Download Investigation on spectrum of the adjacency matrix and

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