vol6_pp62-71. 168KB Jun 04 2011 12:06:15 AM
The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society.
Volume 6, pp. 62-71, March 2000.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela
ELA
TIGHT BOUNDS ON THE ALGEBRAIC CONNECTIVITY
OF A BALANCED BINARY TREE
JASON J. MOLITIERNOy , MICHAEL NEUMANNy , AND BRYAN L. SHADERz
Abstra
t. In this paper, quite tight lower and upper bounds are obtained on the algebrai
onne
tivity, namely, the se
ond-smallest eigenvalue of the Lapla
ian matrix, of an unweighted balan
ed
binary tree with k levels and hen
e n = 2k 1 verti
es. This is a
omplished by
onsidering the
inverse of a matrix of order k 1 readily obtained from the Lapla
ian matrix. It is shown that the
algebrai
onne
tivity is 1=(2k 2k + 3) + O(1=22k ).
Key words. Binary trees, Lapla
ian matrix, Algebrai
onne
tivity
AMS subje
t
lassi
ations. 5C50, 15A48
1. Introdu
tion. In this paper we improve known lower and upper bounds on
the algebrai
onne
tivity, namely, the se
ond-smallest eigenvalue of the Lapla
ian
matrix1 , of the balan
ed binary tree Bk of k levels and hen
e n = 2k 1 verti
es.
Spe
i
ally, in [9, Lemma 6.1℄, Guattery and Miller quote an earlier result of theirs
in [8℄ in whi
h they have shown that the algebrai
onne
tivity, (Bk ), of Bk satis es
1
n
(Bk )
2
n
:
Our new upper bound is
(Bk )
1
(2k
2k + 3)
2k 2
2k 1 1
and our new lower bound is
(2k
2k
2k
2k + 2)
1
p
2 2k 1 2k 1
p k 1
+
1
2 (2
1)
3
p
1
2 2
os
2k
(Bk ) :
1
We
omment that for large k the di eren
e between the denominators in the lower
bound and the upper bound is approximately
1
p
2
2
p
2
+
p
1
p = 1 + 2 = 2:414 213 6:::;
3 2 2
Re
eived
by the editors on 8 September 1999. A
epted for publi
ation on 16 Mar
h 2000.
Handling editor: Stephen J. Kirkland
y Department of Mathemati
s, University of Conne
ti
ut, Storrs, Conne
ti
ut 06269-3009, USA
(jjmolmath.u
onn.edu, neumannmath.u
onn.edu). The work of the se
ond author was supported
in part by NSF Grant No. DMS9973247.
z Department of Mathemati
s, University of Wyoming, Laramie, Wyoming 82071-3036, USA
(bshaderuwyo.edu).
1 For pre
ise de nitions of all
on
epts mentioned in this se
tion, see Se
tion 2.
62
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
63
whi
h implies that the algebrai
onne
tivity of the balan
ed binary tree is 1=(2k
2k + 3) + O(1=22k ).
We prove our main results in Se
tion 3, while in Se
tion 2 we review ne
essary
preliminaries and also des
ribe some motivation for our work here.
We
omment that two other lower estimates for (Bk )
an be obtained as spe
ial
ases of two general lower estimates on the algebrai
onne
tivity of a graph G .
The rst lower estimate, equaling 1=[2(2k 3)2 ℄, is due to Friedland [6, Theorem 2.6℄
and involves the so-
alled Cheeger lower bound. The se
ond lower bound, equaling
9 1=(2k 1 1)2 , is due to Berman and Zhang [1, Theorem 2.2℄. It
an be
3
readily
he
ked that our new lower bound is better than the two bounds just mentioned.
p
2. Preliminaries. Let G be a graph with verti
es 1, 2, . . . , n. Denote the degree
of vertex i by deg(i). The Lapla
ian (matrix) of G is the n n matrix L = [`ij ℄ with
`ij
8< deg(i)
=
: 01
if i = j ,
if i 6= j and i is adja
ent to j ,
otherwise:
The Lapla
ian L of a graph G is a useful algebrai
tool for assessing
ertain
properties of the graph. Perhaps the most well-known property of L is the matrixtree theorem due to Cayley [2℄ (see also Chaiken [3℄), whi
h relates the Lapla
ian
to the number of spanning trees of G . Numerous other properties of G , related to
the
onne
tivity and the isoperimetri
number of G , are re
e
ted by the spe
trum
of L (see [14, 16, 17℄ and the referen
es therein). Sin
e L is a symmetri
, positive
semide nite, and singular matrix, its eigenvalues are nonnegative real numbers, and
so they
an be arranged in nonde
reasing order:
0 = 1
2
n :
Fiedler [5℄ observed that the se
ond-smallest eigenvalue, (G ) := 2 , of L provides
a measure of
onne
tivity and he
alled (G ) the algebrai
onne
tivity of G . In
parti
ular he has shown that (G ) > 0 if and only if G is a
onne
ted graph.
The algebrai
onne
tivity of a graph G and its appli
ations have been extensively studied in the literature; we
ite the following papers and the referen
es quoted
therein: Grone and Merris [7℄, Merris [14, 15℄, Powers [19℄, Pothen, Simon, and Liou
[18℄, Guattery and Miller [9℄, and Kirkland, Neumann, and Shader [12, 11, 13℄.
As an example of the usage of algebrai
onne
tivity we give the problem of
nding separators for graphs. Separators are edges or verti
es whi
h, if removed
from the graph, break it into separate
omponents. This problem is an important
omponent of many graph algorithms. Many popular separator algorithms involve
spe
tral methods, in parti
ular the Lapla
ian L of the underlying graph G . We refer
the reader to [9℄ for a more detailed dis
ussion. Typi
ally, the algebrai
onne
tivity
(G ) and a
orresponding eigenve
tor u are
omputed. The best threshold
ut method
an be des
ribed as follows. Let n be the number of verti
es of G . Given a nonempty,
ELA
64
J. J. Molitierno, M. Neumann, and B. L. Shader
G we de ne the
of X to be
jX j ;
minfjX j; jX jg
where jX j is the number of edges in G with exa
tly one vertex in X . The minimum
of the
ut quotients is the
i(G )
G . It is well known that
(G )
i( G )
:
proper subset X of verti
es of
ut quotient
isoperimetri
number,
, of
2
The best threshold
ut method strives to nd a set X for whi
h the
ut quotient of
X is
lose to the minimum i(G ). This is a
hieved as follows.
(i) Asso
iate with ea
h vertex, i, the value of the ith entry of u.
(ii) Sort the verti
es a
ording to their value. For ea
h index 1 i n 1,
ompute the
ut quotient for the separator obtained by splitting the verti
es
into those with sorted index less than i and those with sorted index greater
than i.
(iii) Choose the split from (ii) that provides the smallest
ut quotient.
As noted in [9℄, until re
ently, there has not been a rigorous analysis of the quality
of separators produ
ed by su
h algorithms. In [9℄, the
omplete balan
ed binary trees
are used as building blo
ks to
onstru
t graphs for whi
h the best threshold
ut
method does poorly. An essential ingredient in their work is their bound
1
n
(B )
k
2
n
on the algebrai
onne
tivity for a
omplete balan
ed binary tree.
3. Tight Bounds on (Bk ). As before, let Bk denote the balan
ed binary tree
with k 2 levels so that Bk has n := 2k 1 verti
es. We now relabel the verti
es
of Bk so that 1 is the root vertex, the verti
es on the left bran
h of 1 pre
ede those
on the right bran
h, and the verti
es on the ith level pre
ede those on level i + 1 for
i = 2; 3; : : : ; k 1. For example, for k = 3, we have the labeling illustrated below.
1
q
2
q
q
q
5
7
6
q
q
q
3
4
The following notation is used throughout this se
tion. If N is an m m matrix,
ea
h of whose eigenvalues is real, then the eigenvalues of N are denoted by i (N ),
i = 1; 2; : : : ; m, where
1 (N )
2 (N )
m 1 (N )
m (N ):
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
65
The tra
e and adjoint of N are denoted by tr(N ) and adj(N ), respe
tively.
We begin by showing that the problem of determining (Bk )
an be transformed
to that of determining the smallest eigenvalue of a prin
ipal submatrix of the Lapla
ian
of order 2k 1 1.
Proposition 3.1. Let Lk be the Lapla
ian matrix of Bk . Then
(Bk ) = 1 (Lk (1; 1));
(1)
where Lk (1; 1) is the prin
ipal submatrix of Lk obtained by deleting its rst row and
olumn.
Proof. Note that Lk has the form
2
66
66
66
66
66
66
66
4
1 0
2
1
0
..
.
0
1
0
..
.
0
0
1 0
C
O
O
C
0
3
77
77
77
77
77 ;
77
77
5
where C is a matrix of order 2k 1 1. It follows that ea
h eigenvalue of Lk (1; 1)
has multipli
ity at least two and hen
e, by the Cau
hy interla
ing property (see [10℄),
1 (Lk (1; 1)) = 2 (Lk ).
Next, let Q = (qi;j ) be the (n 1) (k 1) matrix with
qi;j =
and de ne the (k
Fk
Con
erning Fk
1
1 if vertex i + 1 is on level j + 1 of
0 otherwise,
1) (k
2
66
66
66
:= 6
66
66
64
B
,
1) tridiagonal matrix
3
2
0
1
3
0
..
.
..
.
0
0
1
..
.
2
..
.
..
.
..
.
..
..
..
..
0
.
.
.
.
..
..
..
1
0
.
.
.
..
.
..
.
..
3
0
..
.
..
.
..
.
..
.
2
1
.
1
we
laim the following proposition.
Let Lk = [`i;j ℄ be the Lapla
ian matrix of
(a) Lk (1; 1) Q = QFk 1 ,
1
Proposition 3.2.
k
B
k
3
77
77
77
77 :
77
77
5
. Then
ELA
66
J. J. Molitierno, M. Neumann, and B. L. Shader
(b) ea
h eigenvalue of Fk 1 is an eigenvalue of Lk (1; 1),
(
) ea
h eigenvalue of Fk 1 is real,
(d) 1 (Fk 1 ) = 1 (Lk (1; 1)), and
1
(e) tr(Fk 1 ) = 2k
(k + 1).
Proof. Part (a) follows from the observation that for 1
(i) If
jj
in level
j.
ij > 1,
i j k,
i of B
then there are no edges joining a vertex in level
k
to a vertex
j i = 1, then ea
h vertex in level i of Bk is joined to exa
tly two verti
es in
j.
(iii) If j
i = 1, then ea
h vertex in level i of Bk is joined to exa
tly one vertex in
level j .
(iv) If v is in level i, then
(ii) If
level
`v;v
(2
if
1
if
3
otherwise.
=
i = 1,
i = k,
Now let (; x) be an eigenpair of
(Qx).
Sin
e the
olumns of
Q
Hen
e (; Qx) is an eigenpair of
Lk (1; 1)
and the fa
t that
Fk 1 . Then (a) implies that Lk (1; 1)(Qx) =
are linearly independent, it follows that Qx = 0.
Lk (1; 1)
6
and so (b) holds. Part (
) follows from (b)
is a real symmetri
matrix. Adopting the notation in the
Lk (1; 1) is the dire
t sum of two
opies of the
C is a prin
ipal submatrix of a Lapla
ian matrix and is irredu
ible,
C is a nonsingular M -matrix. Hen
e C has a nonnegative eigenve
tor v
orresponding
to the eigenvalue 1 (C ) = 1 (Lk (1; 1)).
proof of Proposition 3.1, we see that
matrix
C.
Sin
e
It is now easy to verify that
[v
T
0℄Lk (1; 1) =
1 (LK (1; 1))[v T
0℄:
v is a nonnegative eigenve
tor implies that [v T 0℄Q 6= 0. This along with
(a) implies that [v T 0℄Q is a left-eigenve
tor of Fk 1
orresponding to the eigenvalue
1 (Lk (1; 1)). Part (d) now follows from (b).
Finally, we prove (e). As
an be readily
he
ked, the matrix Fk 1 admits the
The fa
t that
following fa
torization as the produ
t of an upper triangular matrix and a lower
triangular matrix:
Fk
1 =
2
66
66
66
66
66
66
4
1
0
2
:: ::
0
1
2
0
0
1
2
0
1
.
.
.
..
.
0
::
0
:::
.
.
.
:: ::
..
0
0
.
::
2
0
0
1
2
0
1
32
77 6
77 66
77 66
77 66
77 66
77 66
54
1
0
1
1
1
:: ::
0
1
:: ::
:: ::
:: ::
::
.
.
.
1
0
1
1
3
77
77
77
77 ;
77
75
ELA
67
Algebrai
Conne
tivity of a Balan
ed Tree
and thus we have the LU-fa
torization
2
3
1
6 1 1
7
6
7
6
7
1
6
7
6 .
7
1
F 1 = 6 ..
7
::
6
7
6
7
::
6
7
4 1
1 5
1
:::
1 1
k
But then
Fk 11 i;i = 2i
1, so that, easily, tr
2
6
6
6
6
6
6
6
6
4
1 2 22 23
1 2 22
1 ::
::
::
::
:: ::
::
P
Fk 11 = ik=11
::
::
::
::
::
1
2
2
2
3
k
k
::
23
22
2
1
3
7
7
7
7
7:
7
7
7
5
2 1 = 2 (k +1).
i
k
It follows from Proposition 3.2 that to either determine exa
tly or approximately
1 (L(1; 1)), it suÆ
es to
onsider the problem of determining exa
tly or approximately
the smallest eigenvalue of Fk 1 . We begin by noting
that for any m 1, if D is the
m m diagonal matrix whose (i; i)th entry is 2i=2 , i = 1; 2; : : : ; m, then the matrix
3 p2 0 0 3
p
6 p
. . . ... 77
6
2
3
2 ...
6
p
6
... 777
... ... ...
6
2
0
6
. . . . . . . . . . . . . . . ... 777
(2) G := DF D 1 = 66 ...
6
7
6
...
. . . . . . . . . . . . ... 77
6
6
p 75
p
4
2 p3
2
0
0
0 0
2 1
is (diagonally) similar to F and hen
e it has the same eigenvalues as F .
Now let S 1 = G (m; m) be the leading prin
ipal submatrix of G of order
m 1. Note that S 1 is a symmetri
, tridiagonal matrix ea
h
of whose diagonal
entries is 3 and ea
h of whose super- and subdiagonal entries is p2. Some additional
properties of the matri
es S1, S2, . . ., are the following.
(a) det(S ) = 2 +1
1.
2
t+2
1
(b) tr(S ) = t
2 + 2 +1 1 , provided that t 2.
Proof. We prove (a) by indu
tion on t. If t = 1, then det S1 = 3 = 22
1.
Assume that t 2 and pro
eed by indu
tion. Note that by Lapla
e expansion
along the last row,
p
det(S ) = 3 det(S 1) ( 2)2 det(S 2)
= 3(2 1) 2(2 1 1)
= 2 +1 1;
as desired.
2
m
m
m
m
m
m
m
m
Lemma 3.3.
t
t
t
t
t
t
t
t
t
t
ELA
68
J. J. Molitierno, M. Neumann, and B. L. Shader
Next note that if we set det S0 = 1, then
tr (adj(St ))
=
1
=0 det(Sj ) det(St 1 j )
Pt 1 j +1
(2
1)(2t j
1)
j =0
Pt 1 t+1
2
+1
2j +1
2t j
j =0
t2t+1 + t 2(2 + 22 + + 2t )
=
t2t+1 + t
=
(t
Pt
=
j
=
=
2(2t+1
2)
2)2t+1 + t + 4:
Statement (b) now follows from (a).
1 (Gm ). Sin
e the
Sm 1 interla
e the eigenvalues of Gm , and as the eigenvalues of Gm are
We shall now use Lemma 3.3 to obtain a lower bound on
eigenvalues of
those of
Fm , we
an write that
1 (Fm )
1 (Sm 1 )
2 (Fm )
m 1 (Sm 1 )
m (Fm ):
Gm is a nonsingular M -matrix; ea
h of its eigenvalues is positive, as are
Sm 1 . Hen
e the eigenvalues of Fm 1 interla
e those of Sm1 1 . That is,
Furthermore
those of
1 (Fm 1 )
1 (Sm1 1 )
2 (Fm 1 )
m 1 (Sm1 1 )
m (Fm 1 ):
Thus,
tr(Fm 1 )
=
m
X
i
=1
i (Fm 1 )
X1
m
i
=1
i (Sm1 1 )
1
=
tr(Sm
1
tr
!
1
+ m (Fm )
1
1 ) + m (Fm ):
Hen
e we have that
tr(Fm )
(3)
Sm1 1
Fm 1 ):
m(
This essentially proves the following lemma.
Lemma 3.4. Let
(4)
Gm
be the matrix given in
1 (Gm )
(2)
1
2m+1
2m + 1
with
m 3.
:
2m
2m
1
Then
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
69
Proof. Re
all that that the eigenvalues of Gm are the re
ipro
als of the eigenvalues
of the matrix Fm 1 used above. Thus (4) follows from (3) and the tra
e formulas in
Proposition 3.2 (with m = k 1) and Lemma 3.3b (with t = m 1).
To obtain an upper bound on 1 (Gm ), we
onsider the following t t matrix
whi
h, for t = m, is readily seen to be a perturbation by a positive semide nite
matrix of Gm :
2
(5)
Ht :=
6
6
6
6
6
6
6
6
6
6
6
6
4
p
2
3
p
2
p
3
..
2
..
.
..
.
..
.
p
2
..
.
0
..
.
..
.
0
0
0
..
..
..
0
.
.
..
.
.
..
.
p.
..
.
2
0
..
.
..
.
..
.
3
p
2
0
..
.
..
.
..
.
..
p.
p2
3
2
3
7
7
7
7
7
7
7:
7
7
7
7
7
5
Thus, from Weyl's theorem on the perturbation of eigenvalues of symmetri
matri
es
(see p. 181 of Horn and Johnson [10℄), we have that
i (Gm )
(6)
i (Hm ); i = 1; : : : ; m:
We
omment that from Elliot [4℄, it
an be dedu
ed that the eigenvalues of Ht
are given by
i (Ht ) = 3
(7)
p
2 2
os
(2i 1)
; i = 1; : : : ; t:
2t + 1
We pro
eed now to prove for Ht a result similar to Lemma 3.3.
For Ht aspgiven in (5), the following
onditions hold:
(a) det(Ht ) = 2t+1
1
2(2t p 1).
2 (2t + 1 2t )
1 = t 2 + 2t + 2
p t
, provided that t 2.
(b) tr Ht
2t+1 1
2(2 1)
Proof. Statement (a)
learly holds if t = 1.pIf t 2, then by the linearity of
the determinant we see that det(Ht ) = det(St )
2 det(St 1 ). Thus part (a) follows
from the formula for the determinant of St 1 and St , whi
h is given in Lemma 3.3a.
For t 2, the (i; i)th entry of adj(Ht ) is det(St 1 ) det(Ht i ) (we set det(S0 ) = 1
and det(H0 ) = 1 here). Hen
e it follows that
Lemma 3.5.
tr(adj(Ht )) = 2t
= (t
p
t i+1
1 + ti=11 (2i 1)(2
1
2(2t i
p
t+1
t
2)2 + t + 4
2((t 3)2 + t + 3);
P
1))
with the last inequality
oming from arithmeti
simpli
ation using formulas for geometri
sums. Statement (b) now follows from the above tra
e formula, (a), and some
additional arithmeti
simpli
ation.
ELA
70
J. J. Molitierno, M. Neumann, and B. L. Shader
1 (Gm ),
To obtain an upper bound on
we on
e again resort to working with
Fm
inverses. From (6) and the fa
t that the eigenvalues of
Gm , we have that
i (Fm 1 )
are the same as those of
i (Hm1 ); i = 1; 2; : : : ; m:
This implies that
m (Fm 1 ) + tr(Hm1 )
m (Hm1 )
1
tr(Fm ):
Substituting in the appropriate tra
e formulas from Lemmas 3.3 and 3.5 yields
m (Fm 1 )
2
m
+1
p
2m + 2
2m
2m+1
2 (2m + 1
p
1
1
2m )
2 (2m
+ m ( Hm ) :
1)
m (Hm1 ) is the re
ipro
al of the smallest eigenvalue of Hm , we arrive, using (7), at the following lower bound on the 1 (Gm ).
Lemma 3.6. Let Gm be the matrix given in (2) with m 3. Then
If we now make use of the fa
t that
1 (Gm )
2m+1
p
2 (2m + 1
2m + 2
2m
1
2m+1
p
1
2 (2m
2m )
1)
3
2
:
1
p
+
2
os
2m + 1
The goal of our paper is to provide good lower and upper bounds on
(Bk )
=
1 (Lk (1; 1))
=
Thus, by using Lemmas 3.4 and 3.6 (with
this paper.
Theorem 3.7. Let
1 (Fk 1 )
m=k
=
1), we obtain the main result of
B
k be the balan
ed binary tree on
(Bk )
1
(2k
2k
2k + 3)
1 (Gk 1 ):
2k
1
k4
levels. Then
2
1
and
(Bk )
(2
k
2k + 2)
2k
2k
p
1
2 2k
1
p
1
2 (2k
1
2k
1
1)
+
3
2
p
1
2
os
:
2k
1
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
71
REFERENCES
[1℄ A. Berman and X. Zhang. Lower bounds for the eigenvalues of Lapla
ian matri
es. Linear
Algebra Appl., submitted, 1999.
[2℄ A. Cayley. A theorem on trees. Quart. J. Pure Appl. Math., 23:376{378, 1889.
[3℄ S. Chaiken. A
ombinatorial proof of the all minors matrix tree theorem. SIAM J. Algebrai
Dis
rete Meth., 3:319{329, 1982.
[4℄ J. F. Elliot. The Chara
teristi
Roots of Certain Real Symmetri
Matri
es. Master's Thesis,
University of Tennessee, Knoxville, Tennessee, 1953.
[5℄ M. Fiedler. Algebrai
onne
tivity of graphs. Cze
hoslovak Math. J., 23:298{305, 1973.
[6℄ S. Friedland. Lower bounds for the rst eigenvalue of
ertain M-matri
es asso
iated with
graphs. Linear Algebra Appl., 172:71{84, 1992.
[7℄ R. Grone and R. Merris. Algebrai
onne
tivity of trees. Cze
hoslovak Math. J., 37(112):660{
670, 1987.
[8℄ S. Guattery and G. L. Miller. On the Performan
e of Spe
tral Graph Partitioning Methods.
Te
h. report CMU-CS-94-228, Carnegie Mellon University, Pittsburgh, PA, 1994.
[9℄ S. Guattery and G. Miller. On the quality of spe
tral separators. SIAM J. Matrix Anal. Appl.,
19:701{719, 1998.
[10℄ R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, New York, 1985.
[11℄ S. J. Kirkland, M. Neumann, and B. Shader. Distan
es in weighted trees and group inverses
of Lapla
ian matri
es. SIAM J. Matrix Anal. Appl., 18:827{841, 1997.
[12℄ S. J. Kirkland, M. Neumann, and B. Shader. Bounds on the subdominant eigenvalue involving
group inverses with appli
ations to graphs. Cze
hoslovak Math. J., 48:1{20, 1998.
[13℄ S. J. Kirkland, M. Neumann, and B. Shader. On a bound on algebrai
onne
tivity: The
ase
of equality. Cze
hoslovak Math. J., 48:65{76, 1998.
[14℄ R. Merris. Lapla
ian matri
es of graphs: a survey. Linear Algebra Appl., 197, 198:143{176,
1994.
[15℄ R. Merris. A note on Lapla
ian graph eigenvalues. Linear Algebra Appl., 285:33{35, 1998.
[16℄ B. Mohar. Eigenvalues, diameter, and mean distan
e in graphs. Graphs Combin., 7:53{64,
1991.
[17℄ B. Mohar and S. Poljak. Eigenvalues in Combinatorial Optimization. IMA Vol. Math. Appl.,
(R. Brualdi, S. Friedland and V. Klee, eds.), 50:107{152, 1993.
[18℄ A. Pothen, H. D. Simon, and K. Liou. Partitioning sparse matri
es with eigenve
tors of graphs.
SIAM J. Matrix Anal. Appl., 11:431{452, 1990.
[19℄ D. L. Powers. Graph partitioning by eigenve
tors. Linear Algebra Appl., 101:121{133, 1988.
A publication of the International Linear Algebra Society.
Volume 6, pp. 62-71, March 2000.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela
ELA
TIGHT BOUNDS ON THE ALGEBRAIC CONNECTIVITY
OF A BALANCED BINARY TREE
JASON J. MOLITIERNOy , MICHAEL NEUMANNy , AND BRYAN L. SHADERz
Abstra
t. In this paper, quite tight lower and upper bounds are obtained on the algebrai
onne
tivity, namely, the se
ond-smallest eigenvalue of the Lapla
ian matrix, of an unweighted balan
ed
binary tree with k levels and hen
e n = 2k 1 verti
es. This is a
omplished by
onsidering the
inverse of a matrix of order k 1 readily obtained from the Lapla
ian matrix. It is shown that the
algebrai
onne
tivity is 1=(2k 2k + 3) + O(1=22k ).
Key words. Binary trees, Lapla
ian matrix, Algebrai
onne
tivity
AMS subje
t
lassi
ations. 5C50, 15A48
1. Introdu
tion. In this paper we improve known lower and upper bounds on
the algebrai
onne
tivity, namely, the se
ond-smallest eigenvalue of the Lapla
ian
matrix1 , of the balan
ed binary tree Bk of k levels and hen
e n = 2k 1 verti
es.
Spe
i
ally, in [9, Lemma 6.1℄, Guattery and Miller quote an earlier result of theirs
in [8℄ in whi
h they have shown that the algebrai
onne
tivity, (Bk ), of Bk satis es
1
n
(Bk )
2
n
:
Our new upper bound is
(Bk )
1
(2k
2k + 3)
2k 2
2k 1 1
and our new lower bound is
(2k
2k
2k
2k + 2)
1
p
2 2k 1 2k 1
p k 1
+
1
2 (2
1)
3
p
1
2 2
os
2k
(Bk ) :
1
We
omment that for large k the di eren
e between the denominators in the lower
bound and the upper bound is approximately
1
p
2
2
p
2
+
p
1
p = 1 + 2 = 2:414 213 6:::;
3 2 2
Re
eived
by the editors on 8 September 1999. A
epted for publi
ation on 16 Mar
h 2000.
Handling editor: Stephen J. Kirkland
y Department of Mathemati
s, University of Conne
ti
ut, Storrs, Conne
ti
ut 06269-3009, USA
(jjmolmath.u
onn.edu, neumannmath.u
onn.edu). The work of the se
ond author was supported
in part by NSF Grant No. DMS9973247.
z Department of Mathemati
s, University of Wyoming, Laramie, Wyoming 82071-3036, USA
(bshaderuwyo.edu).
1 For pre
ise de nitions of all
on
epts mentioned in this se
tion, see Se
tion 2.
62
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
63
whi
h implies that the algebrai
onne
tivity of the balan
ed binary tree is 1=(2k
2k + 3) + O(1=22k ).
We prove our main results in Se
tion 3, while in Se
tion 2 we review ne
essary
preliminaries and also des
ribe some motivation for our work here.
We
omment that two other lower estimates for (Bk )
an be obtained as spe
ial
ases of two general lower estimates on the algebrai
onne
tivity of a graph G .
The rst lower estimate, equaling 1=[2(2k 3)2 ℄, is due to Friedland [6, Theorem 2.6℄
and involves the so-
alled Cheeger lower bound. The se
ond lower bound, equaling
9 1=(2k 1 1)2 , is due to Berman and Zhang [1, Theorem 2.2℄. It
an be
3
readily
he
ked that our new lower bound is better than the two bounds just mentioned.
p
2. Preliminaries. Let G be a graph with verti
es 1, 2, . . . , n. Denote the degree
of vertex i by deg(i). The Lapla
ian (matrix) of G is the n n matrix L = [`ij ℄ with
`ij
8< deg(i)
=
: 01
if i = j ,
if i 6= j and i is adja
ent to j ,
otherwise:
The Lapla
ian L of a graph G is a useful algebrai
tool for assessing
ertain
properties of the graph. Perhaps the most well-known property of L is the matrixtree theorem due to Cayley [2℄ (see also Chaiken [3℄), whi
h relates the Lapla
ian
to the number of spanning trees of G . Numerous other properties of G , related to
the
onne
tivity and the isoperimetri
number of G , are re
e
ted by the spe
trum
of L (see [14, 16, 17℄ and the referen
es therein). Sin
e L is a symmetri
, positive
semide nite, and singular matrix, its eigenvalues are nonnegative real numbers, and
so they
an be arranged in nonde
reasing order:
0 = 1
2
n :
Fiedler [5℄ observed that the se
ond-smallest eigenvalue, (G ) := 2 , of L provides
a measure of
onne
tivity and he
alled (G ) the algebrai
onne
tivity of G . In
parti
ular he has shown that (G ) > 0 if and only if G is a
onne
ted graph.
The algebrai
onne
tivity of a graph G and its appli
ations have been extensively studied in the literature; we
ite the following papers and the referen
es quoted
therein: Grone and Merris [7℄, Merris [14, 15℄, Powers [19℄, Pothen, Simon, and Liou
[18℄, Guattery and Miller [9℄, and Kirkland, Neumann, and Shader [12, 11, 13℄.
As an example of the usage of algebrai
onne
tivity we give the problem of
nding separators for graphs. Separators are edges or verti
es whi
h, if removed
from the graph, break it into separate
omponents. This problem is an important
omponent of many graph algorithms. Many popular separator algorithms involve
spe
tral methods, in parti
ular the Lapla
ian L of the underlying graph G . We refer
the reader to [9℄ for a more detailed dis
ussion. Typi
ally, the algebrai
onne
tivity
(G ) and a
orresponding eigenve
tor u are
omputed. The best threshold
ut method
an be des
ribed as follows. Let n be the number of verti
es of G . Given a nonempty,
ELA
64
J. J. Molitierno, M. Neumann, and B. L. Shader
G we de ne the
of X to be
jX j ;
minfjX j; jX jg
where jX j is the number of edges in G with exa
tly one vertex in X . The minimum
of the
ut quotients is the
i(G )
G . It is well known that
(G )
i( G )
:
proper subset X of verti
es of
ut quotient
isoperimetri
number,
, of
2
The best threshold
ut method strives to nd a set X for whi
h the
ut quotient of
X is
lose to the minimum i(G ). This is a
hieved as follows.
(i) Asso
iate with ea
h vertex, i, the value of the ith entry of u.
(ii) Sort the verti
es a
ording to their value. For ea
h index 1 i n 1,
ompute the
ut quotient for the separator obtained by splitting the verti
es
into those with sorted index less than i and those with sorted index greater
than i.
(iii) Choose the split from (ii) that provides the smallest
ut quotient.
As noted in [9℄, until re
ently, there has not been a rigorous analysis of the quality
of separators produ
ed by su
h algorithms. In [9℄, the
omplete balan
ed binary trees
are used as building blo
ks to
onstru
t graphs for whi
h the best threshold
ut
method does poorly. An essential ingredient in their work is their bound
1
n
(B )
k
2
n
on the algebrai
onne
tivity for a
omplete balan
ed binary tree.
3. Tight Bounds on (Bk ). As before, let Bk denote the balan
ed binary tree
with k 2 levels so that Bk has n := 2k 1 verti
es. We now relabel the verti
es
of Bk so that 1 is the root vertex, the verti
es on the left bran
h of 1 pre
ede those
on the right bran
h, and the verti
es on the ith level pre
ede those on level i + 1 for
i = 2; 3; : : : ; k 1. For example, for k = 3, we have the labeling illustrated below.
1
q
2
q
q
q
5
7
6
q
q
q
3
4
The following notation is used throughout this se
tion. If N is an m m matrix,
ea
h of whose eigenvalues is real, then the eigenvalues of N are denoted by i (N ),
i = 1; 2; : : : ; m, where
1 (N )
2 (N )
m 1 (N )
m (N ):
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
65
The tra
e and adjoint of N are denoted by tr(N ) and adj(N ), respe
tively.
We begin by showing that the problem of determining (Bk )
an be transformed
to that of determining the smallest eigenvalue of a prin
ipal submatrix of the Lapla
ian
of order 2k 1 1.
Proposition 3.1. Let Lk be the Lapla
ian matrix of Bk . Then
(Bk ) = 1 (Lk (1; 1));
(1)
where Lk (1; 1) is the prin
ipal submatrix of Lk obtained by deleting its rst row and
olumn.
Proof. Note that Lk has the form
2
66
66
66
66
66
66
66
4
1 0
2
1
0
..
.
0
1
0
..
.
0
0
1 0
C
O
O
C
0
3
77
77
77
77
77 ;
77
77
5
where C is a matrix of order 2k 1 1. It follows that ea
h eigenvalue of Lk (1; 1)
has multipli
ity at least two and hen
e, by the Cau
hy interla
ing property (see [10℄),
1 (Lk (1; 1)) = 2 (Lk ).
Next, let Q = (qi;j ) be the (n 1) (k 1) matrix with
qi;j =
and de ne the (k
Fk
Con
erning Fk
1
1 if vertex i + 1 is on level j + 1 of
0 otherwise,
1) (k
2
66
66
66
:= 6
66
66
64
B
,
1) tridiagonal matrix
3
2
0
1
3
0
..
.
..
.
0
0
1
..
.
2
..
.
..
.
..
.
..
..
..
..
0
.
.
.
.
..
..
..
1
0
.
.
.
..
.
..
.
..
3
0
..
.
..
.
..
.
..
.
2
1
.
1
we
laim the following proposition.
Let Lk = [`i;j ℄ be the Lapla
ian matrix of
(a) Lk (1; 1) Q = QFk 1 ,
1
Proposition 3.2.
k
B
k
3
77
77
77
77 :
77
77
5
. Then
ELA
66
J. J. Molitierno, M. Neumann, and B. L. Shader
(b) ea
h eigenvalue of Fk 1 is an eigenvalue of Lk (1; 1),
(
) ea
h eigenvalue of Fk 1 is real,
(d) 1 (Fk 1 ) = 1 (Lk (1; 1)), and
1
(e) tr(Fk 1 ) = 2k
(k + 1).
Proof. Part (a) follows from the observation that for 1
(i) If
jj
in level
j.
ij > 1,
i j k,
i of B
then there are no edges joining a vertex in level
k
to a vertex
j i = 1, then ea
h vertex in level i of Bk is joined to exa
tly two verti
es in
j.
(iii) If j
i = 1, then ea
h vertex in level i of Bk is joined to exa
tly one vertex in
level j .
(iv) If v is in level i, then
(ii) If
level
`v;v
(2
if
1
if
3
otherwise.
=
i = 1,
i = k,
Now let (; x) be an eigenpair of
(Qx).
Sin
e the
olumns of
Q
Hen
e (; Qx) is an eigenpair of
Lk (1; 1)
and the fa
t that
Fk 1 . Then (a) implies that Lk (1; 1)(Qx) =
are linearly independent, it follows that Qx = 0.
Lk (1; 1)
6
and so (b) holds. Part (
) follows from (b)
is a real symmetri
matrix. Adopting the notation in the
Lk (1; 1) is the dire
t sum of two
opies of the
C is a prin
ipal submatrix of a Lapla
ian matrix and is irredu
ible,
C is a nonsingular M -matrix. Hen
e C has a nonnegative eigenve
tor v
orresponding
to the eigenvalue 1 (C ) = 1 (Lk (1; 1)).
proof of Proposition 3.1, we see that
matrix
C.
Sin
e
It is now easy to verify that
[v
T
0℄Lk (1; 1) =
1 (LK (1; 1))[v T
0℄:
v is a nonnegative eigenve
tor implies that [v T 0℄Q 6= 0. This along with
(a) implies that [v T 0℄Q is a left-eigenve
tor of Fk 1
orresponding to the eigenvalue
1 (Lk (1; 1)). Part (d) now follows from (b).
Finally, we prove (e). As
an be readily
he
ked, the matrix Fk 1 admits the
The fa
t that
following fa
torization as the produ
t of an upper triangular matrix and a lower
triangular matrix:
Fk
1 =
2
66
66
66
66
66
66
4
1
0
2
:: ::
0
1
2
0
0
1
2
0
1
.
.
.
..
.
0
::
0
:::
.
.
.
:: ::
..
0
0
.
::
2
0
0
1
2
0
1
32
77 6
77 66
77 66
77 66
77 66
77 66
54
1
0
1
1
1
:: ::
0
1
:: ::
:: ::
:: ::
::
.
.
.
1
0
1
1
3
77
77
77
77 ;
77
75
ELA
67
Algebrai
Conne
tivity of a Balan
ed Tree
and thus we have the LU-fa
torization
2
3
1
6 1 1
7
6
7
6
7
1
6
7
6 .
7
1
F 1 = 6 ..
7
::
6
7
6
7
::
6
7
4 1
1 5
1
:::
1 1
k
But then
Fk 11 i;i = 2i
1, so that, easily, tr
2
6
6
6
6
6
6
6
6
4
1 2 22 23
1 2 22
1 ::
::
::
::
:: ::
::
P
Fk 11 = ik=11
::
::
::
::
::
1
2
2
2
3
k
k
::
23
22
2
1
3
7
7
7
7
7:
7
7
7
5
2 1 = 2 (k +1).
i
k
It follows from Proposition 3.2 that to either determine exa
tly or approximately
1 (L(1; 1)), it suÆ
es to
onsider the problem of determining exa
tly or approximately
the smallest eigenvalue of Fk 1 . We begin by noting
that for any m 1, if D is the
m m diagonal matrix whose (i; i)th entry is 2i=2 , i = 1; 2; : : : ; m, then the matrix
3 p2 0 0 3
p
6 p
. . . ... 77
6
2
3
2 ...
6
p
6
... 777
... ... ...
6
2
0
6
. . . . . . . . . . . . . . . ... 777
(2) G := DF D 1 = 66 ...
6
7
6
...
. . . . . . . . . . . . ... 77
6
6
p 75
p
4
2 p3
2
0
0
0 0
2 1
is (diagonally) similar to F and hen
e it has the same eigenvalues as F .
Now let S 1 = G (m; m) be the leading prin
ipal submatrix of G of order
m 1. Note that S 1 is a symmetri
, tridiagonal matrix ea
h
of whose diagonal
entries is 3 and ea
h of whose super- and subdiagonal entries is p2. Some additional
properties of the matri
es S1, S2, . . ., are the following.
(a) det(S ) = 2 +1
1.
2
t+2
1
(b) tr(S ) = t
2 + 2 +1 1 , provided that t 2.
Proof. We prove (a) by indu
tion on t. If t = 1, then det S1 = 3 = 22
1.
Assume that t 2 and pro
eed by indu
tion. Note that by Lapla
e expansion
along the last row,
p
det(S ) = 3 det(S 1) ( 2)2 det(S 2)
= 3(2 1) 2(2 1 1)
= 2 +1 1;
as desired.
2
m
m
m
m
m
m
m
m
Lemma 3.3.
t
t
t
t
t
t
t
t
t
t
ELA
68
J. J. Molitierno, M. Neumann, and B. L. Shader
Next note that if we set det S0 = 1, then
tr (adj(St ))
=
1
=0 det(Sj ) det(St 1 j )
Pt 1 j +1
(2
1)(2t j
1)
j =0
Pt 1 t+1
2
+1
2j +1
2t j
j =0
t2t+1 + t 2(2 + 22 + + 2t )
=
t2t+1 + t
=
(t
Pt
=
j
=
=
2(2t+1
2)
2)2t+1 + t + 4:
Statement (b) now follows from (a).
1 (Gm ). Sin
e the
Sm 1 interla
e the eigenvalues of Gm , and as the eigenvalues of Gm are
We shall now use Lemma 3.3 to obtain a lower bound on
eigenvalues of
those of
Fm , we
an write that
1 (Fm )
1 (Sm 1 )
2 (Fm )
m 1 (Sm 1 )
m (Fm ):
Gm is a nonsingular M -matrix; ea
h of its eigenvalues is positive, as are
Sm 1 . Hen
e the eigenvalues of Fm 1 interla
e those of Sm1 1 . That is,
Furthermore
those of
1 (Fm 1 )
1 (Sm1 1 )
2 (Fm 1 )
m 1 (Sm1 1 )
m (Fm 1 ):
Thus,
tr(Fm 1 )
=
m
X
i
=1
i (Fm 1 )
X1
m
i
=1
i (Sm1 1 )
1
=
tr(Sm
1
tr
!
1
+ m (Fm )
1
1 ) + m (Fm ):
Hen
e we have that
tr(Fm )
(3)
Sm1 1
Fm 1 ):
m(
This essentially proves the following lemma.
Lemma 3.4. Let
(4)
Gm
be the matrix given in
1 (Gm )
(2)
1
2m+1
2m + 1
with
m 3.
:
2m
2m
1
Then
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
69
Proof. Re
all that that the eigenvalues of Gm are the re
ipro
als of the eigenvalues
of the matrix Fm 1 used above. Thus (4) follows from (3) and the tra
e formulas in
Proposition 3.2 (with m = k 1) and Lemma 3.3b (with t = m 1).
To obtain an upper bound on 1 (Gm ), we
onsider the following t t matrix
whi
h, for t = m, is readily seen to be a perturbation by a positive semide nite
matrix of Gm :
2
(5)
Ht :=
6
6
6
6
6
6
6
6
6
6
6
6
4
p
2
3
p
2
p
3
..
2
..
.
..
.
..
.
p
2
..
.
0
..
.
..
.
0
0
0
..
..
..
0
.
.
..
.
.
..
.
p.
..
.
2
0
..
.
..
.
..
.
3
p
2
0
..
.
..
.
..
.
..
p.
p2
3
2
3
7
7
7
7
7
7
7:
7
7
7
7
7
5
Thus, from Weyl's theorem on the perturbation of eigenvalues of symmetri
matri
es
(see p. 181 of Horn and Johnson [10℄), we have that
i (Gm )
(6)
i (Hm ); i = 1; : : : ; m:
We
omment that from Elliot [4℄, it
an be dedu
ed that the eigenvalues of Ht
are given by
i (Ht ) = 3
(7)
p
2 2
os
(2i 1)
; i = 1; : : : ; t:
2t + 1
We pro
eed now to prove for Ht a result similar to Lemma 3.3.
For Ht aspgiven in (5), the following
onditions hold:
(a) det(Ht ) = 2t+1
1
2(2t p 1).
2 (2t + 1 2t )
1 = t 2 + 2t + 2
p t
, provided that t 2.
(b) tr Ht
2t+1 1
2(2 1)
Proof. Statement (a)
learly holds if t = 1.pIf t 2, then by the linearity of
the determinant we see that det(Ht ) = det(St )
2 det(St 1 ). Thus part (a) follows
from the formula for the determinant of St 1 and St , whi
h is given in Lemma 3.3a.
For t 2, the (i; i)th entry of adj(Ht ) is det(St 1 ) det(Ht i ) (we set det(S0 ) = 1
and det(H0 ) = 1 here). Hen
e it follows that
Lemma 3.5.
tr(adj(Ht )) = 2t
= (t
p
t i+1
1 + ti=11 (2i 1)(2
1
2(2t i
p
t+1
t
2)2 + t + 4
2((t 3)2 + t + 3);
P
1))
with the last inequality
oming from arithmeti
simpli
ation using formulas for geometri
sums. Statement (b) now follows from the above tra
e formula, (a), and some
additional arithmeti
simpli
ation.
ELA
70
J. J. Molitierno, M. Neumann, and B. L. Shader
1 (Gm ),
To obtain an upper bound on
we on
e again resort to working with
Fm
inverses. From (6) and the fa
t that the eigenvalues of
Gm , we have that
i (Fm 1 )
are the same as those of
i (Hm1 ); i = 1; 2; : : : ; m:
This implies that
m (Fm 1 ) + tr(Hm1 )
m (Hm1 )
1
tr(Fm ):
Substituting in the appropriate tra
e formulas from Lemmas 3.3 and 3.5 yields
m (Fm 1 )
2
m
+1
p
2m + 2
2m
2m+1
2 (2m + 1
p
1
1
2m )
2 (2m
+ m ( Hm ) :
1)
m (Hm1 ) is the re
ipro
al of the smallest eigenvalue of Hm , we arrive, using (7), at the following lower bound on the 1 (Gm ).
Lemma 3.6. Let Gm be the matrix given in (2) with m 3. Then
If we now make use of the fa
t that
1 (Gm )
2m+1
p
2 (2m + 1
2m + 2
2m
1
2m+1
p
1
2 (2m
2m )
1)
3
2
:
1
p
+
2
os
2m + 1
The goal of our paper is to provide good lower and upper bounds on
(Bk )
=
1 (Lk (1; 1))
=
Thus, by using Lemmas 3.4 and 3.6 (with
this paper.
Theorem 3.7. Let
1 (Fk 1 )
m=k
=
1), we obtain the main result of
B
k be the balan
ed binary tree on
(Bk )
1
(2k
2k
2k + 3)
1 (Gk 1 ):
2k
1
k4
levels. Then
2
1
and
(Bk )
(2
k
2k + 2)
2k
2k
p
1
2 2k
1
p
1
2 (2k
1
2k
1
1)
+
3
2
p
1
2
os
:
2k
1
ELA
Algebrai
Conne
tivity of a Balan
ed Tree
71
REFERENCES
[1℄ A. Berman and X. Zhang. Lower bounds for the eigenvalues of Lapla
ian matri
es. Linear
Algebra Appl., submitted, 1999.
[2℄ A. Cayley. A theorem on trees. Quart. J. Pure Appl. Math., 23:376{378, 1889.
[3℄ S. Chaiken. A
ombinatorial proof of the all minors matrix tree theorem. SIAM J. Algebrai
Dis
rete Meth., 3:319{329, 1982.
[4℄ J. F. Elliot. The Chara
teristi
Roots of Certain Real Symmetri
Matri
es. Master's Thesis,
University of Tennessee, Knoxville, Tennessee, 1953.
[5℄ M. Fiedler. Algebrai
onne
tivity of graphs. Cze
hoslovak Math. J., 23:298{305, 1973.
[6℄ S. Friedland. Lower bounds for the rst eigenvalue of
ertain M-matri
es asso
iated with
graphs. Linear Algebra Appl., 172:71{84, 1992.
[7℄ R. Grone and R. Merris. Algebrai
onne
tivity of trees. Cze
hoslovak Math. J., 37(112):660{
670, 1987.
[8℄ S. Guattery and G. L. Miller. On the Performan
e of Spe
tral Graph Partitioning Methods.
Te
h. report CMU-CS-94-228, Carnegie Mellon University, Pittsburgh, PA, 1994.
[9℄ S. Guattery and G. Miller. On the quality of spe
tral separators. SIAM J. Matrix Anal. Appl.,
19:701{719, 1998.
[10℄ R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, New York, 1985.
[11℄ S. J. Kirkland, M. Neumann, and B. Shader. Distan
es in weighted trees and group inverses
of Lapla
ian matri
es. SIAM J. Matrix Anal. Appl., 18:827{841, 1997.
[12℄ S. J. Kirkland, M. Neumann, and B. Shader. Bounds on the subdominant eigenvalue involving
group inverses with appli
ations to graphs. Cze
hoslovak Math. J., 48:1{20, 1998.
[13℄ S. J. Kirkland, M. Neumann, and B. Shader. On a bound on algebrai
onne
tivity: The
ase
of equality. Cze
hoslovak Math. J., 48:65{76, 1998.
[14℄ R. Merris. Lapla
ian matri
es of graphs: a survey. Linear Algebra Appl., 197, 198:143{176,
1994.
[15℄ R. Merris. A note on Lapla
ian graph eigenvalues. Linear Algebra Appl., 285:33{35, 1998.
[16℄ B. Mohar. Eigenvalues, diameter, and mean distan
e in graphs. Graphs Combin., 7:53{64,
1991.
[17℄ B. Mohar and S. Poljak. Eigenvalues in Combinatorial Optimization. IMA Vol. Math. Appl.,
(R. Brualdi, S. Friedland and V. Klee, eds.), 50:107{152, 1993.
[18℄ A. Pothen, H. D. Simon, and K. Liou. Partitioning sparse matri
es with eigenve
tors of graphs.
SIAM J. Matrix Anal. Appl., 11:431{452, 1990.
[19℄ D. L. Powers. Graph partitioning by eigenve
tors. Linear Algebra Appl., 101:121{133, 1988.