
|λI − G| =
|λI
− T1 |
|λI − T2 |
..
.
|λI
− T2 |
..
.
|λI −
Tm−1 |
..
.
|λI − Tm−1 |
|λI
− Tm |
.
|λI −
Tm−1 |
|λI − Tm |
We observe that λI
− Ti = λI − Ti,ki −1 . We now use Lemma 4 to obtain
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
P1 (λ)
P2,k2 −1 (λ)
|λI − G| =
P1,k1 −1 (λ)
P2 (λ)
P3,k3 −1 (λ)
P2,k2 −1 (λ)
..
.
..
.
= P (λ).
The proof is complete.
..
..
.
.
Pm,km −1 (λ)
495
Pm−1,km−1 −1 (λ)
Pm (λ)
Theorem 2. (a)
σ (L(Pm {Bi })) = (∪m
i=1 ∪j ∈i σ (Ti,j )) ∪ σ (G).
(b) The multiplicity of each eigenvalue of the matrix Ti,j , as an eigenvalue of L(Pm {Bi }), is
at least (ni,j − ni,j +1 ) for j ∈ i .
(c) The matrix G is singular.
Proof. It is known that the eigenvalues of a symmetric tridiagonal matrix with nonzero codiagonal
entries are simple [2]. This fact and Theorem 1, Lemma 4 and Lemma 5 imply (a) and (b). One
can easily check that |Ti,j | = 1 for 1 i m and 1 j ki − 1. This fact and part (a) imply
that 0 is an eigenvalue of G.
Example 2. Let P4 {Bi : i = 1, 2, 3, 4} be the tree in Example 1. For this tree
⎡
⎤
√
⎤
⎡
1
1
√
2 √
⎢1
⎥
√1
2
2
√
√ ⎥ , T2 = ⎣ 2
T1 = ⎢
3⎦
⎣
√3
2 √3
2⎦
3
5
2
3
√
⎤
⎡
√
⎡
⎤
3 √
√1
2 √
1
⎥
⎢ 3
√
2
⎥ , T4 = ⎣ 2
√4
T3 = ⎢
2⎦ .
⎣
√3
2
3
1⎦
2
3
1
3
We have
1 = {2, 3},
2 = {1, 2},
3 = {1, 2},
4 = {1, 2}.
From Theorem 2, the eigenvalues of L(Pm {Bi }) are those of
T1,2 , T1,3 , T2,1 , T2,2 , T3,1 , T3,2 , T4,1 , T4,2 and G,
where
⎤
⎡
T1
F1
⎥
⎢F T T2
F2
1
⎥.
G=⎢
T
⎣
F2
T 3 F3 ⎦
F3T T4
Let ρ(A) be the spectral radius of a matrix A.
Lemma 6 [8]. If A is a irreducible nonnegative matrix and B is any principal submatrix of A
then ρ(B) < ρ(A).
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Theorem 3. The spectral radius of L(Pm (Bi )) is the largest eigenvalue of the matrix G.
Proof. From Theorem 2, the eigenvalues of L(Pm (Bi )) are the eigenvalues of the matrices Ti,j
for i ∈ i and 1 j ki − 1 together with the eigenvalues of G. These matrices are irreducible
nonnegative matrices and each Ti,j is a principal submatrix of G. Thus the result follows from
Lemma 6.
Lemma 7 [3, Corollary 4.2]. 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).
be a subtree of the tree T. Then
Corollary 1. Let T
a(T) a(T).
Proof. From Lemma 7, it follows that the algebraic connectivity of a graph do not increase if a
is a subtree T, we can construct T
pendant vertex and its edge are added to the graph. Since T
by successively adding in pendants vertices and edges. Hence, from the above mentioned
from T
fact, we obtain the result.
Theorem 4. The algebraic connectivity of Pm (Bi ) is the smallest positive eigenvalue of G.
Proof. From Theorem 2
σ (L(Pm {Bi })) = ∪m
i=1 ∪j ∈i σ (Ti,j ) ∪ σ (G).
If i = φ for all i then σ (L(Pm {Bi })) = σ (G) and there nothing to prove. Suppose i =
/ φ. Let
⎤
⎡
di,2 − 1
di,1
⎥
⎢ di,2 − 1
di,2
⎥
⎢
⎥
⎢
.
..
Li = ⎢
⎥
di,ki −1 − 1
⎥
⎢
⎣
di,ki −1 − 1
di,ki −1
di,ki ⎦
di,ki
di,ki
of order ki × ki . In [6] we prove
σ (L(Bi )) = ∪j ∈i σ (Ti,j ) ∪ σ (Li ).
(22)
Let j ∈ i and λ ∈ σ (Ti,j ). From (22), λ is a Laplacian eigenvalue of Bi . Then λ a(Bi ). We
now use Corollary 1, to obtain a(Bi ) a(Pm {Bi }). Hence, λ a(Pm {Bi }). This inequality has
been obtained for any i =
/ φ, any j ∈ i and any λ ∈ σ (Ti,j ). Therefore λ a(Pm {Bi }) and
thus the proof is complete.
4. The spectrum of the adjacency matrix
From (4) we have
⎤
···
0
E1
⎥
⎢
..
⎢0
.
A2
E2 ⎥
⎥
⎢
.
.. ⎥ ,
..
..
A(Pm {Bi }) = ⎢
⎥
⎢ ..
.
.
0
.
⎥
⎢
⎣0
0
Am
Em ⎦
T
E1T E2T · · · Em
Am+1
where the diagonal blocks Ai (1 i m) are given by (6) and Am+1 is given by (8).
⎡
A1
0
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
497
We may apply Lemma 2 to X = λI − A(Pm {Bi }). For this matrix αi,j = λ for 1 i m
and 1 j ki − 1, and αi = λ for 1 i m.
Definition 3. For i = 1, 2, . . . , m, let
Qi,0 (λ) = 1,
Qi,1 (λ) = λ
and for j = 2, 3, . . . , ki − 1, let
ni,j −1
Qi,j (λ) = λQi,j −1 (λ) −
Qi,j −2 (λ).
ni,j
Moreover, let
Qi (λ) = λQi,ki −1 (λ) − ni,ki −1 Qi,ki −2 (λ)
for i = 1, 2, . . . , m.
Theorem 5. We have
(a)
m
|λI − A(Pm {Bi })| = Q(λ)
(b)
n
Qi,ji,j
−ni,j +1
(λ).
i=1 j ∈i
σ (A(Pm {Bi })) = ∪m
i=1 ∪j ∈i {λ : Qi,j (λ) = 0} ∪ {λ : Q(λ) = 0},
where
Q1 (λ)
Q2,k2 −1 (λ)
Q(λ) =
Q1,k1 −1 (λ)
Q2 (λ)
Q3,k3 −1 (λ)
Q2,k2 −1 (λ)
..
.
..
.
..
..
.
.
.
Qm−1,km−1 −1 (λ)
Qm (λ)
Qm,km −1 (λ)
Proof. Similar to the proof of Theorem 1.
Definition 4. For i = 1, 2, 3, . . . , m, let
⎡
di,2 − 1
0
⎢
..
⎢ di,2 − 1
.
0
⎢
.
.
Si = ⎢
..
..
⎢
⎢
⎣
di,k −1 − 1
i
⎤
di,ki −1 − 1
0
di,ki
di,ki
0
⎥
⎥
⎥
⎥
⎥
⎥
⎦
of order ki × ki . Moreover, for i = 1, 2, . . . m and for j = 1, 2, 3, . . . , ki − 1, let Si,j be the j × j
leading principal submatrix of Si .
Lemma 8. For i = 1, 2, . . . , m and for j = 1, 2, . . . , ki − 1, we have
|λI − Si,j | = Qi,j (λ).
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Moreover,
|λI − Si | = Qi (λ)
for i = 1, 2, . . . , m.
Proof. Similar to the proof of Lemma 4.
Lemma 9. Let r = m
i=1 ki . Let H be the symmetric matrix of order r × r defined by
⎡
⎤
S1 F1
⎢ T
⎥
..
⎢F
⎥
.
S2
⎢ 1
⎥
⎢
.
.
.
H =⎢
.. ⎥
..
..
⎥.
⎢
⎥
⎣
Sm−1 Fm−1 ⎦
T
· · · Fm−1
Sm
Then
|λI − H | = Q(λ).
Proof. Similar to the proof of Lemma 5.
Theorem 6. (a)
σ (A(Pm {Bi })) = ∪m
i=1 ∪j ∈i σ (Si,j ) ∪ σ (H ).
(b) The multiplicity of each eigenvalue of the matrix Ai,j , as an eigenvalue of A(Pm {Bi }), is
at least (ni ,j −ni,j +1 ) for j ∈ i .
(c) The largest eigenvalue of H is the largest eigenvalue of A(Pm {Bi }).
Proof. The proof for (a) and (b) is similar to the proof of Theorem 2. Finally, (c) follows from
part (a) and Lemma 6.
Example 3. Let P4 {Bi : i = 1, 2, 3, 4} be the tree in Example 1. For this tree
⎡
⎤
√
⎤
⎡
0
1
√
0
2 √
⎢1
⎥
√
2 √ ⎥
√0
S1 = ⎢
, S2 = ⎣ 2 √0
3⎦ ,
⎣
2 √0
2⎦
3
0
2
0
√
⎤
⎡
√
⎡
⎤
3 √
√0
2 √0
⎥
⎢ 3
√0
0
2
⎥ , S4 = ⎣ 2
√
S3 = ⎢
2⎦ .
⎣
√0
2
0
1⎦
0
2
0
1
0
We have
1 = {2, 3},
2 = {1, 2},
3 = {1, 2},
4 = {1, 2}.
From Theorem 6, the eigenvalues of A(Pm {Bi }) are those of
S1,2 , S1,3 , S2,1 , S2,2 , S3,1 , S3,2 , S4,1 , S4,2 and S
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
where
⎡
S1
⎢F T
1
S=⎢
⎣
F1
S2
F2T
F2
S3
F3T
⎤
⎥
⎥.
F3 ⎦
S4
5. Copies of a generalized Bethe tree attached to a path
In this section, we assume B1 = B2 = · · · = Bm = B, where B is a generalized Bethe of k
levels in which the numbers dk−j +1 and nk−j +1 are the degree of the vertices and the number of
them at level j . Then
k1 = k2 = · · · = km = k,
1 = · · · = m = = j : 1 j k − 1, nj +1 > nj ,
F1 = F2 = · · · = Fm = F,
where F is a k × k matrix whose entries are 0 except F (k, k) = 1,
T2 = T3 = · · · = Tm−1 = T + F,
where
⎡
√ d1
⎢ d2 − 1
⎢
⎢
T = T1 = Tm = ⎢
⎢
⎣
⎤
√
d2 − 1
d2
√
and for i = 1, 2, . . . , m, j = 1, 2, . . . , k − 1,
√
⎡
d2 − 1
√ d1
⎢ d2 − 1
d2
⎢
⎢
..
Ti,j = ⎢
.
⎢
√
⎣
dk−1 − 1
..
.
dk−1 − 1
√
√
dk−1 − 1
dj −1
dj − 1
dk−1 − 1
d√k−1
dk
⎥
⎥
⎥
⎥
√ ⎥
dk ⎦
dk + 1
⎤
⎥
⎥
⎥
⎥.
⎥
dj − 1 ⎦
dj
Definition 5. For s = 1, . . . , m, let
√
⎡
⎤
d2 − 1 √
√ 1
⎢ d2 − 1
⎥
d2
d3 − 1
⎢
⎥
⎢
⎥
√
.
.
⎢
⎥
.
d3 − 1
d3
⎥
L(s) = ⎢
⎢
⎥
√
..
..
⎢
⎥
.
.
d
−
1
k−1
⎢
⎥
√
√
⎣
⎦
dk−1 − 1
d√k−1
dk
dk
dk + 2 + 2 cos πs
m
of order k × k and, j = 1, 2, . . . , k − 1, let Lj the j × j leading principal submatrix of L(s).
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
We recall that the Kronecker product [9] of two matrices A = (ai,j ) and B = (bi,j ) of sizes
m × m and n × n, respectively, is defined to be the (mn) × (mn) matrix A ⊗ B = (ai,j B). For
matrices A, B, C and D of appropiate sizes
(A ⊗ B)(C ⊗ D) = (AC ⊗ BD).
We write Pm {B} instead of Pm {Bi }.
Theorem 7. The set of eigenvalues of L(Pm {B}) is
σ (L(Pm {B})) = ∪j ∈ σ (Lj ) ∪ ∪m
s=1 σ (L(s)) .
Proof. From Theorem 2, we have
σ (L(Pm {B})) = ∪m
i=1 ∪j ∈i σ (Ti,j ) ∪ σ (G)
We observe that Lj = Ti,j . Then
σ (L(Pm {B})) = ∪j ∈ σ (Lj ) ∪ σ (G).
It remains to prove that σ (G) = ∪m
s=1 σ (L(s)). We have
√
⎡
d2 − 1 √
√ 1
⎢ d2 − 1
d2
d3 − 1
⎢
⎢
√
..
⎢
.
d3 − 1
d3
⎢
L(m) = ⎢
.
.
..
..
⎢
⎢
√
⎣
dk−1 − 1
⎤
√
dk−1 − 1
d√k−1
dk
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
√ ⎥
dk ⎦
dk
For brevity, let L = L(m). Then L(s) = L + 2 + 2 cos πs
m F and T = L + F . Hence
⎡
⎤ ⎡
T
F
L+F
F
⎢F T + F F
⎥ ⎢ F
L
+
2F F
⎢
⎥ ⎢
⎢
⎥
⎢
.
.
..
..
..
..
⎥=⎢
.
.
F
F
G=⎢
⎢
⎥ ⎢
⎢
⎥ ⎢
.
.
.
.
⎣
. T + F F⎦ ⎣
. L + 2F
F
T
F
We may write
⎤
⎥
⎥
⎥
⎥.
⎥
⎥
F⎦
L
G = Im ⊗ L + Cm ⊗ F,
where
⎡
1
⎢
⎢1
⎢
⎢
Cm = ⎢
⎢
⎢
⎣
⎤
1
2
..
.
..
.
..
.
..
.
..
.
2
1
⎥
⎥
⎥
⎥
⎥
⎥
⎥
1⎦
1
of order m × m. The eigenvalues of Cm are known [4]. They are γs = 2 + 2 cos πs
m , 1 s m.
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O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
Let
V = v1
v2
vm−1
···
vm
be a orthogonal matrix whose columns v1 , v2 , . . . , vm are eigenvectors corresponding to the
eigenvalues γ1 , γ2 , .., γs . Therefore
(V ⊗ Ik )G(V T ⊗ Ik ) = (V ⊗ Ik )(Im ⊗ L + Cm ⊗ F )(V T ⊗ Ik )
= Im ⊗ L + (V Cm V T ) ⊗ F.
We have
⎡
⎢
⎢
(V Cm V ) ⊗ F = ⎢
⎢
⎣
T
⎡
Therefore
⎢
⎢
⎢
=⎢
⎢
⎣
⎤
γ1
γ2
γm−1
γm
γ1 F
⎥
⎥
⎥⊗F
⎥
⎦
⎤
γ2 F
..
.
γm−1 F
γm F
⎡
⎢
⎢
⎢
(V ⊗ Ik )G(V T ⊗ Ik ) = ⎢
⎢
⎣
L + γ1 F
⎡
⎢
⎢
⎢
=⎢
⎢
⎣
L + γ2 F
..
⎥
⎥
⎥
⎥.
⎥
⎦
⎤
.
L + γm−1 F
⎤
L(1)
L (2)
..
.
L (m − 1)
L + γm F
⎥
⎥
⎥
⎥
⎥
⎦
⎥
⎥
⎥
⎥.
⎥
⎦
L (m)
Since G and (V ⊗ Ik )G(V T ⊗ Ik ) are similar matrices, we conclude that
σ (G) = ∪m
s=1 σ (L(s)).
This completes the proof.
Corollary 2. The largest eigenvalue of L(1) is the spectral radius of L(Pr (B)).
Proof. From Theorem 3, the spectral radius of L(Pr (B)) is the largest eigenvalue in ∪m
s=1 σ (L(s)).
That is
ρ(L(Pr (B))) = max{ρ(L(s) : s = 1, 2, . . . , m)}.
We have
502
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
π
πs
> cos
for s = 2, 3, . . . , m.
(23)
m
m
Since the spectral radius of a irreducible nonnegative matrix strictly increases when any entry of
the matrix strictly increases [8, Theorem 2.1.], it follows from the inequality (23) that ρ(L(1)) is
the spectral radius of L(Pr (B)).
cos
Corollary 3. The smallest eigenvalue of L(m − 1) is the algebraic connectivity of Pr (B).
Proof. We begin observing that L(m) is a singular. From Theorem 4, the algebraic connectivity
of Pr (B) is the smallest eigenvalue in ∪m
s=1 σ (L(s)). Then
a(Pr (B)) = min{λ1 (L(s)) : s = 1, 2, . . . , m − 1},
where λ1 (L(s)) denotes the smallest eigenvalue of L(s). We may write
L(s) = L(s + 1) + E
where
⎡
0
⎢ ..
⎢.
E=⎢
⎢ ..
⎣.
0
···
..
.
···
⎤
···
0
..
.
0
0
0
−2 cos π(s+1)
+ 2 cos πs
m
m
⎥
⎥
⎥.
⎥
⎦
From the inequality
πs
π(2s + 1)
π
π(s + 1)
+ cos
= 2 sin
sin
>0
− cos
m
m
2m
2m
for s = 1, 2, . . . , m − 1, it follows that E is a positive semidefinite matrix. Since that the eigenvalues of a Hermitian matrix do not decrease if a positive semidefinite matrix is added to it,
we have λ1 (L(s + 1)) λ1 (L(s)) for s = 1, 2, . . . , m − 2. It follows that min{λ1 (L(s)) : s =
1, 2, . . . , m − 1} = λ1 (L(m − 1)).
We now consider the adjacency matrix of Pr (B).
Definition 6. For s = 1, . . . , m, let
√
⎡
d2 − 1
0
⎢√
⎢ d2 − 1
0
⎢
.
A(s) = ⎢
..
⎢
⎢
⎣
√
..
.
..
.
dk−1 − 1
⎤
√
dk−1 − 1
√0
dk
√
dk
2 + 2 cos πs
m
⎥
⎥
⎥
⎥
⎥
⎥
⎦
of order k × k and, j = 1, 2, . . . , k − 1, let Aj the j × j leading principal submatrix of A(s).
The following results can be proved in a way that is similar to the proofs of Theorem 7 and
Corollary 3.
Theorem 8. The set of eigenvalues of A(Pm {B}) is
σ (A(Pm {B})) = ∪j ∈ σ (Aj ) ∪ ∪m
s=1 σ (A(s)).
O. Rojo, L. Medina / Linear Algebra and its Applications 430 (2009) 483–503
503
Corollary 4. The largest eigenvalue of A(1) is the spectral radius of A(Pr (B)).
Acknowledgement
The authors wish to thank the referee for the comments which led to an improved version of
the paper.
References
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[2] G.H. Golub, C.F. Van Loan, Matrix Computations, second ed., Johns Hopkins University Press, Baltimore, 1989.
[3] R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph, SIAM Matrix Anal. Appl. 11 (12) (1990)
218–238.
[4] S. Kouachi, Eigenvalues and eigenvectors of tridiagonal matrices, Electron. J. Linear Algebra 15 (2006) 115–133.
[5] O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra Appl. 428 (2008) 2961–2979.
[6] O. Rojo, R. Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra
Appl. 403 (2005) 97–117.
[7] L.N. Trefethen, D. Bau III, Numerical linear algebra, Society for Industrial and Applied Mathematics (1997).
[8] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall Inc., 1965.
[9] F. Zhang, Matrix Theory, Springer, 1999.
Oscar Rojo