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# THE INDEPENDENT DOMINATION NUMBER OF A RANDOM GRAPH

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(1)

n,p

2

n,p

n,p

n,p

e(Gn,p)

n2

−e(Gn,p)

n,p

n,p

2

2

2

2

2

2

n,p

(2)

2

2

2

2

2

2

2

2

n,p

b

b

b

b

b

b

b

b

b

n,p

b

2

n,p

b

b

b

b

b

b

b

b

n,p

1q

k

s

s

s

k=1

k

k

k

n−k

k2

s

s

k=1

k

s

k=1

k

n−k

k2

(3)

b

b

b

b

s

elnn2n

b

b

b

elnn2n

r

s

k=r+1

k

k

k

n−k

k2

k

2

k

b

b

b

b

b

b

b

b

b

b

b

b

s

b

k

2

b

(4)

b

b

b

b

2

1q

elnn2n

p642

ln (

lnn p )

ln n

s

b

b

b

b

2

k

k

k

k

n−k

k2

k

n

k

2 2

k

k

2

k

k

k

2

2

k

12

k

k

2

k

k

2

2

ln n

n12

b

b

b

b

d

dk

12

1−qnqkk

k22

k

(5)

b

b

b

b

12

s

b

12

k

b

b

b

b

b

b

b

b

12

s

b

b

b

b

b

b

b

b

L

18

1q

1 q

1 q

ln n

n

1 2

b

ln np

logbnnln n

b

12

s

b

b

12

q

1

2logbnln n

2n

b

b

12

pq

1 2

2

b

b

pq

1 2

2

12

pq

1 2

2

b

x2

12

(6)

b

12

pq

1 2

2

b

b

p 2

pq

1 2

2

b

2

b

b

2

b

b

C8

b

b

b

ln2n p

b

b

2

ln n p

ln n

n

1 2

(7)

p642

ln

lnn p



ln n

EV arX2(Xss)

b

b

b

b

s

s

2

s

s−1

m=0

s

2s−m

n−2s+m

2

2s

m2

b

b

b

b

b

b

b

b

b

b

b

b

s

s

2

s

s

2s

n−2s

2

s2

2

s

2

s

s−1

m=1

s

2s−m

n−2s+m

2

s2

m2

2

s

s

2s

n−2s

2

s2

2

s

2

s

s

−2s

2

s

s

s

(8)

ln

3 2n

n12

1−q2sqss

s

s

s

2s−m

n−2s+m

2

s2

m2

s−m

s

2s−m

n−2s+m

2

s2

m2

s−m

s

2s−m

n

2

s2

m2

s−m

s

2s−m

2

s2

m2

ln

3 2n n

1

2

s−m

s

2s−m

2

2s

m2

2

m

2s−m−1

0

0

s−1

m=1

2

2

2s

b2

b

2

2−2ǫ

(9)

2

2

4

2−2ǫ

ln n

n

1 2

s−1

s

1−ǫ

b2

ǫ

b

1−ǫ

2 1q

3

ǫ

2

1−ǫ

2

3

1+ǫ

2

2

m

2s−m−1

2s−m−1

m

2

b

n(m+1)qm (s−m)2

2s−1

b

n(m+1)qm (s−m)2

b2

2

b

b

m

2

b

b

(10)

b

b

m

2

b

b

2

1

ln

1 q



1

ln

1 q



2

ln

1 q



n(m+1)qm (s−m)2

1

ln

1 q



1

ln

1 q



2

1

ln

1 q



8

ln

1 q



1

ln

1 q



3

(s+1) ln

1 q



2

8

(s+1) ln2

1 q



2

2

32

2

2

2

2

32

1

(s−1) ln1

q



1

ln

1 q



2

ln

1 q



n(m+1)qm (s−m)2

2

1 q

6

(s+1) ln1

q



1

(s+1)2ln21

q



32

(11)

pln n1

ln(n)1

−1 1q

−1 1q

−1 1q

−1 1q

1

−1 1q

2

−1 1q

1

2−δ1

2

1−δ2

2

1−δ

0

0

0

0

2−δ1

1−δ2

2

0

0

0

0

2−δ1

1

2

0

0

0

0

0

0

0

0

m0

0

2

s−1

b

1−ǫ

b

4n p

b

b

0

0

s−1

2s−1

s+1

s−12

s

s+1

2s−1

s−12

2

(12)

s

s+1

2s−1

s−12

s

p2logb8nln n

b

2

ln n24

s−1

m=1

s

2s−m

n−2s+m

2

2s

m2

ns

2

s

ns

2

s

2

s−1

2s−1

s

n

s

s

2(n−s)

3

2s−1

s

s

2n

2

3

2s−1

s

s

s

3

2s−1

s

b3

b2

2

s

2ǫ−1

ln n

n13

b

b

b

b

b

b

b

b

s

2

s

p2 64

ln (

lnn p ) ln n

p642

ln

lnn p



ln n

b

b

b

b

b

b

b

b

(13)

b

b

b

b

s

s

b

b

b

b

s

s

s

s

s

2

s

6

(14)

1

2

n

k

## } are highly dependent posing difficulty in verifying the conditions needed in many central limit theorems for dependent sums.

### Accepted 13 April 2010

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