On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n

On the Line Graph of the Complement Graph for the Ring

of Gaussian Integers Modulo

n

Manal Ghanem1, Khalida Nazzal2

1_{Department of Mathematics, Irbid National University, Irbid, Jordan}

2_{Department of Mathematics, Palestine Technical University (Kadoorie), Tulkarm, Palestine} Email: [email protected], [email protected]

Received October 13,2011; revised November 2, 2011; accepted November 22, 2011

ABSTRACT

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally H and locally connected





_{[ ]}





n

L   i is given. The chromatic number when n is a power

of a prime is computed. Further properties for



_

_{[ ]}

_



n

L   i and 



n[ ]i



are also discussed.

Keywords: Complement of a Graph; Chromatic Index; Diameter; Domination Number; Eulerian Graph; Gaussian

Integers Modulo n; Hamiltonian Graph; Line Graph; Radius; Zero Divisor Graph

1. Introduction

The line graph L G

 

of a graph G is defined to be

the graph whose vertex set constitutes of the edges of G,

Where two vertices are adjacent if the corresponding edges have a common vertex in G. The importance of

line graphs stems from the fact that the line graph transforms the adjacency relations on edges to adjacency relations on vertices. For example, the chromatic index of a graph leads to the chromatic number of its line graph. The zero divisor graph of a commutative ring R ,

denoted by 

 

R , is defined as the graph whose vertex

set is the set of all non-zero zero divisors of R and

edge set E





 

R



=



xy x y: ,  R

 

0 and xy= 0



. This type of graphs provides an example showing that algebraic methods could be applied to problems about graphs. The set of Gaussian integers, denoted by [ ]i , is defined as

the set of complex numbers a bi , where a b,  . If x is a prime Gaussian integer, then x is either

1)



1i



or



1i



, or

2) q where q is a prime integer and q3 mod 4





, or

3) a bi , a bi where a2b2 = p, p is a prime

integer and p1 mod 4





.

Throughout this paper, p and p_i denote prime

integers which are congruent to 1 modulo 4, while q

and and qi denote prime integers which are congruent

to 3 modulo 4. All rings in this paper are assumed to be commutative with unity. The zero divisor graph for the ring of Gaussian integers modulo n is studied in [1]

and [2], the complement of this graph is discussed in [3]. While the line graph of the zero divisor graph for the ring

of Gaussian integers modulo n is investigated in [4]. In

this paper it should be kept in mind that







2[ ] = {1 }



V   i i , and hence, its line graph is K0,

[ ]

q i

 is an integral domain, so 



q[ ] =i



K0. Further,

 

_q2[ ]i

  is a complete graph whose complement is totally disconnected and thus its line graph is K0. While



p[ ] =i



Kp1,p1

  , so its complement is disconnected with two components each of which is isomorphic to

1

p

K  . Finally, note that the graph 



2q[ ]i



is bipartite,

[1] and



_{1 2}



2 2 1 1,2 1 [ ] =

q q i K_{q  q}

  .

In this paper, we investigate properties of the graph







n[ ]



L   i . We find the diameter, the radius of







n[ ]



L   i . We determine which L







_n[ ]i





is Eu-

lerian, Hamiltonian, regular, locally H , locally connec-

ted or planer. Furthermore, the chromatic index and the edge domination number of 



n[ ]i



where n is a

power of a prime are computed. While the domination number of 



n[ ]i



is given. On the other hand, a for- mula which gives the degree of each vertex in 



n[ ]i



is derived, thus the degree of its complement as well as its line graph could easily be found.

2. When Is

L









_n

[ ]

i





Eulerian or

Planner

only if every vertex of G has even degree. For a finite

ring R, the line graph L

 



 

R of a connected graph

 

R

 is Eulerian if and only if all vertices of 

 

R

have the same parity ( see the proof of Lemma 3.10, [5]). On the other hand, if G has both even and odd vertices,

then so is its complement. So, for a connected graph



n[ ]i



  , the graph L







n[ ]i





is Eulerian if and

only if all vertices in 



n[ ]i



are either even or all

vertices in 



n[ ]i



are all odd. But 



n[ ]i



is con-

nected if , 2 , , 1 2

m m

n p q q q [3] and 



_n[ ]i



is

Eulerian if n= 2,p or n is a product of distinct odd

primes [1]. It is easy to show that all vertices of



n[ ]i



  are odd if and only if ₌ 2

n q . This proves the

following theorem.

Theorem 2.1 L







n[ ]i





is Eulerian if and only if

n is a product of distinct odd primes.

A planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way

that its edges intersect only at their endpoints.

Next we determine when the graph L







n[ ]i





is

planar.

In a graph G the maximum vertex degree and the

minimum vertex degree will be denoted by 

 

G and

 

G

 , respectively.

The following theorem characterizes graphs G whose

line graph L G

 

is planer.

Theorem 2.2 [6]

A nonempty graph G has a planer line graph L G

 

if and only if 1) G is planer.

2)  



 

G



4, and

3) if degG

 

v = 4, then v is a cut vertex.

The graph 



n[ ]i



is planer if and only if n= 2,5

or 2

q [3]. For n= 2, q2, L







n[ ] =i





K₀. While

for n= 5, 



n[ ] =i



K₄K₄, this graph is regular of

degree 3.

Thus we obtain the following.

Theorem 2.3 The graph L







n[ ]i





is planer if

and only is n= 5.

3. The Diameter of

L









_n

_{[ ]}

i





For a connected graph G, the distance, d u v

 

, ,

between two vertices u and v is the minimum of the

lengths of all u v paths of G. The eccentricity of a

vertex v in G is the maximum distance from

v

to any vertex in G. The diameter of G, diam G

 

, is the

maximum eccentricity among the vertices of G. Since



n[ ]i



  is connected if n p, 2 ,m qm,q q1 2 and each

of 



_p[ ]i



and





1 2[ ] q q i

  is the union of two complete graphs, while





2m[ ]i

  and



_qm[ ] ,i



m 3

   are the union of a nullgraph and a connected graph [3], we have the following.

Theorem 3.1 L







n[ ]i





is connected if and only if

2 1 2

2, , ,

n p q q q .

Theorem 3.2 If n= 2 ,m m2 or n=qm,m3, then











n[ ]



= 2

diam L   i .

Proof. 1) Assume that n= 2 ,m m2 and



x= x1x i y2, = y1y i2

 

, z=z1z i w2, =w1w i2



are two nonadjacent vertices in V L









n[ ]i







. Since

for every

2m[ ]

a bi  i , a and b are both even or

odd [1], we have three cases:

Case I: for = 1, 2i , x y zi, ,i i and wi are odd. Then

we have the path [ , ]x y   [ , ]x z   [ , ]z w .

Case II: for = 1, 2i , xi or yi is odd(even) and zi

or w_i is even (odd). Assume that x x₁, ₂ are even and 1, 2

z z are odd. Then we have the path

[ , ]x y   [ , ]x z   [ , ]z w .

Case III: for = 1, 2i , x y zi, ,i i and wi are even.

Then 1 1 2 2

1 1 2 2

[ , ] = 2t 2 ,s 2t 2s

x y _  i  i_ and

3 3 4 4

3 3 4 4

[ , ] = 2t 2 ,s 2t 2s

z w _  i  i_ where  _i, _i

are odd and 1t si, im for 1 i 4. If t s t1, , ,1 2 or 2< ₂

m

s  , say t₁, then t s t₃, ,_{3 4} or s₄<m t ₁, say t₃.

So, we have the path [ , ]x y   [ , ]x z   [ , ]z w . Now

suppose that m is odd. Then

a) If = = 1, 2

i i i i

m

t s    , for i= 1 or 2, say for

= 1

i , then t s t₃, ,_{3 4} or s₄<m t ₁, say t₃. Hence, we

have the path [ , ]x y   [ , ]x z   [ , ]z w .

b) If ti or

1 =

2

i

m

s  and t_i s_i, for i= 1 or 2, say

for i= 1, then we have a path [ , ]x y   [ , ]x z   [ , ]z w or

[ , ]x y   [ , ]x w   [ , ]z w .

c) If = = 1, = 2

i i i i

m

t s    , for i= 1 or 2, say for

= 1

i , then ₂= ₂= 1

2

m

t s  implies that ₂₂. Other-

wise t2 or 2

1 2

m

s   . Then we have a path

[ , ]x y   [ , ]y z   [ , ]z w or

[ , ]x y   [ , ]y w   [ , ]z w .

2) Assume that _n=_qm,_m3 _and













1 1 2 2

1 1 2 2

3 3 4 4

3 3 4 4

, ,

, [ ]

t s t s

t s t s

n

q q i q q i

q q i q q i V L i

   

   

   

 

    

Then t s t1, ,1 2 or 2 <

2

m s  _{ }

 , say t1. Hence t s t3, ,3 4 or 4< 1

s m t , say t₃. Then we have the path

1 1 2 2 1 1 2 2

3 3 1 1

1 1 3 3 3 3 4 4 3 3 4 4

,

[ , ]

,

t s t s

t s

t s

t s t s

q q i q q i

q q i q q i

q q i q q i

   

   

   

     

 

    

   

 

.



Theorem 3.3 Let R be a ring that is a product of two rings R1 and R2 with at least one of them is not ID with more than one regular element and the other has more than two regular elements. Then

 

 





= 3

diam L  R .

Proof. Suppose that R=R1R2 and R1 is not ID,

 

1 2

reg R  and reg R

 

₂ 3 . Let z₁V





 

R₁



and u₂reg R

   

₂  1 . Clearly,

  

   





1,0 , 1,0 , 0,1 , 0, 2



= 3

d _ z  _{ } u _ inL

 



 

R . So,

 

 





3

diam L  R  . Now, let



a a1, 2

 

, b b1, 2

 

, c c1, 2

 

, d d1, 2



V L



 



 

R



   

    ,

then a b_{1 1}0 or a b_{2 2} 0 and c d_{1 1}0 or c d_{2 2}0.

So, we have three cases:

Case I: a b_{1 1}0 and c d_{1 1} 0. Then

 

1, 1 1

a c reg R implies that



z1,0 ,

 

a a1, 2

 

z1,0 ,

 

c c1, 2



E L



 



 

R



   

    .

And a1 or c1Z R

 

1 , say a1 implies that



u1,0 ,

 

a a1, 2

 

u1,0 ,

 

c c1, 2



E L



 



 

R



   

   

where u₁reg R

   

₁  c₁ .

Case II: a b_{2 2} 0 and c d_{2 2} 0. Then there exists

  



2 2 2, 2

v reg R  a c and hence



a a1, 2

 

, 0,v2

 

c c1, 2

 

, 0,v2



E L



 



 

R



   

    .

Case III: a b1 10 and c d2 2 0 or a b2 20 and 1 1 0

c d  . Let a b_{1 1}0 and c d_{2 2} 0. Then

 

1 1

a reg R implies that



a a1, 2

 

, z c1, 2



V L



 



 

R



 

  and



z c1, 2

 

= d d1, 2



or 



d d1, 2

 

, z c1, 2



V L



 



 

R



.

And if a1Z R

 

1 , then



a c1, 2

 

= b b1, 2



or



a c1, 2

 

, b b1, 2



V L



 



 

R



 

  and



a c1, 2

 

= d d1, 2



or 



a c1, 2

 

, d d1, 2



V L



 



 

R



.



For = m

n p , _n[ ] _{m m}

p p

i  

   [7] and for

1 2

=

n n n with g c d n n. .



₁, ₂



= 1,

1 2

[ ] [ ] [ ]

n i  n i  n i

   .

Moreover reg



₂[ ] = 2i



and reg



_m[ ]i



3 for

2

m . An immediate consequence of Theorem 3.3 is the

following.

Theorem 3.4 Let _n= _pm,_m2

or n is a composite such that nq q_{1 2}. Then











n[ ]



= 3

diam L   i .

4. The Radius and the Girth of the Graph







_n

[ ]



L





i

For a connected graph G, the radius of G, rad G

 

, is

the minimum eccentricity among the vertices of G. So,

 

 

rad G diam G . Since for any













[ , ]a b V L  n[ ]i , [ , ]a b and [ , ]ai bi are non ad-

jacent, rad L









n[ ]i







> 1. Using Theorem 3.2 gives for = 2 ,m 2

n m or n=qm,m3,











n[ ]



= 2

rad L   i .

Theorem 4.1 If _n= _pm,_m2

or = m

n t s where

1

m , t is prime integer, g c d t s. .

 

, = 1 and nq q1 2, then rad L









n[ ]i







= 2.

Proof. Since rad L









n[ ]i







> 1 to show that











n[ ]



= 2

rad L   i it is enough to find a vertex











n[ ]



vV L   i with eccentricity 2. If

2 2

= m, = , 2

n p p a b m , then



[ , ],[ , ]



2

d abi a bi x y  for every













[ , ]x y V L  n[ ]i . So rad L



_pm[ ]i



= 2

 _ 

 _ _

   .

Now, assume that = m , 1

n t s m and



( , ),( , )x y w z



V L









n[ ]i







.

Then we have four cases: Case I: = 2t . Then



     





1 ,1 , 1,0 , , , ,



2

d _ i  _{ } x y w z _  .

Case II: =t p. Then



 

   





,1 , ,1 , , , ,



2

d _ abi a bi  _{ } x y w z _  .

Case III: t=q1 and m= 1. Then sq2 and hence

there exists aV







s[ ]i





. So,

      





0,1 , 1, , , , ,



2

d _ a  _{ } x y w z _  .

Case IV: = ,t q m2. Then

      





,1 , 1,0 , , , ,



2

d _ q  _{ } x y w z _  .



Theorem 4.2 rad L









n[ ]i







= 2 if and only if 2

2, , ,

Vising [8], proved that for a connected simple graph

G with n-vertices and radius 2, the upper bound of the

number of edges of G is



2



2

n n

. Then Golberg [9] proved that the lower bound of numbers of edges of a simple connected graph G with radius 2 is 3



1



4

n

. So we can conclude the following.

Theorem 4.3 For n2, , ,p q q2 or q q1 2,







n[ ]



=

L   i t implies that





_

_

_

_

_

_





3 1 2

[ ]

4 n 2

t t t

E L i

 

    .

The girth of a graph G, g G

 

is the length of a

shortest cycle contained in the graph. If the graph does not contain any cycles (i.e.. it’s an acyclic graph), its

girth is defined to be infinity. If a b c a, , , is a cycle of

length three in G. Then [ , ],[ , ],[ , ],[ , ]a b b c c a a b is a

cycle of length 3 in L G

 

. So, g L G



 



= 3 when-

ever g G

 

= 3. In [3] it is proved that the girth of



n[ ]i



  equals 3 for _{n2, ,q q}2_{. So, we have the fol-}

lowing.

Theorem 4.4 For n2, ,q q2, g L









_n[ ]i







= 3.

5. The Locally Connected Property of the

Graphs







_n

[ ]

i



and

L









_n

[ ]

i





We say that a vertex v is locally connected if the

neighborhood of v, N v

 

, is connected; and G is

locally connected if every vertex of G is locally con-

nected.

Theorem 5.1 If R=R1R2, reg R

 

i 2 for

= 1, 2

i and either R1 or R2 is not ID, then 

 

R is locally connected.

Proof. Suppose that R1 is not ID and

 

x y, V

 



 

R . Then we have two cases:

Case I: = 0x or = 0y . If = 0x , then there exists

 





1 1

z V  R . So

  

z1,1 a b, E





 

R



for all

 

a b, N



 

x y,



. And if y= 0, then there exists

 

1 1 { }

u reg R  x such that



u1,0



N



 

x y,



. There-

fore,



u1,0

 

a b, E

 



 

R for every

 

a b, N



 

x y,



. So N



 

x y,



is connected.

Case II: x0 and y0. Then there exist

 

1 1 { }

v reg R  x , v2reg R

 

2 { }y and

 





1 1

z V  R such that



v1,0 ,

 

z v1, 2



and



0,v2



N



 

x y,



. Moreover,



v1,0



z v1, 2

 

, 0,v2



z v1, 2



E

 



 

R . And for every

 

t s, Z R

 

,



v₁,0

 

t s, or



0,v₂

 

t s, E

 



 

R .

So N



 

x y,



is connected.



Theorem 5.2 If R=R₁R₂, reg R

 

i 2 for

= 1, 2

i and either R1 or R2 is not ID, then

 

 

L  R is locally connected.

Proof. Suppose that R1 is not ID, z1V





 

R1



and _

  

x y, , ,z w



_V L



 



 

R



, then we have three

cases:

Case I: x= = 0z . Then

      

z1,1 , ,x y z1,1 , ,z w



E L



 



 

R



   

    .

Case II: y=w= 0. If x z, reg R

 

₁ , then

      

z1,1 , ,x y z1,1 , ,z w



E L



 



 

R



   

    .

Otherwise there exists u₁reg R

 

₁ { , }x z . So,



u1,0 , ,

   

x y u1,0 , ,

 

z w



E L



 



 

R



   

    .

Case III: ,x y0 or ,z w0. Assume that xy0,

then z0 implies that there exists u1reg R

 

1 { }x

satisfies



u1,0 , ,

   

x y u1,0 , ,

 

z w



E L



 



 

R



   

    .

While w0 implies that that there exists

 

2 2 { }

v reg R  w satisfies



0,v2

   

, ,x y 0,v2

 

, ,z w



E L



 



 

R



   

    .



From Theorem 5.1 and Theorem 5.2 we conclude the following.

Theorem 5.3 If n= pm,m1 or n is a composite integer such that nq q1 2, then both 



n[ ]i



and







n[ ]



L   i are locally connected.

6. When Is

L









_n

[ ]

i





Hamiltonian?

A Hamiltonian cycle is a cycle that visits each vertex exactly once (except the vertex which is both the start and end, and so is visited twice). A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. The line graph of a graph G with more than 4 vertices and

diameter 2 is Hamiltonian [10]. But





2m[ ] ,i m 2

   is disconnected with one isolated vertex



₂m1 ₂m1



i  _ 

and the other component, call this component H, with

diameter 2 [3]. So,





 

2m[ ]

L_  i _L H . Similarly,



_qm[ ]i



  has a connected subgraph H with diameter

2 and



_m[ ]



 

q

L_ i _L H

   . Hence, the following

result is obtained.

Theorem 6.1 If n= 2 ,m m2 or n=qm,m3, then







n[ ]



L   i is Hamiltonian.

a complete bipartite graph K1,3) is hamiltonian. Since

the line graph is claw free, using Theorem 5.3, we get the following.

Theorem 6.2 If n= pm, m2 or n is a composite integer such that nq q_{1 2}, then L







n[ ]i





is hamil-

tonian.

7. The Chromatic Number of the Graph







_n

[ ]



L





i

The edge coloring of a graph G is an assignment of

colors to the edges of the graph so that no two adjacent edges have the same color. The minimum required num- ber of colors for the edges of a given graph is called the chromatic index of the graph denoted by 

 

G .

Lemma 7.1 [12]

If G has order 2s and 

 

G = 2s1, then

 

G =

 

G

  .

Theorem 7.2 If n= 2 ,m m2, then







_{[ ] = 2}



2m1 ₃ n i

_{  }  _

. Proof. Note that in





2m[ ]i

  , the induced subgraph,

H , with

 







1 1



2

= [ ] 2m 2m

m

V H V_ i _   i

   is con-

nected,

 

_{= 2}2m1 ₂

V H   , [1] and

 

H =



₂m[ ]i



    

  . Since the vertex 1i is adja-

cent to all other vertices in H , we have

 

₌



₁



_{= 2}2m1 ₃

H deg i 

   . Using Lemma 6.1,





2 1

2 [ ] = 2 3

m m i

_{ }   _

 

   .



Since 



q[ ]i



is empty graph and

 



2



2[ ] = 1 1

q i q K

   is edgeless with 2 ₁

q  verti-

ces, we consider the case 



_qm[ ] ,i



q3.

Theorem 7.3 If = m, 3

n q m , then







_{[ ] =}



2m 2 2 ₁

n i q q

_{  }  _{ }

Proof. Let A=



qm1 qm1i: ,  q



 

0 .

 _  _{ }

Then A is the set of all isolated vertices in



_m[ ]



q i

  .

So the induced subgraph, H, with the vertices

 

=



_qm[ ]



V H V_ i _A

   is a connected graph,

 

₌ 2m 2 2

V H q  q . Clearly the vertex q is adjacent to

all other vertices in H and hence,

 

₌ 2m 2 2 ₁

deg q q  q  . Using Lemma 7.1,



_{[ ] =}



2m 2 2 _1.

m

q i q q

_{ }   _{ }

 

  



Finally we find the chromatic index of



_m[ ] ,



2

p i m

   .

A subset D of the vertex set V G

 

is said to be

independent if no two vertices in this set are adjacent. A clique of a graph is a maximal complete subgraph. A graph G is said to be split if it’s vertex set can be

partitioned into two subsets A and B such that A

induces a clique and B is independent in G.

Lemma 7.4 [13] Let G be a split graph. If 

 

G is odd, then 

 

G =

 

G .

Theorem 7.5 If n= p2, then







_{[ ] = 2}



3 2 ₁

n i p p p

      .

Proof. Since ₂[ ] _{2 2}

p i  p  p

   , it is enough to find



_p2 _p2



   . First, we’ll show that



_{2 2}



p p

   is a split graph. Let





 









 





2

2

= , : and }

, : and ,

p p

p p

A u p u U p v v U

 

 

 _{ }

 



   _



 

 











 



= , : and p 0,0

B  p p     .

Clearly, V



_p2_p2



=AB, A induces a clique

and B is independent. Therefore,



_{2 2}



p p

   is a split graph. Moreover,





 

 







 





 



2 2

2 3 2

= 1,

= [ ] 0, : 0 1,

= 2 1

p p

p p

deg p

i p p

p p p

 

 

 _  _

 

    

  

 

 

is odd. From Lemma 7.4,



_{[ ] = 2}



3 2 ₁

m

p i p p p

 _ _   

   .



A graph G is said to be critical if G is connected

and 

 

G =

 

G 1 and for every edge e of G,

we have 



G\{ } <e





 

G . The well-known Vizing’s

theorem states that for a simple graph G,

 

G =

 

G

  or 

 

G 1.

Lemma 7.6 [14]

If G is a critical graph, then G has at least

   

G  G 2

   of vertices of maximum degree. Therefore, if G is a simple graph such that for every

vertex v of maximum degree there exists an edge vu

of vertices with maximum degree in G, we have

 

G =

 

G

  [13].

Theorem 7.7 If n= pm,m3, then







_{[ ] = 2}



2m 1 2m 2 ₁

n i p p p

_{  }  _  _{ }

.

Proof. Let  , V





 

p



and u v, U

 

_pm .

Then the vertices of 



_pm_pm



with maximum de-

gree have the form



p v,



or



u,p



where  0

and  0 and































1



, =

,0 : 0 ,

m m

p p

m

N p v V

p p v



   

_{ } 

 

 

  

 

and































1



, =

0, : 0 , .

m m

p p

m

N u p V

p u p



   

_{ } 

 

 

  

 

So,





_{= 2} 2m1 2m 2 ₁

m m

p p p p p

 

 

 _  _   

    . And

the vertices of 



_pm_pm



with minimum degree

have the form



m1_,0



p

  _or



_0, m1



p

  _where







m1_{,0 =}







_, i



_{: 1}



N p  up i and







_0, m1



₌





i_,



_{: 1}



N p  p v i . So



_{[ ] =}





m m1





₂



m

p i p p mp m p

_  _  _{  }

 

   .

Therefore,























1 1 2 1

1

[ ] 2

= 2 1

> 2 1

m m m

p p p

m m m m

m m

i

p p p mp m p p p

p p p



  



   

 _{ } _  _

   

       

 

  

.

But the graph 



_pm_pm



has only



1







2 m m 1

p p  p vertices of maximum degree. So,





_{= 2} 2m1 2m 2 ₁

m m

p p p p p

_{ } _   _  _{ }

 

    .

Since _pm[ ]i _pm_pm, the result holds.



Since the edge coloring of any graph leads to a vertex coloring of its line graph, we obtain the following.

Corollary 7.8 1) If n= 2 ,m m2, then











_{[ ]}



_{= 2}2m 1 ₃ n

L i

 _{ }  _

.

2) If = m, 3

n q m , then











_{[ ]}



₌ 2m 2 2 ₁ n

L i q q

 _{ }  _{  .}

3) If = m, 2

n p m , then











_{[ ]}



_{= 2} 2m 1 2m 2 ₁ n

L i p p p

 _{ }  _  _{  .}

8. The Domination Number of







_n

[ ]

i



A subset D of the vertex set V G

 

of a graph G is

a dominating set in G if each vertex of G, not in D,

is adjacent to at least one vertex of D. The minimum

cardinality of all dominating sets in G, 

 

G , is called

the domination number of G.

In





2m[ ] ,i m 2

   , the vertex ₂m1 ₂m1 i

 _  _{is an}

isolated vertex while the vertex 1i dominates all

vertices in the second component. Therefore,



2m[ ] = 2i



  

   . The graph

 





2 2[ ] = 1 1

q i q K

   ,

thus

 

2

2[ ] = 1

q i q

 _ _ 

   . In 



_qm[ ] ,i



m3 the

vertices m 1 m1

q q i

  _  _{are isolated while the vertex}

q

is adjacent to all other vertices in



_{[ ]}





m1 m1 _{: ,}



m q

q

V_ i _ q  q i  

    ,

so



_{[ ] =}



2

m

q i q

 _ _

   . Since



_{1 2}



2 ₁ 2 ₁ 1 2

[ ] =

q q i K_{q } K_{q }

  

and 



p[ ] =i



Kp1Kp1,



q q1 2[ ] =i







p[ ] = 2i





_  _    .

The set D= 1,0 , 0,1



   



is a minimum dominating set for 



_pm_pm



. And if n=n n1 2, where



1 2



. . , = 1

g c d n n , then









1 2

[ ] [ ] [ ]

n i n i n i

      . This

graph is connected and the set D= 1,0 , 0,1



   



is a minimum dominating set for





1[ ] 2[ ]

n i n i

   .

Theorem 8.1 1) If n2,qm, then







n[ ] = 2i



   .

2)

 

2

2[ ] = 1

q i q

 _ _ 

   and



_{[ ] =}



2_{, 3.}

m

q i q m

 _ _ 

9. The Domination Number of

L









_n

[ ]

i





The independence number of G, 

 

G , is the maxi-

mum cardinality of all independent sets in G. A subset D of the edge set V G

 

of a graph G is an edge

dominating set in G if each edge of G, not in D, is

adjacent to at least one edge of D . The minimum

cardinality of all edge dominating sets in G, 

 

G , is

called the edge domination number of G. The minimum

cardinality of all independent edge dominating sets,

 

i G

 , is called the independence edge domination number of G. The study of the domination number of

the line graph of G leads to the study of edge or line

domination number of G, i.e. 



L G

 



= 

 

G . On

the other hand, for any graph G, i

 

G =

 

G [15].

If S is an independent set in G, then S induces a

complete graph in G. While if S induces a complete

graph in G, then it is independent in G. Recall that 2

2m[ ]i  2m

  [2]. Then the sets,







₂2



= 2 :j

j m j

A  U   , j= 1, 2, , 2 m1 form a partition for the set V





 

₂2m



. Clearly, the set

2 1 =

= m j j m

T



A is the maximum independent set in

 

₂2m

  , while the set = m₌₁1 j j

S



A induces a maxi-

mum complete subgraph in

 

₂

2 m

  . There are some edges joining S to T , no other adjacency exists in

 

₂2m

  . Any edge dominating set for 

 

₂2m must

contain at least _S 2_ element in order to dominate

S

 . On the other hand, this dominating set for  S do-

minates all other edges in

 

₂

2m

  . Since

2 1

= 2 m j j

A   , then S and T , could easily be com-

puted to get the following theorem.

Theorem 9.1 For = 2 ,m 2

n m .

1)





_{[ ] = 2 2}





m



m1 ₁



n i

 _{ }  _{ .}

2)





[ ] = 2





m 1

n i

    .

3)





























1



1



[ ] = [ ]

= [ ] = 2 2 1 .

n i n

m m

n

L i i

i              

To study the graph



_{[ ] ,}



₃

m

q i m

   , consider the

partition of 



_qm[ ]i



given by

 

 





= : and ,

1 , .

k j

kj _qm k _qm j

A q q i U U

k j m

   _  _

 

 

and not both j k, =m. The set

= =

2 2

= m m

kj mm

m m

k j

T _{   }A A

            _  _  _{ } 







is the maximum inde-

pendent set, while 2 1 2 1 =1 =1 = m m kj j k S A  _  _       _   _    





induces a

maximum complete subgraph in 



_qm[ ]i



. There are

some edges joining S to T , and



_m[ ]



q i

  has no other adjacency. Easy calculations give





2 _{2 2}

= 1 m k j

kj

A q q    when 1k j,  m 1, 1

= m j m j

mj

A q  q   and Akm =qm k qm k 1

 _   _when

,

k jm. While

2 2 = 1 m T q    

 _{ and}

2 2

2 2

= 1

m m

S q q

         _  _ _

 

  .

Thus we obtain the following theorem.

Theorem 9.2 If n=qm,m3, then

1)









2 2

2 2

[ ] = 1

m m

n i q q

 _    _    _ _

 

 

 .

2)





[ ] =





m 1

ni q

    if m is even and m 1

q  if m is odd.

3)









































2 2

2 2

[ ] = [ ]

1

= [ ] = 1 .

2

n i n

m m

n

L i i

i q q

                               

Now, we move to the case = m

n p . Let





 

 





= k, j : and

kj _pm k _pm j

A p p U  _ U  _ .

Clearly, the sets A_kj where 0k j, m and not

both k j, =m or 0, partition the vertices of



_pm _pm



   and ₌ 2m k j 2



₁



2 kj

A p    p . Let



 

1



1 ₌₁ 0 ₌₁ 0 1 1

2 2

2 3

=1 =1 _{= =} 2 2 1 1 2 2 4 5 =1 =1 = = 2 2 = = , = ,

= and = .

m m k k k k m m m m kj kj m m k j k j m m m m kj kj m m j k k j

S A A

S A S A

S A S A

  _  _                      _  _                                    _ _                      





 





 

 



_pm _pm



   . Vertices in =11 0 m

k k A





are adjacent to all vertices except some vertices in m₌₁1

km k A





. Similarly, vertices in m₌₁1 ₀

k k A





are adjacent to all vertices except some vertices in m₌₁1

mk k A





, and vertices in Am₀ are

adjacent to all vertices except vertices in A₀m. On the

other hand A₀m induces a complete subgraph and

vertices in this set are adjacent to all other vertices except those of Am₀. Clearly S2 induces a complete subgraph.

Vertices in S₃ form an independent set, and are

adjacent to some vertices in S₁S₂S₄S₅A₀m.

Each of S₄ and S₅ induces a complete subgraph and

are adjacent to some vertices in S₁S₂S₃A₀m.

Besides, there are some edges between S₄ and S₅. On

the other hand,

3

= = 2 2

= m m kj mm.

m m

k j

S A A

           



 

The above argument shows that

















3

2

2 1 2 2 2

[ ] = [ ]

1

= [ ] = [ ]

2 1

= 2 2 .

2

m i m

p p

m m

p p

m m

m m

L i i

i i S

P p p

 



       

 _  __ 

    

   

 

 

 

 _{  }  _

   

 

    

 

 

 

 

10. The Degree of the Vertices in







_n

[ ]

i



and

L









_n

[ ]

i





Now, we determine the cardinality of the annihilator of the element a bi , ann a



bi



in n[ ]i . This helps

find the degree of each vertex in 



n[ ]i



, its complement, as well as the degree of each vertex in their corresponding line graphs.

Theorem 10.1 If a bi n[ ]i , then





₌ 2 2

ann abi c d where g c d a. .



bi n,



=cdi. Proof. Let a bi n[ ]i and





. . , =

g c d abi n cdi. Then





=



[ ] :





0 mod







ann abi xi x abi  n .









So, .x a bi 0 modn x.a bi 0 mod n

c di c di

  

   _{  }

   .

But _n [ ]

c di

a bi

U i

c di _

 



 _ _

 _ _. So, 0 mod

n x

c di

 

 _{ } _

and hence there exists m[ ]i such that =x n m cdi .

Since m=t c



di



r where t r, [ ]i and the norm

of r is less than the norm of cdi,



 

= : c di[ ] =



c di[ ]

ann a bi r r  i   i . By Theorem

2 of [7], 2 2

2 2

[ ] = =

c di i _{c d} c d

  , so the result

holds.



Theorem 10.2 Let vV







n[ ]i





and

 

. . , =

g c d v n cdi. Then

 

= 2₂ 2₂ 1, if 2₂ 0

2, if = 0

c d v

deg v

c d v

   



 

 .

The order of 



n[ ]i



can be easily computed using

formulas given in [1]. Thus we can find the degree of each vertex in the complement of 



n[ ]i



, here we

give the degree of each vertex in the line graph of



_n[ ]i



  , an analogous formula for the degree of vertices in L







n[ ]i





could be obtained.

Corollary 10.3 Let

 

u v, V L









n[ ]i







,

 

. . , =

g c d u n abi and g c d v n. .

 

, =cdi. Then





2 2 2 2 2 2

2 2 2 2 2 2

2 2 2 2 2 2

[ , ]

4, if 0 and 0

= 5, if = 0 and 0

6, if = 0 and = 0

deg u v

a b c d u v

a b c d u v

a b c d u v

      

     



     

.

Proof. Note that, for any graph G and uvE G

 

,





 

 

( ) [ , ] = 2

L G G G

deg u v deg u deg v  .



In the following we determine the degree of every vertex in the graphs 



_n[ ]i



when

= 2 ,m 2, = m, 3

n m n q m and _n= _pm,_m1_. Theorem 10.4 Let = 2 ,m 3

n m and  , are odd. Then in 



n[ ]i



,

1)





2

2

2 1

2 1

2 1, if 1 < < and < 1 = < and

2 2

2 2, if < < or = < and

2 2

2 2 =

2 2, if = < and = 2

2 1, if 1 = < and = 2

k

k

k s

k

k

m m

k s m k or k s

m m

k s m k s m

deg i

m

k s m m k s

 

 

 

 

 





 _{ }   _   _{ }

  _{ }  _{ }



 _{   }

    

 _{   }

    

 _

 

 _{  }

 

 _{ }



 

 _{  }

 

 _{ }

