On the largest eigenvalues of bipartite graphs which are nearly complete

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Linear Algebra and its Applications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / l a a

On the largest eigenvalues of bipartite graphs which are

nearly complete

Yi-Fan Chen

a

, Hung-Lin Fu

a

, In-Jae Kim

b,∗

, Eryn Stehr

b

, Brendon Watts

c

a_{Department of Applied Mathematics, National Chiao Tung University, Hsin Chu 30050, Taiwan, ROC} b_{Department of Mathematics and Statistics, Minnesota State University, Mankato, MN 56001, United States} c_{Department of Mathematics, University of Oklahoma, Norman, OK 73019-0315, United States}

A R T I C L E I N F O A B S T R A C T Article history:

Received 10 June 2009 Accepted 4 September 2009 Available online 14 October 2009

Submitted by S. Kirkland AMS classiﬁcation: 05C50 15A18 Keywords: Bipartite graph Eigenvector Largest eigenvalue

For positive integers p, q, r, s and t satisfying rtp and stq, let G(p, q;r, s;t)be the bipartite graph with partite sets{u1,. . ., up}

and{v1,. . ., vq}such that uiand vjare not adjacent if and only if

there exists a positive integer k with 1kt such that(k−1)r+

1ikr and(k−1)s+1jks. In this paper we study the

largest eigenvalues of bipartite graphs which are nearly complete. We first compute the largest eigenvalue (and all other eigenvalues) of G(p, q;r, s;t), and then list nearly complete bipartite graphs ac-cording to the magnitudes of their largest eigenvalues. These results give an affirmative answer to [1, Conjecture 1.2] when the number of edges of a bipartite graph with partite sets U and V , having |U| =p and|V| =q for pq, is pq−2. Furthermore, we refine [1, Conjecture 1.2] for the case when the number of edges is at least

pq−p+1.

1. Introduction and preliminary

Let G be a (simple) graph with vertex set V

(

G

) = {

v1,

. . .

, vn

}

and edge set E

(

G

) = {

vi, vj

|

viand vj are adjacent

}

. If H is a graph with V

(

H

) ⊆

V

(

G

)

and E

(

H

) ⊆

E

(

G

)

, then H is a subgraph of G. If H

/=

G, then H is a proper subgraph. If H is a proper subgraph of G, and there is no proper subgraph Hof G such that H is a proper subgraph of H, then H is a maximal proper subgraph of G.

∗Corresponding author.

E-mail address: in-jae.kim@mnsu.edu (I.-J. Kim).

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The adjacency matrix of G on n vertices is the n by n matrix A

(

G

)

whose entries aijare given by aij

=

1, if vi, vj

∈

E

(

G

);

0, otherwise

.

Note that A

(

G

)

is symmetric. Hence, all the eigenvalues of A

(

G

)

are real. The eigenvalues of A

(

G

)

are called eigenvalues of the graph G. We can list the eigenvalues of graph G in non-increasing order

λ

1

(

G

)

λ

2

(

G

)

· · ·

λ

n

(

G

).

By [4, Theorem 8.4.5], we have the following result.

Proposition 1. If H is a subgraph of G, then

λ

1

(

H

)

λ

1

(

G

).

A graph G is bipartite if its vertex set can be partitioned into two parts U and V , so called partite sets, such that every edge has one end in U and the other in V . Let G be a bipartite graph with partite sets U

= {

u1,

. . .

, up

}

and V

= {

v1,

. . .

, vq

}

, and A be the

(

p

+

q

)

by

(

p

+

q

)

adjacency matrix of G of the form O B BT O , (1)

where B

= [

bij

]

is the p by q matrix such that bij

=

1, if uiand vjare adjacent

;

0, otherwise

.

If B is the p by q matrix with each entry equal to 1, then the bipartite graph G is a complete bipartite graph, and denoted by Kp,q. The following two results describe spectral properties of bipartite graphs (Theorem2; see [8, Theorem 8.6.9]) and the matrix product of the form BBT(Proposition3; see [4]).

Theorem 2. Let G be a bipartite graph, and A be its adjacency matrix

.

Then the eigenvalues of A are symmetric about the origin, i

.

e

.

, if

λ ∈ σ(

A

)

, then

−λ ∈ σ (

A

).

Moreover,

μ(

0

)

is an eigenvalue of A2if and only if

±√μ

are eigenvalues of A

.

For each x

∈

R

n, if an n by n real symmetric matrix M satisﬁes xTMx 0, then M is a positive semidef-inite matrix. It is well known that each eigenvalue of a positive semideﬁnite matrix is a nonnegative real number (see [4]).

Proposition 3. Let B be a p by q matrix

.

Then

(

a

)

The rank of B is equal to that of the p by p matrix BBT

.

(

b

)

The matrix BBTof order p is positive semideﬁnite

.

(

c

)

The number of nonzero eigenvalues of BBTis equal to the rank of BBT

.

(

d

)

If p q, then the q by q matrix BTB has the same eigenvalues as BBTof order p, counting multiplicity, together with additional

(

q

−

p

)

eigenvalues equal to 0

.

For positive integers p, q, r, s and t satisfying rt p and st q, we deﬁne G

(

p, q

;

r, s

;

t

)

to be the bipartite graph with partite sets

{

u1,

. . .

, up

}

and

{

v1,

. . .

, vq

}

such that uiand vjare not adjacent if and only if there exists a positive integer k with 1 k t such that

(

k

−

1

)

r

+

1 i kr and

(

k

−

1

)

s

+

1 j ks. In Fig. 1 there is an edge between uiand vjif and only if they are not connected by a dotted line.

In [2,3] the largest and the second largest eigenvalues of certain types of trees are obtained. This motivates our study of the largest eigenvalues of bipartite graphs which are close to a complete bipartite graph. In this paper we ﬁnd all the eigenvalues of the bipartite graph G

(

p, q

;

r, s

;

t

)

by computing eigen-values of the square of the adjacency matrix of G

(

p, q

;

r, s

;

t

)

, and list its eigenvalues according to their

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Fig. 1. G(5, 6;1, 2;2).

magnitudes. Using this result, we list nearly complete bipartite graphs according to the magnitudes of their largest eigenvalues. These results give an afﬁrmative answer to [1, Conjecture 1.2] when the number of edges of a bipartite graph with partite sets U and V , having

|

U

| =

p and

|

V

| =

q for p q, is pq

−

2. Furthermore, by Theorem7, we reﬁne [1, Conjecture 1.2] when the number of edges is at least pq

−

p

+

1 (see Conjecture11).

2. Eigenvalues of G

(

p, q

;

r, s

;

t

)

Let A be the adjacency matrix of G

(

p, q

;

r, s

;

t

)

of the form (1). If we can compute eigenvalues of A2, then, by Theorem2, we can ﬁnd the eigenvalues of A. Note that

A2

=

O B BT O O B BT O

=

BBT O O BTB

.

By Proposition3(d), it sufﬁces to compute eigenvalues of BBT.

Let

(

a

)

r×sbe the r by s matrix each of whose entries is a. When r

=

s, we use

(

a

)

rto denote

(

a

)

r×r. Then B

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(

0

)

r×s

(

1

)

r×s

· · ·

(

1

)

r×s

(

1

)

r×s

(

0

)

r×s . ..

...

. ._. . ._.

₍

₁

₎

r×s

(

1

)

rt×(q−st)

(

1

)

r×s

· · ·

(

1

)

r×s

(

0

)

r×s

(

1

)

₍p−rt)×st

(

1

)

(p−rt)×(q−st) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , and BBT

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(

q

−

s

)

r

(

q

−

2s

)

r

· · ·

(

q

−

2s

)

r

(

q

−

2s

)

r

(

q

−

s

)

r . ..

...

. ._. . ._.

₍

_q

₋

_2s

₎

r

(

q

−

s

)

rt×(p−rt)

(

q

−

2s

)

r

· · ·

(

q

−

2s

)

r

(

q

−

s

)

r

(

q

−

s

)

₍p−rt)×rt

(

q

)

(p−rt) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.

Assume that p

>

rt and q

>

st. Then it can be shown that, by elementary row and column opera-tions, B has the same rank as that of the

(

t

+

1

)

by

(

t

+

1

)

matrix

C

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1

· · ·

1 1 1 0 . ..

... ...

...

. ._. . ._. ₁

...

1

· · ·

1 0 1 1

· · ·

1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.

(2)

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If t

=

1, then C

=

0 1 1 1 ,

which is nonsingular. For t 2, note that the

(

1, 1

)

-block of C is

(

1

)

t

−

It, which is nonsingular. Let Ri be the ith row of the matrix (2). By using the row operation

(

t

−

1

)

Rt+1and then Rt+1

−

Rifor each i

=

1, 2,

. . .

, t, we can obtain the following matrix whose rank is equal to that of C:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1

· · ·

1 1 1 0 . ..

... ...

...

. ._. . ._. ₁

...

1

· · ·

1 0 1 0

· · ·

0

−

1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.

This implies that the rank of C (and hence BBT) is t

+

1.

If p

=

rt (resp. q

=

st), then the matrix C is obtained by deleting the last row (resp. the last column) from the matrix in the form (2). By Proposition3(a) and (c), we get the following result.

Proposition 4. If p

>

rt and q

>

st, then BBThas t

+

1 nonzero eigenvalues, and if p

=

rt or q

=

st, then BBThas t nonzero eigenvalues

.

The p by p matrix BBTcan be rewritten as follows:

BBT

= (

q

−

2s

)(

1

)

p

+

sM, (3) where M

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

(

1

)

r O . ._.

₍

₁

₎

rt×(p−rt) O

(

1

)

r

(

1

)

₍p−rt)×rt

(

2

)

p−rt ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ p×p

.

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In the following we ﬁnd some eigenvalues and their corresponding eigenvectors of M and use them to ﬁnd the eigenvalues of BBT. The vector eidenotes the vector with exactly one nonzero entry that is 1 and located in the ith position. For a square matrix M of order p and a p by 1 vector x, if Mx

=

0, then x is a nullvector of M.

Proposition 5. For p

>

rt and q

>

st, let BBTbe of the form

(

3

)

, and M be the p by p matrix of the form

(

4

).

Then the following hold

:

(

a

)

The p

− (

t

+

1

)

nonzero vectors in the set

{

erk+1

−

erk+j

|

k

=

0, 1,

. . .

, t

−

1 and j

=

2,

. . .

, r

} ∪ {

ert+1

−

ej

|

j

=

rt

+

2,

. . .

, p

}

(5) are linearly independent nullvectors of M

.

(

b

)

The p

− (

t

+

1

)

linearly independent vectors in

(

5

)

are nonzero nullvectors of BBT

.

(

c

)

For t 2, the scalar r is an eigenvalue of M with multiplicity t

−

1, and the t

−

1 nonzero vectors in the set ⎧ ⎨ ⎩xk xk

=

r i=1 ei

−

r j=1 erk+j, k

=

1,

. . .

, t

−

1 ⎫ ⎬ ⎭ (6)

are linearly independent eigenvectors corresponding to the eigenvalue r

.

(

d

)

When t 2, the

(

t

−

1

)

nonzero vectors in

(

6

)

are linearly independent eigenvectors of BBT corre-sponding to the eigenvalue rs

.

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Proof. A direct computation proves (a) and (c). Since the nonzero vectors in

(

5

)

and (6) are nullvectors of the matrix

(

1

)

pin (3), (b) and (d) follow.

By Proposition4, when p

>

rt and q

>

st, there are t

+

1 nonzero eigenvalues of BBT. Hence, by Proposition5, there are two more nonzero eigenvalues of BBT to be computed. Let W1and W2 be subspaces of

R

p. We say that W1and W2are perpendicular (denoted by W1

⊥

W2) provided that for any vectors w1

∈

W1and w2

∈

W2, w1is perpendicular to w2, i.e., w1Tw2

=

0. For a subspace W of

R

p_{, we deﬁne W}⊥_{as follows:}

W⊥

= {

z

|

zTw

=

0 for each w

∈

W

}.

Let S be a p by p matrix. We say that W is invariant under S if Sw

∈

W for each w

∈

W . The following result can be found in [6, Theorems 4.2 and 4.3].

Theorem 6. Let S be a p by p real symmetric matrix

.

Then the following hold

:

(

a

)

If E1and E2are the eigenspaces of S corresponding to distinct eigenvalues, then E1

⊥

E2

.

(

b

)

If W is a subspace of

R

pwhich is invariant under S, then W⊥is also invariant under S

.

Let W be the vector space spanned by the vectors in

(

5

)

and (6). It can be veriﬁed that W⊥is spanned by z1and z2, i.e.,

W⊥

=

z1, z2

, where z1

=

₍

1

)

rt×1

(

0

)

₍p−rt)×1 and z2

=

₍

0

)

rt×1

(

1

)

₍p−rt)×1

.

Moreover, by Theorem6, the eigenvectors of BBTcorresponding to the remaining two nonzero eigen-values are in W⊥, and

(

BBT

)

z

∈

W⊥for every z

∈

W⊥. LetZ

= {

z1, z2

}

. Then the eigenvalues of the 2 by 2Z-matrix for BBT(see [7, p. 329]) are the remaining two eigenvalues of BBT(see [5, Proposition 1.5.4]). To ﬁnd the_Z-matrix for BBT, we compute

(

BBT

)

zifor each i

=

1, 2:

(

BBT

)

z1

= [

r

(

q

−

s

) +

r

(

t

−

1

)(

q

−

2s

)]

z1

+ [

rt

(

q

−

s

)]

z2,

(

BBT

)

z2

= [(

p

−

rt

)(

q

−

s

)]

z1

+ [

q

(

p

−

rt

)]

z2

.

Hence, the 2 by 2_Z-matrix for BBTis

r

(

q

−

s

) +

r

(

t

−

1

)(

q

−

2s

) (

p

−

rt

)(

q

−

s

)

rt

(

q

−

s

)

q

(

p

−

rt

)

.

By computing the eigenvalues of theZ-matrix for BBT, we get the eigenvalues of BBT:

pq

−

2rst

+

rs

±

(

pq

−

2rst

+

rs

)

2

−

_4rs

(

_p

−

_rt

)(

_q

−

_st

)

2

.

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By Theorem2, Propositions3,4and5along with the eigenvalues of BBTin (7), we have found all the eigenvalues of G

(

p, q

;

r, s

;

t

)

.

Theorem 7. Let A be the adjacency matrix of the bipartite graph G

(

p, q

;

r, s

;

t

).

Then the following hold

:

(6)

(

a

)

For p

>

rt, q

>

st and t 2, the eigenvalues of A are

λ

1

=

 pq

−

2rst

+

rs

+

(

pq

−

2rst

+

rs

)

2

−

_4rs

(

_p

−

_rt

)(

_q

−

_st

)

2

λ

2

=

√

rs

λ

3

=

 pq

−

2rst

+

rs

−

(

pq

−

2rst

+

rs

)

2

−

_4rs

(

_p

−

_rt

)(

_q

−

_st

)

2

λ

4

=

0

− λ

3

−λ

2

−λ

1

.

Moreover, the multiplicities of

±λ

1 and

±λ

3are 1, the multiplicities of

±λ

2 are t

−

1, and the multiplicity of

λ

4

=

0 is

(

p

+

q

) −

2

(

t

+

1

).

(

b

)

If p

=

rt or q

=

st with t 2, then the eigenvalues of A are

λ

1

=

√

pq

−

2rst

+

rs

λ

2

=

√

rs

λ

3

=

0

−λ

2

−λ

1

.

(

c

)

If t

=

1, then the eigenvalues of A are

λ

1

=

 pq

−

rs

+

(

pq

−

rs

)

2

₋

_4rs

₍

_p

₋

_r

₎₍

_q

₋

_s

₎

2

λ

2

=

 pq

−

rs

−

(

pq

−

rs

)

2

₋

_4rs

₍

_p

₋

_r

₎₍

_q

₋

_s

₎

2 0

−λ

2

−λ

1

.

Proof. We here show that for nonnegative

λ

1,

λ

2and

λ

3in (a),

λ

21

λ

2 2

λ

2

3when p rt, q st and t 2. Then the orders of eigenvalues according to their magnitudes in (a), (b) and (c) follow.

We ﬁrst show that

λ

2₁

λ

2₂. Since p rt and q rt, it follows that pq

−

2rst

+

rs rst2

−

2rst

+

rs

=

rs

(

t2

−

2t

+

1

)

=

rs

(

t

−

1

)

2

.

Hence, for t 3,

λ

2₁

λ

2₂. Let t

=

2. Consider

pq

−

3rs

+

(

pq

−

3rst

)

2

−

_4rs

(

_p

−

_2r

)(

_q

−

_2s

)

2

−

rs

.

(8)

If pq 5rs, then (8) is nonnegative and hence

λ

2₁

λ

2 2. Suppose that pq

<

5rs, i.e., 5rs

−

pq

>

0. We show that

(

pq

−

3rs

)

2

−

4rs

(

p

−

2r

)(

q

−

2s

) − (

5rs

−

pq

)

2 (9)

is nonnegative. By a simple calculation, it can be shown that (9) is equal to 8rs2p

+

8r2sq

−

32r2s2

.

Since p 2r and q 2s, we have 8rs2p

+

8r2sq

−

32r2s2 16r2

s2

+

16r2s2

−

32r2s2

=

0

.

Hence,

λ

2₁

λ

2₂.

(7)

Next, we show that

λ

2₂

λ

2

3when p rt, q st and t 2. The difference

λ

2 2

− λ

2 3is equal to rs

+

2rst

−

pq

+

(

pq

−

2rst

+

rs

)

2

₋

_4rs

₍

_p

₋

_rt

₎₍

_q

₋

_st

₎

2

.

If rs

+

2rst

−

pq 0,

λ

2 2

λ

2 3.

Suppose that rs

+

2rst

−

pq

<

0, i.e., pq

−

2rst

−

rs

>

0. Then, in order to show

λ

2₂

λ

2

3, it sufﬁces to show that

(

pq

−

2rst

+

rs

)

2

−

4rs

(

p

−

rt

)(

q

−

st

) − (

pq

−

2rst

−

rs

)

2 (10)

is nonnegative. By a simple calculation, it can be shown that (10) is equal to 4rs2tp

+

4r2stq

−

4r2s2t2

−

8r2s2t

.

Since p rt, q st and t 2, we have

4rs2tp

+

4r2stq

−

4r2s2t2

−

8r2s2t 4r2s2t2

+

4r2s2t2

−

4r2s2t2

−

8r2s2t

=

4r2s2t

(

t

−

2

)

0

.

Hence, the result follows.

3. List of nearly complete bipartite graphs

We now list bipartite graphs, missing at most two edges from a complete bipartite graph, according to the magnitudes of their largest eigenvalues

λ

1. We denote by G(i)the bipartite graph with the ith largest

λ

1among all bipartite graphs with 2n vertices.

Theorem 8. For n 3, G(1)

₌

_K

n,n, G(2)

=

Kn−1,n+1, G(3)

=

G

(

n, n

;

1, 1

;

1

)

, G(4)

=

G

(

n

−

1, n

+

1

;

1, 1

;

1

)

, G(5)

=

G

(

n, n

;

2, 1

;

1

)

, G(6)

=

G

(

n, n

;

1, 1

;

2

)

, G(7)

=

Kn−2,n+2, G(8)

=

G

(

n

−

1, n

+

1

;

2, 1

;

1

)

, G(9)

=

G

(

n

−

1, n

+

1

;

1, 2

;

1

)

and G(10)

=

G

(

n

−

1, n

+

1

;

1, 1

;

2

).

Proof. Let H be a subgraph of Kp,qwith 1 p q and p

+

q

=

2n. Note that

λ

1

(

Kp,q

) =

√

pq (11)

and

√

pq n. Furthermore, by Proposition1,

λ

1

(

H

)

√

pq. Hence, G(1)

=

Kn,n

.

By Proposition1and the fact that

√

pq with p

+

q

=

2n is increasing as the value of p grows from 1 to n, it is sufﬁcient to consider G

(

n, n

;

1, 1

;

1

)

and Kn−1,n+1for G(2). By Theorem7and (11), we have

λ

1

(

G(2)

) =

max 1 √ 2

[(

n 2

₋

₁

_{) + (}

_n4

₋

_6n2

₊

_8n

₋

₃

₎

1/2

_]

1/2_,

₍

_n2

₋

₁

₎

1/2 . Note that √1 2

[(

n 2

₋

₁

₎

+ (

n4

−

6n2

+

8n

−

3

)

1/2

]

1/2_√1 2

[(

n 2

₋

₁

_{) + (}

_n4

₋

_6n2

₊

_8n

₋

₃

₊

_4n2

₋

_8n

₊

₄

₎

1/2

_]

1/2

₍

_n2

−

1

)

1/2. Hence, G(2)

=

Kn−1,n+1

.

Similarly, in order to ﬁnd G(3), it sufﬁces to consider G

(

n, n

;

1, 1

;

1

)

, G

(

n

−

1, n

+

1

;

1, 1

;

1

)

and Kn−2,n+2. We compute the largest largest eigenvalues of these three bipartite graphs by Theorem7and (11), and then use the facts, n4

−

6n2

+

8n

−

3

+

4n2

−

8n

+

4 n4

₋

_8n2

₊

_8n

₊

_{4 for n}_{3 and}

(

n

−

4

)

1/2

=

√1 2

[(

n

2

₋

₁

_{) + (}

_n4

₋

_14n2

₊

₄₉

₎

1/2

_]

1/2 _{in order to compare the largest eigenvalues.} This gives

(8)

Next, we consider the maximal proper subgraphs of G(1), G(2), G(3), and the complete bipartite graph Kn−2,n+2for G(4), using Theorem7and (11). By repeating this process, considering the maximal subgraphs of G(1),

. . .

, G(i), and a complete bipartite graph Kp,q with p

+

q

=

2n (which was not considered in the previous steps), we can get G(i+1)for i

=

3,

. . .

, 9.

Example 9. The following is the list of bipartite graphs with 40 vertices according to the magnitudes of the largest eigenvalues:

Graph

λ

1 1. K20,20 20 2. K19,20 19.97498436 3. G

(

20, 20

;

1, 1

;

1

)

19.95227248 4. G

(

19, 21

;

1, 1

;

1

)

19.92720282 5. G

(

20, 20

;

2, 1

;

1

)

19.90663008 6. G

(

20, 20

;

1, 1

;

2

)

19.90432602 7. K18,20 19.89974874 8. G

(

19, 21

;

2, 1

;

1

)

19.88164226 9. G

(

19, 21

;

1, 2

;

1

)

19.88138664 10. G

(

19, 21

;

1, 1

;

2

)

19.87920160

The computation of

λ

1 can be done by the open source mathematical software SAGE (see

http://www.sagemath.org). The following is a SAGE code for

λ

1

(

G

(

20, 20

;

2, 1

;

1

))

:

p

=

20

;

q

=

20

;

r

=

2

;

s

=

1

;

t

=

1 def a

(

i, j

)

: for k in

[

1

..

t

]

: if (i in

[

r

∗ (

k

−

1

) +

1

..

r

∗

k

]

) and (j in

[

s

∗ (

k

−

1

) +

1

..

s

∗

k

]

): return 0 else: return 1

B

=

matrix

([[

a

(

i, j

)

for j in

[

1

..

q

]]

for i in

[

1

..

p

]])

E

= (

B

∗

B

.

transpose

()).

eigenvalues

()

print sqrt

(

E

[

p

−

1

])

From Theorem8it follows that

λ

1

(

G

(

p, q

;

2, 1

;

1

))

λ

1

(

G

(

p, q

;

1, 2

;

1

))

λ

1

(

G

(

p, q

;

1, 1

;

2

))

for

(

p, q

) ∈ {(

n, n

)

,

(

n

−

1, n

+

1

)}

. This can be generalized to the case for any positive integers p, q with 2 p q.

Proposition 10. Let p and q be positive integers

.

If 2 p q, then

λ

1

(

G

(

p, q

;

2, 1

;

1

))

λ

1

(

G

(

p, q

;

1, 2

;

1

))

λ

1

(

G

(

p, q

;

1, 1

;

2

)).

Proof. This can be shown by ﬁrst computing the eigenvalues, using Theorem7, and then using a direct comparison.

Proposition10gives an afﬁrmative answer to [1, Conjecture 1.2] when the number of edges of a bipartite graph with partite sets U and V , having

|

U

| =

p and

|

V

| =

q for p q, is pq

−

2. By Theorem

(9)

Conjecture 11. For positive integers p, q and k satisfying p q and k

<

p, let G be a bipartite graph with partite sets U and V , having

|

U

| =

p and

|

V

| =

q, and

|

E

(

G

)| =

pq

−

k. Then

λ

1

(

G

)

λ

1

(

G

(

p, q

;

k, 1

;

1

)) =

 pq

−

k

+

p2_q2

₋

_6pqk

₊

_4pk

₊

_4qk2

₋

_3k2 2

.

Acknowledgements

The work of Yi-Fan Chen and Hung-Lin Fu was supported in part by NSC 94-2115-M-009-017. The work of In-Jae Kim was supported in part by the Presidential Teaching Scholar Fellowship from the Minnesota State University, Mankato, and the work of Brendon Watts was done as a part of SAMER activity while he was a PSEO student at the Minnesota State University, Mankato.

References

[1] A. Bhattacharya, S. Friedland, U.N. Peled, On the ﬁrst eigenvalue of bipartite graphs, Electron. J. Combin. 15 (2008) #R144. [2] Xiang En Chen, On the largest eigenvalues of trees, Discrete Math. 285 (2004) 47–55.

[3] M. Hofmeister, On the two largest eigenvalues of trees, Linear Algebra Appl. 260 (1997) 43–59. [4] R. Horn, C.R. Johnson, Matrix Analysis, Cambridge Univeristy Press, 1985.

[5] I. Gohberg, P. Lancaster, L. Rodman, Invariant Subspaces of Matrices with Applications, SIAM, 2006. [6] S. Lang, Linear Algebra, Springer, 2001.

[7] D.C. Lay, Linear Algebra and its Applications, Addison Wesley, 2006. [8] D.B. West, Introduction to Graph Theory, Prentice-Hall, 1996.