On the bounds for the largest Laplacian eigenvalues of weighted graphs

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Discrete Optimization

journal homepage:www.elsevier.com/locate/disopt

On the bounds for the largest Laplacian eigenvalues of weighted graphs

Sezer Sorgun

a,∗

, Şerife Büyükköse

b

a_{Nevşehir University, Faculty of Sciences and Arts, Departments of Mathematics, Nevşehir, Turkey} b_{Ahi Evran University, Faculty of Sciences and Arts, Department of Mathematics, 40100 Kırşehir, Turkey}

a r t i c l e i n f o Article history:

Received 28 March 2011

Received in revised form 19 January 2012 Accepted 1 March 2012

Available online 22 March 2012 MSC: 05C50 Keywords: Weighted graph Laplacian matrix Upper bound

a b s t r a c t

We consider weighted graphs, such as graphs where the edge weights are positive definite matrices. The Laplacian eigenvalues of a graph are the eigenvalues of the Laplacian matrix of a graph G. We obtain an upper bound for the largest Laplacian eigenvalue and we compare this bound with previously known bounds.

1. Introduction

We consider simple graphs, such as graphs which have no loops or parallel edges. Hence a graph G

=

(

V

,

E

)

consists of a finite set of vertices, V , and a set of edges, E, each of whose elements is an unordered pair of distinct vertices. Generally V is taken as V

= {

1

,

2

, . . . ,

n

}

.

A weighted graph is a graph each edge of which has been assigned to a square matrix called the weight of the edge. All the weight matrices are assumed to be of the same order and to be positive matrix. In this paper, by ‘‘weighted graph’’ we mean ‘‘a weighted graph with each of its edges bearing a positive definite matrix as weight’’, unless otherwise stated.

The following are the notations to be used in this paper. Let G be a weighted graph on n vertices. Denote by

w

i,jthe

positive definite weight matrix of order p of the edge ij, and assume that

w

ij

=

w

ji. We write i

∼

j if vertices i and j are

adjacent. Let

w

i

=



j:j∼i

w

ij.

The Laplacian matrix of a graph G is defined as L

(

G

) = (

lij

)

, where

li,j

=



w

i

;

if i

=

j

,

−

w

_ij

;

if i

∼

j

,

0

;

otherwise

.

Let

λ

1denote the largest eigenvalue of L

(

G

)

. If V is the disjoint union of two nonempty sets V1and V2such that every

vertex i in V1has the same

λ

1

(w

i

)

and every vertex j in V2has the same

λ

1

(w

j

)

, then G is called a weight-semiregular graph.

If

λ

1

(w

i

) = λ

1

(w

j

)

in a weight semiregular graph, then G is called a weight-regular graph.

In the definitions above, the zero denotes the p

×

p zero matrix. Hence L

(

G

)

is a square matrix of order np.

Upper and lower bounds for the largest Laplacian eigenvalue for unweighted graphs have been investigated to a great extent in the literature [1–10]. For most of the bounds, Pan [11] has characterized the graphs which achieve the upper bounds of the largest Laplacian eigenvalues for unweighted graphs.

∗_{Corresponding author.}

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Theorem 1 (Rayleigh–Ritz [12]). Let A

∈

Mnbe Hermitian, and let the eigenvalues of A be ordered such that

λ

n

≤

λ

n−1

≤

· · · ≤

λ

₁. Then,

λ

nxTx

≤

xTAx

≤

λ

1xTx

λ

max

=

λ

1

=

max x̸=0 xTAx xT_x

=

_xmaxT_x₌₁x T_Ax

λ

min

=

λ

n

=

min x̸=0 xTAx xT_x

=

_xminT_x₌₁x T_Ax for all x

∈

_Cn_.

Proposition 1 ([13]). Let A

∈

Mnhave eigenvalues

{

λ

i

}

. Even if A is not Hermitian, one has the bounds

min x̸=0



xT_Ax xT_x



≤ |

λ

_i

| ≤

max x̸=0



xT_Ax xT_x



(1.1) for i

=

1

,

2

, . . . ,

n.

Corollary 1 ([13]). Let A

∈

Mnhave eigenvalues

{

λ

i

}

. Even if A is not Hermitian, one has the bounds

min x̸=0,y̸=0



xTAy xT_y



≤ |

λ

_i

| ≤

max x̸=0,y̸=0



xTAy xT_y



(1.2)

for any

¯

x

∈

Rn

_(¯

_x

_{̸= ¯}

₀

_{), ¯}

_y

_∈

_Rn

_(¯

_y

_{̸= ¯}

₀

₎

_{and for i}

₌

₁

_,

₂

_{, . . . ,}

_n.

Lemma 1 (Horn and Johnson [12]). Let B be a Hermitian n

×

n matrix with eigenvalues

λ

1

≥

λ

2

≥ · · · ≥

λ

n, then for any

¯

x

∈

Rn

_(¯

_x

_{̸= ¯}

₀

_{), ¯}

_y

_∈

_Rn

_(¯

_y

_{̸= ¯}

₀

₎

_,



 ¯

xTBy

¯



 ≤

λ

₁

√

¯

xT

_¯

_x



_¯

_yT_y

_¯

.

_(1.3)

Equality holds if and only ifx is an eigenvector of B corresponding to

¯

λ

1andy

¯

=

α¯

x for some

α ∈

R.

Some upper bounds on the largest Laplacian eigenvalue for weighted graphs, where the edge weights are positive definite matrices, are known as below. Then, we also give an upper bound on the largest Laplacian eigenvalue for weighted graphs in Section2and compare our bound with other bounds.

Theorem 2 (Das and Bapat [14]). Let G be a simple connected weighted graph. Then

λ

1

≤

max i∼j



λ

1





k:k∼i

w

ik



+



k:k∼j

λ

1

(w

jk

)



(1.4)

where

w

ijis the positive definite weight matrix of order p of the edge ij. Moreover equality holds in(1.4)if and only if

(i) G is a weight-semiregular bipartite graph;

(ii)

w

ijhave a common eigenvector corresponding to the largest eigenvalue

λ

1

(w

ij

)

for all i

,

j. Theorem 3 (Das [15]). Let G be a simple connected weighted graph. Then

λ

1

≤

max i∼j















k:k∼i

λ

1

(w

ik

)





r:r∼i

λ

1

(w

ir

) +



s:s∼k

λ

1

(w

ks

)



+



k:k∼j

λ

1

(w

jk

)





r:r∼j

λ

1

(w

jr

) +



s:s∼k

λ

1

(w

ks

)









(1.5)

where

w

ijis the positive definite weight matrix of order p of the edge ij. Moreover equality holds in(1.5)if and only if

(i) G is a bipartite semiregular graph; and

(ii)

w

ijhave a common eigenvector corresponding to the largest eigenvalue

λ

1

(w

ij

)

for all i

,

j. Theorem 4 (Das [15]). Let G be a simple connected weighted graph. Then

λ

1

≤

max i∼j

 λ

1

(w

i

) + λ

1

(w

j

) + (λ

1

(w

i

) − λ

1

(w

j

))

2

+

4

γ

¯

i

γ

¯

j 2



(1.6) where

γ

¯

_i

=

 k:k∼iλ1(wik)λ1(wk)

λ1(wi) and

w

ijis the positive definite weight matrix of order p of the edge ij. Moreover equality holds in(1.6)if and only if

(i) G is a weighted-regular graph or G is a weight-semiregular bipartite graph;

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2. An upper bounds on the largest Laplacian eigenvalue of weighted graphs

Theorem 5. Let G be a simple connected weighted graph. Then

λ

1

≤

max i









λ

2 1

(w

i

) +



k:k∼i

λ

2 1

(w

ik

) +



k:k∼i

λ

1

(w

i

w

ik

+

w

ik

w

k

) +



1≤i,t≤n



s∈Ni∩Nt

λ

1

(w

is

w

st

)







(2.1)

where

w

ikis the positive definite weight matrix of order p of the edge ik and Ni

∩

Nkis the set of common neigbours of i and k.

Moreover equality holds in(2.1)if and only if

(i) G is a weighted-regular graph or G is a weight-semiregular bipartite graph;

(ii)

w

ikhave a common eigenvector corresponding to the largest eigenvalue

λ

1

(w

ik

)

for all i

,

k. Proof. Let_X

¯

₌

_(¯

_xT

1

, ¯

xT2

, . . . , ¯

xTn

)

Tbe an eigenvector corresponding to the largest eigenvalue

λ

1of L

(

G

)

.We assume thatx

¯

iis

the vector component ofX such that

¯

xT_i

¯

xi

=

max k∈V

 ¯

xT_k

¯

xk



.

(2.2) SinceX is nonzero, so is

¯

x

¯

i. The

(

i

,

j

)

-th element of L

(

G

)

is



w

i

;

if i

=

j

−

w

_i_,_j

;

if i

∼

j 0

;

otherwise

.

Now we consider the matrix L2

₍

_G

₎

_{. The}

₍

_i

_,

_j

₎

_{-th element of L}2

₍

_G

₎

_is











w

2 i

+



k∈Ni

w

2 ik

;

if i

=

j

−

w

_i

w

_ij

−

w

_ji

w

_j

+



k∈Ni∩Nj

w

ik

w

kj

;

otherwise

.

We have L2

(

G

) ¯

X

=

λ

2₁X

¯

.

(2.3)

From the i-th equation of(2.3), we have

λ

2 1

¯

xi

=

w

2ix

¯

i

+



k:k∼i

w

2 ik

¯

xi

+



k:k∼i

−

(w

_i

w

_ik

+

w

_ik

w

_k

)¯

xk

+



1≤i,t≤n





s∈Ni∩Nt

w

is

w

stx

¯

t



i.e.

λ

2 1

¯

x T ix

¯

i

= ¯

xTi

w

2 ix

¯

i

+



k:k∼i

¯

xT_i

w

2_ikx

¯

i

+



k:k∼i

−¯

xT_i

((w

i

w

ik

+

w

ki

w

k

)) ¯

xk

+



1≤i,t≤n





s∈Ni∩Nt

¯

xT_i

w

is

w

stx

¯

t



.

(2.4)

Taking the modulus on both sides of(2.4), we get



λ

2₁



 ¯

xT_ix

¯

_i

≤



 ¯

xT_i

w

_i2x

¯

_i



 +



k:k∼i



 ¯

xT_i

w

_ik2x

¯

_i



 +



k∼i



 ¯

xT_i

(w

_i

w

_ik

+

w

_ki

w

_k

)¯

x_k



 +



1≤i,t≤n





s∈Ni∩Nt



 ¯

xT_i

w

_is

w

_stx

¯

_t





.

(2.5) Since

w

i,kis the positive definite matrix for every i

,

k

, w

2i,kmatrices are also positive definite. So, we have

≤

λ

₁

(w

_i2

)¯

xT_i

¯

xi

+



k:k∼i

λ

1

(w

2ik

)¯

xTix

¯

i

+



k∼i



 ¯

xT_i

(w

_i

w

_ik

+

w

_ik

w

_k

)¯

x_k



 +



1≤i,t≤n





s∈Ni∩Nt



 ¯

xT_i

w

_is

w

_stx

¯

_t





(2.6) from(1.3).

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Now let examine whether

(w

i

w

ik

+

w

ki

w

k

)

for k

∼

i and

w

is

w

stfor s

∈

Ni

∩

Ntare Hermitian in the inequality of(2.6).

Case 1:

(w

i

w

ik

+

w

ki

w

k

)

and

w

is

w

stare Hermitian matrices.

Then using inequality in(1.3), we get(2.6)as

≤

λ

₁

(w

2_i

)¯

xT_ix

¯

i

+



k:k∼i

λ

1

(w

2ik

)¯

x T i

¯

xi

+



k∼i

λ

1

(w

i

w

ik

+

w

ki

w

k

)



¯

xT ix

¯

i



¯

xT kx

¯

k

+



1≤i,t≤n





s∈Ni∩Nt

λ

1

(w

is

w

st

)



¯

xT_i

¯

xi



¯

xT_tx

¯

t



.

(2.7)

Case 2:

(w

i

w

ik

+

w

ki

w

k

)

is Hermitian for k

∼

i and

w

is

w

stis not a Hermitian matrix for s

∈

Ni

∩

Nt

,

1

≤

i

,

t

≤

n.

Since

(w

i

w

ik

+

w

ki

w

k

)

is Hermitian, from(1.3)we have



k∼i



 ¯

xT_i

(w

_i

w

_ik

+

w

_ki

w

_k

)¯

x_k



 ≤

λ

₁

((w

_i

w

_ik

+

w

_ki

w

_k

))



¯

xT_ix

¯

i



¯

xT_kx

¯

k

.

(2.8)

Now, let

w

is

w

stnot be a Hermitian matrix for s

∈

Ni

∩

Nt

,

1

≤

i

,

t

≤

n. Let us take the ratio of



¯

xT k

w

ks

w

stx

¯

t

¯

xT_kx

¯

t



(2.9)

for 1

≤

k

,

t

≤

n. If Nk

∩

Nt

=

∅, this ratio is zero. So let us consider Nk

∩

Nt

̸=

∅. Then we get



¯

xT k

w

ks

w

st

¯

xt

¯

xT k

¯

xt



=



 ¯

xT_k

w

ks

w

stx

¯

t



 ¯

xT_kx

¯

_t



and using the Cauchy Schwarz inequality we have

≥



 ¯

xT_k

w

ks

w

st

¯

xt





¯

xT k

¯

xk



¯

xT tx

¯

t

.

From(2.2), we get



¯

xT_k

w

ks

w

stx

¯

t

¯

xT_kx

¯

t



≥



 ¯

xT_k

w

_ks

w

_stx

¯

_t





¯

xT_ix

¯

i



¯

xT_t

¯

xt

.

(2.10)

Since(2.10)implies for eachx

¯

kandx

¯

t

min ¯ xk̸=0,¯xt̸=0





¯

xT_k

w

ks

w

stx

¯

t

¯

xT kx

¯

t





=



 ¯

xT_i

w

_ks

w

_stx

¯

_t





¯

xT ix

¯

i



¯

xT tx

¯

t

,

from inequality of(1.2)and since

λ

1

(w

is

w

st

)

is the largest eigenvalue of

w

is

w

stmatrix for s

∈

Ni

∩

Nt

,

1

≤

i

,

t

≤

n we have



 ¯

xT_i

w

_is

w

_st

¯

x_t





¯

xT_i

¯

xi



¯

xT_t

¯

xt

≤ |

λ

_i

|

(w

_is

w

_st

) ≤ λ

₁

(w

_is

w

_st

) ,

(2.11) i.e.



1≤i,t≤n



s∈Ni∩Nt



 ¯

xT_i

w

_is

w

_stx

¯

_t



 ≤

λ

₁

(w

_is

w

_st

)



¯

xT_ix

¯

i



¯

xT_t

¯

xt

.

(2.12)

If we arrange the expressions(2.8)and(2.12)in the inequality of(2.6), we can again get the inequality in(2.7). Case 3:

(w

i

w

ik

+

w

ki

w

k

)

is not Hermitian for k

∼

i and

w

is

w

stis a Hermitian matrix for s

∈

Ni

∩

Nt

,

1

≤

i

,

t

≤

n.

Since the matrix of

w

is

w

stis Hermitian, we get



1≤i,t≤n





s∈Ni∩Nt



 ¯

xT_i

w

_is

w

_stx

¯

_t





≤



1≤i,t≤n





s∈Ni∩Nt

λ

1

(w

is

w

st

)



¯

xT ix

¯

i



¯

xT tx

¯

t



.

(2.13)

On the other hand, let

(w

i

w

ik

+

w

ki

w

k

)

not be a Hermitian matrix for k

∼

i. By a similar argument to Case 2 we have



 ¯

xT_i

(w

i

w

ik

+

w

ki

w

k

)¯

xt





¯

xT ix

¯

i



¯

xT kx

¯

k

≤ |

λ

_i

|

(w

_i

w

_ik

+

w

_ki

w

_k

) ≤ λ

₁

(w

_i

w

_ik

+

w

_ki

w

_k

) ,

(2.14)

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i.e.



k∼i



 ¯

xT_i

(w

_i

w

_ik

+

w

_ki

w

_k

)¯

x_k



 ≤



k∼i

λ

1

(w

i

w

ik

+

w

ki

w

k

)



¯

xT_i

¯

xi



¯

xT_kx

¯

k

.

(2.15)

If we arrange the expressions(2.15)and(2.13)in the inequality of(2.6), we can again get the inequality in(2.7). Case 4: The matrices of

(w

i

w

ik

+

w

ki

w

k

)

for k

∼

i and

w

is

w

stfor s

∈

Ni

∩

Nt

,

1

≤

i

,

t

≤

n are not Hermitian matrices.

By applying the same methods as Cases 2 and 3, we have also(2.7). Therefore, we see that

λ

2 1

¯

x T ix

¯

i

≤

λ

1

(w

i2

)¯

x T i

¯

xi

+



k:k∼i

λ

1

(w

2ik

)¯

x T ix

¯

i

+



k∼i

λ

1

(w

i

w

ik

+

w

ki

w

k

)



¯

xT_ix

¯

i



¯

xT_k

¯

xk

+



1≤i,t≤n





s∈Ni∩Nt

λ

1

(w

is

w

st

)



¯

xT i

¯

xi



¯

xT tx

¯

t



(2.16)

in all situations. If we use(2.2), we have

≤

λ

₁

(w

_i2

)¯

xT_i

¯

xi

+



k:k∼i

λ

1

(w

2ik

)¯

x T ix

¯

i

+



k∼i

λ

1

(w

i

w

ik

+

w

ki

w

k

)¯

xTix

¯

i

+



1≤i,t≤n



s∈Ni∩Nt

λ

1

(w

is

w

st

)¯

xTix

¯

i

.

(2.17) Thus we obtain

λ

1

≤



λ

1

(w

2i

) +



k:k∼i

λ

1

(w

i2,k

) +



k∼i

λ

1

(w

i

w

ik

+

w

ki

w

k

) +



1≤i,t≤n



s∈Ni∩Nt

λ

1

(w

i,s

w

s,t

),

i.e.

λ

1

≤

max i∈V















λ

1

(w

2i

) +



k:k∼i

λ

1

(w

ik2

) +



k∼i

λ

1

(w

i

w

ik

+

w

ki

w

k

) +



1≤i,t≤n



s∈Ni∩Nt

λ

1

(w

is

w

st

)









.

(2.18) Now suppose that equality in(2.1)holds. Then all the equalities in the above argument must be equalities. From equality in(2.17)we have

¯

xT_i

¯

xi

= ¯

xTk

¯

xk (2.19)

for all k

,

k

∼

i and for all k

,

k

∼

p

,

p

∼

i. From this we sayx

¯

k

̸= ¯

0.

From equality in(2.16)and usingLemma 1we get thatx

¯

i is eigenvector of

w

i,k

, (w

i

w

ik

+

w

ki

w

k

), w

is

w

sk such that

s

∈

Ni

∩

Nkfor the largest eigenvalues

λ

1

(w

ik

), λ

1

(w

i

w

ik

+

w

ki

w

k

) , λ

1

(w

is

w

sk

)

respectively and for any k

¯

xk

=

bikx

¯

i (2.20)

for some bik. Similarly, from equality in(2.16)and usingLemma 1we also get thatx

¯

iis an eigenvector of

w

is

w

stsuch that

s

∈

Ni

∩

Ntfor the largest eigenvalue

λ

1

(w

is

w

st

)

for any 1

≤

i

,

t

≤

n

¯

xt

=

cit

¯

xi (2.21) for some cit. From(2.19)we get



b2_ik

−

1

 ¯

xT_ix

¯

i

=

0



c_it2

−

1

 ¯

xT_ix

¯

i

=

0

,

i.e. bik

= ±

1

,

cit

= ±

1 asx

¯

Tix

¯

i

>

0

.

(2.22)

Now let’s take any vertex i.

Since

w

ikis a positive definite matrix,

w

2i,kand

w

2

i are also positive matrices. Thus, we get

¯

xT_i

w

_i2x

¯

i

>

0

,

¯

xT_i

w

_ik2x

¯

i

>

0

.

(2.23)

From equality in(2.16)we get

−



k∼i bikx

¯

Ti

(w

i

w

ik

+

w

ki

w

k

) ¯

xi

=



k∼i

|

bik

|



 ¯

xT_i

(w

_i

w

_ik

+

w

_ki

w

_k

)¯

x_i



(2.24)

(6)

and



1≤i,t≤n



s∈Ni∩Nt citx

¯

Ti

w

is

w

stx

¯

i

=



1≤i,t≤n



s∈Ni∩Nt

|

cit

|



 ¯

xT_i

w

_is

w

_stx

¯

_i



.

(2.25)

Since bik

= ±

1, therefore from(2.24), we get bik

= −

1 for all k

,

k

∼

i. Hence,

¯

xk

= −¯

xi

for all k

,

k

∼

i.

Since cit

= ±

1, therefore from(2.25)we get cit

=

1 for all 1

≤

i

,

t

≤

n. Hence,

¯

xt

= ¯

xi

for all 1

≤

i

,

t

≤

n.

Let U

= {

k

: ¯

xk

= −¯

xi

}

for all k

,

k

∼

i and W

= {

k

: ¯

xk

= ¯

xi

}

such that k

∈

Ni

∩

Ntfor all 1

≤

i

,

t

≤

n. Moreover, from

equality in(2.17),x

¯

iis a common eigenvector of

w

i,k, corresponding to the largest eigenvalue

λ

1

(w

ik

)

for all i

,

k. Since G is

connected V

=

U

∪

W and the subgraphs induced by U and W respectively are empty graphs. Hence G is bipartite.

Now we have L

(

G

)¯

xi

=

λ

1x

¯

i

,

(2.26) i.e.

w

ix

¯

i

−



k:k∼i

w

ikx

¯

k

=

λ

1x

¯

i

.

(2.27) For i

,

p

∈

U

λ

1x

¯

i

=

w

ix

¯

i

+



k:k∼i

w

ikx

¯

i (2.28) and

λ

1x

¯

i

=

w

p

¯

xi

+



k:k∼p

w

ipx

¯

i

.

(2.29) So, we get



λ

1

(w

i

) − λ

1

(w

p

)

xi

=

0 (2.30)

from(2.28)and(2.29)asx

¯

i is an eigenvector of

w

icorresponding to the largest eigenvalue

λ

1

(w

i

)

for all i. Sincex

¯

i

̸=

0,

therefore

λ

1

(w

i

)

is constant for all i

∈

U. Similarly we can also show that

λ

1

(w

i

)

is constant for all i

∈

W . Hence G is a

bipartite semiregular graph.

Conversely, suppose that conditions (i)–(ii) of the theorem hold for the graph G. We must prove

λ

1

=

max i









λ

2 1

(w

i

) +



k:k∼i

λ

2 1

(w

ik

) +



k:k∼i

λ

1

(w

i

w

ik

+

w

ki

w

k

) +



1≤i,t≤n



s∈Ni∩Nt

λ

1

(w

is

w

st

)







.

Let U

,

W be partite sets of G. Also let

λ

1

(w

i

) = α

for i

∈

U and

λ

1

(w

i

) = β

for i

∈

W .

Since G is a bipartite graph,therefore U

,

W are partite sets of G. Ni

∩

Nt is empty for all 1

≤

i

,

t

≤

n. To prove, let

Ni

∩

Nt

̸=

∅. Thus, there is a vertex s such that s

∼

i

,

s

∼

t. On the other hand, let i

∈

U

,

t

∈

W .

s

∼

i

⇒

s

∈

W (2.31)

s

∼

t

⇒

s

∈

U

.

(2.32)

This is contradiction according to(2.30)and(2.31). Hence, we found that Ni

∩

Nt

=

∅. The following equation can be easily verified:



2

α

2

+

2

αβ







¯

x

¯

x

...

¯

x

−¯

x

−¯

x

...

−¯

x







=

A







¯

x

¯

x

...

¯

x

−¯

x

−¯

x

...

−¯

x







(7)

where A=              w2 1+  k∈N1 w2 1,n −(w1w1,2+w1,2w2) +  k∈N1∩_N2 w1kwk,2· · · −(w1w1,n+w1,nwn) +  k∈N1∩Nn w1,kwk,n −(w₂w₁_,₂+w₁_,₂w₁) +  k∈N1∩_N2 w1,kwk,2 w22+  k∈N2 w2 1,2 . . . −(w2w1,n+w1,nwn) +  k∈N2∩Nn w2,kwk,n .. . ... ... −(w_nw₁_,_n+w₁_,_nw₁) +  k∈N1∩Nn w1,kwk,n −(w2w1,n+w1,nwn) +  k∈N2∩Nn w2,kwk,n. . . w2n+  k∈Nn w2 k,n              . Therefore 2

α

2

+

2

αβ

is an eigenvalue of L2

(

G

)

. So 2

α

2

+

2

αβ ≤ λ

2₁

.

(2.33)

On the other hand, we have

λ

2 1

(w

i

) +



k:k∼i

λ

2 1

(w

ik

) +



k:k∼i

λ

1

(w

i

w

ik

+

w

ik

w

k

) +



1≤i,t≤n



s∈Ni∩Nt

λ

1

(w

is

)λ

1

(w

st

) =

2

α

2

+

2

αβ

(2.34)

for all i

∈

V . We get

λ

2 1

≤

max_i_∈_V



λ

1

(w

i2

) +



k:k∼i

λ

1

(w

ik2

) +



k∼i

λ

1

((w

i

w

ik

+

w

ik

w

k

)) +



1≤i,t≤n



s∈Ni∩Nt

λ

1

(w

is

w

sk

)



=

2

α

2

+

2

αβ

from inequality in(2.18). Hence the theorem is proved by(2.33).

Corollary 2. Let G be a simple connected weighted graph where each edge weight

w

i,jis a positive number. Then

λ

1

≤

max i









w

2 i

+

w

i

+



j



(w

i

w

i,j

+

w

i,j

w

j

) :

i

∼

j

 +



1≤i,t≤n



s



w

is

w

sj

:

s

∈

Ni

∩

Nt









.

(2.35)

Moreover equality holds in(2.35)if and only if G is a bipartite semiregular graph.

Proof. We have

λ

₁

(w

_i

) = w

_iand

λ

1

(w

ij

) = w

ijfor all i

,

j. FromTheorem 5we get the required result. Corollary 3. Let G be a simple connected unweighted graph. Then

λ

1

≤

max i









d2_i

+

di

+



j



di

+

dj

+



N_i

∩

N_j



 :

i

∼

j

 +



j





N_i

∩

N_j



 :

i j









where diis the degree of vertex i and



Ni

∩

Nj



is the number common neighbors of i and j.

Proof. For an unweighted graph,

w

i,j

=

1 for i

∼

j. Therefore

w

i

=

di. UsingCorollary 2we get the required results. Example 1. Let G1

=

(

V1

,

E1

)

and G2

=

(

V2

,

E2

)

be a weighted graph where V1

= {

1

,

2

,

3

,

4

}

,

E1

= {{

1

,

4

}

, {

2

,

3

}

, {

3

,

4

}}

and each weight is a positive definite matrix of three order. Let V2

= {

1

,

2

,

3

,

4

,

5

,

6

,

7

}

,

E2

=



_{₁_,₄_}_{, {}₂_,₄_}_{, {}₃_,₄_}_,

{4,5}, {5,6}, {5,7}



such that each weight is a positive definite matrix of order two. Assume that the Laplacian matrices of G1and G2are as follows:

L

(

G1

) =







3 1

−

1 0 0 0 0 0 0

−

3

−

1 1 1 3

−

1 0 0 0 0 0 0

−

1

−

3 1

−

1

−

1 5 0 0 0 0 0 0 1 1

−

5 0 0 0 5 0 2

−

5 0

−

2 0 0 0 0 0 0 0 5 2 0

−

5

−

2 0 0 0 0 0 0 2 2 5

−

2

−

2

−

5 0 0 0 0 0 0

−

5 0

−

2 6 0 2

−

1 0 0 0 0 0 0

−

5

−

2 0 10 5 0

−

5

−

3 0 0 0

−

2

−

2

−

5 2 5 8 0

−

3

−

3

−

3

−

1 1 0 0 0

−

1 0 0 4 1

−

1

−

1

−

3 1 0 0 0 0

−

5

−

3 1 8 2 1 1

−

5 0 0 0 0

−

3

−

3

−

1 2 8







(8)

and L

(

G2

) =







1 1 0 0 0 0

−

1

−

1 0 0 0 0 0 0 1 2 0 0 0 0

−

1

−

2 0 0 0 0 0 0 0 0 1 1 0 0

−

1

−

1 0 0 0 0 0 0 0 0 1 3 0 0

−

1

−

3 0 0 0 0 0 0 0 0 0 0 1 1

−

1

−

1 0 0 0 0 0 0 0 0 0 0 1 4

−

1

−

4 0 0 0 0 0 0

−

1

−

1

−

1

−

1

−

1

−

1 4 4

−

1

−

1 0 0 0 0

−

1

−

2

−

1

−

3

−

1

−

4 4 14

−

1

−

5 0 0 0 0 0 0 0 0 0 0

−

1

−

1 3 3

−

1

−

1

−

1

−

1 0 0 0 0 0 0

−

1

−

5 3 18

−

1

−

6

−

1 7 0 0 0 0 0 0 0 0

−

1

−

1 1 1 0 0 0 0 0 0 0 0 0 0

−

1

−

6 1 6 0 0 0 0 0 0 0 0 0 0

−

1

−

1 0 0 1 1 0 0 0 0 0 0 0 0

−

1

−

7 0 0 1 7







.

The largest eigenvalues of L

(

G1

)

and L

(

G2

)

are

λ

1

=

22

.

25

, λ

2

=

26

.

16 rounded two decimal places and the above

mentioned bounds give the following results:

(1.4) (1.5) (1.6) (2.1) G1 26

.

41 26

.

38 23

.

17 25

.

03 G2 33

.

98 29

.

65 27

.

11 27

.

22

.

Consequently, we see that the bound in(2.1)is better than the bounds in(1.4)and(1.5). But it is not better than the bound in(1.6)from the above table.

Acknowledgment

Thanks are due to Mr. Kinkar Chandra Das for his kindness in checking this paper.

References

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[3] K.C. Das, A characterization on graphs which achieve the upper bound for the largest Laplacian eigenvalue of graphs, Linear Algebra Appl. 376 (2004) 173–186.

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[7] J.-S. Li, D. Zhang, A new upper bound for eigenvalues of the Laplacian matrix of a graph, Linear Algebra Appl. 265 (1997) 93–100. [8] J.-S. Li, D. Zhang, On Laplacian eigenvalues of a graph, Linear Algebra Appl. 285 (1998) 305–307.

[9] J.-S. Li, Y.-L. Pan, de Caen’s inequality and bounds on the largest Laplacian eigenvalue of a graph, Linear Algebra Appl. 328 (2001) 153–160. [10] R. Merris, A note on Laplacian graph eigenvalues, Linear Algebra Appl. 285 (1998) 33–35.

[11] Y.-L. Pan, Sharp upper bounds for the Laplacian graph eigenvalues, Linear Algebra Appl. 355 (2002) 287–295. [12] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.

[13] S. Sorgun, Ş Büyükköse, H.S. Özarslan, An upper bound on the spectral radius of weighted graphs (submitted for publication).

[14] K.C. Das, R.B. Bapat, A sharp upper bound on the largest Laplacian eigenvalue of weighted graph, Linear Algebra Appl. 409 (2005) 153–165. [15] K.C. Das, Extremal graph characterization from the upper bound of the Laplacian spectral radius of weighted graphs, Linear Algebra Appl. 427 (2007)