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INTEGRAL EDGE SUM OF
Γ
( )
Z
nJ. PERIASWAMY*
1, N. SELVI
21
Part-Time Research Scholar,
Bharathidasan University Tiruchirapalli, Tamil Nadu, India-620 024.
A. V. C. College (Autonomous), Mayiladuthurai, Tamil Nadu, India-609 305.
A. D. M. College for Women (Autonomous), Nagappattinam, Tamil Nadu, India-611 001.
(Received On: 30-12-17; Revised & Accepted On: 17-01-18)
ABSTRACT
T
he first simplification of Beck’s [2] zero divisor graph was introduced by D.F.Anderson and P.S.Livingston[1]. Their motivation was to give a better illustration of the zero divisor structure of the ring. In this paper, we find the integral sum of zero divisor graph and study their properties.Keywords: Integral Edge Sum, Zero divisor graph.
AMS Subject classification: 05C25, 05C69.
1. INTRODUCTION
Let R be a commutative ring and let Z(R) be its set of zero-divisors. We associate a graph
Γ
( )
R
to R with vertices( )
=
( ) { }
−
0
Γ
∗R
Z
R
, the set of non-zero divisors of R and for distinctu
,
v
∈
Z
( )
R
∗, the vertices u and v are adjacent if and only ifuv
=
0
. The zero divisor graph is very useful to find the algebraic structures and properties of rings. The idea of a zero divisor graph of a commutative ring was introduced by I. Beck in [2]. The first simplication of Beck’s zero divisor graph was introduced by D.F.Anderson and P.S.Livingston [1]. Their motivation was to give a better illustration of the zero divisor structure of the ring. D.F.Anderson and P.S.Livinston, and others e.g., [4, 5, 6], investigate the interplay between the graph theoretic properties ofΓ
(
R
)
and the ring theoretic properties of R. Throught this paper, we consider the commutative ring R by Zn and zero divisor graphΓ
(
R
)
byΓ
( )
Z
n . LetΓ
( )
Z
n be a graph. A bijectionf
:
E
(
Γ
( )
Z
n)
→
Z
+, whereZ
+ is a set of positive integers is called an edge mapping of thegraph
Γ
( )
Z
n . Now, we define,F v
( )
=
∑
{
f e
( ) :
e is incident on v onV
(
Γ
( )
Z
n)
}
. Then, F is called the edge summapping of the edge mapping f.
f
:
E
(
Γ
( )
Z
n)
→
N
+ such that f and its corresponding edge sum mapping F on( )
(
n)
V
Γ
Z
satisfy the following conditions: (i) F is into mapping toZ
+. That is,F
(
v
)
=
Z
+, for every( )
(
n)
v
∈
E
Γ
Z
. (ii) Ife e
1,
2,....,
e
n∈
E
(
Γ
( )
Z
n)
such thatf
(
e
1)
+
f
(
e
2)
+
...
+
f
(
e
n)
∈
Z
+, thenn
e
e
e
1,
2,
....,
are incident on a vertex inΓ
( )
Z
n . The edge sum labeling was introduced by Paulraj Joseph et al., [3].2. INTEGRAL EDGE SUM OF
Γ
( )
Z
nIn this section, we evaluate the integral edge sum of
Γ
( )
Z
n and study their properties. The integral edge sum graph is defined as follows,Corresponding Author: J. Periaswamy*
1,
1Part-Time Research Scholar,
Definition 2.1: Let
Γ
( )
Z
n be a graph. A bijectionf
:
E
→
S
where S is a set of integers is called as integral edge function of the graphΓ
( )
Z
n . Definef
(
v
)
=
∑
{
f
(
e
)
:
e
is
incident
on
v
}
on V. Then F is called the integral edge sum function of the integral edge function f.( )
Z
nΓ
is said to be an integral edge sum graph if there exists an integral edge functionf
:
E
→
S
such that f and its corresponding integral edge sum function F on V satisfy the following conditions:i) F is into S. That is,
f
(
v
)
∈
S
for everyv
∈
V
.ii) If
e e
1,
2,....,
e
n∈
E
such thatf e
( )
1+
f e
( ) ...
2+
+
f e
( )
n∈
S
, thene e
1,
2,....,
e
n are incident on a vertex.Definition 2.2: If e is an edge of
Γ
( )
Z
n , then the graph obtained by subdividing ‘e’ exactly once is denoted as G(e).Definition 2.3: Let
Γ
( )
Z
n be an integral edge sum graph. The integral edge functionf
:
E
→
S
and its corresponding integral edge sum function F which make G an integral edge sum graph are called optimal integral edge function and optimal integral edge sum function respectively.For example, Let
V
=
{
v
1,
v
2,
...,
v
p−1}
be the vertex set andE
=
{
vv
i:
1
≤
i
≤
p
−
1
}
be the edge set of the graph( )
Z
2pΓ
, where p is any prime number, greater than two. Let
−
−
−
−
=
2
)
2
)(
3
(
),
3
(
...,
,
2
,
1
,
0
p
p
p
S
. Theintegral edge function
f
:
E
→
S
is defined asf
(
vv
i)
=
i
for
1
≤
i
≤
(
p
−
3
)
and2
)
2
)(
3
(
)
(
vv
2=
−
p
−
p
−
f
.
The corresponding integral edge sum function is,
0
)
(
v
=
f
)
3
(
1
)
(
v
=
i
for
≤
i
≤
p
−
f
i2
)
2
)(
3
(
)
(
2−
−
−
=
−p
p
v
f
p0
)
(
v
p−1=
f
Therefore, F is into S. As all the edges are incident on the second condition becomes trivial. Hence
Γ
( )
Z
2p is anintegral edge graph for every natural number
p
−
1
.For example, In
Γ
( )
Z
2p (e),V
=
{
v
1,
v
2,
...,
v
p−2,
u
,
v
,
w
}
is the vertex set and{
,
} {
∪
:
1
≤
≤
(
−
2
)
}
=
uv
uw
vv
i
p
E
i is the edge set ofΓ
( )
Z
2p .
−
−
−
=
2
)
2
)(
1
(
),
3
(
...,
,
2
,
1
,
0
p
p
p
S
. The integral edge sum functionf
:
E
→
S
is defined asf
(
uv
)
=
0
2
)
2
)(
1
(
)
(
uw
=
p
−
p
−
f
)
2
(
1
,
)
(
vv
=
i
≤
i
≤
p
−
f
i .The corresponding integral edge sum function is as follows:
2
)
2
)(
1
(
)
(
)
(
)
(
u
=
f
v
=
f
w
=
p
−
p
−
f
)
2
(
1
,
)
(
v
=
i
≤
i
≤
p
−
f
i .Except uw all the other edges are incident on v. The value
2
)
2
)(
1
(
)
(
uw
=
p
−
p
−
f
can be got by summing either)
(
vv
1In both the examples 0 is one among the labels. But not all the integral edge sum graphs admit 0 as a label.
Definition 2.4: An integral edge sum graph
Γ
( )
Z
n is called a zero integral sum graph if there exists an optimal integral edge functionf
:
E
→
S
with0
∈
S
. We have proved thatΓ
( )
Z
2p orΓ
( )
Z
2p(
e
)
, where p is any prime number, are zero integral edge sum graphs. In the following theorem, we prove that there is no other zero integral edge sum graph.Theorem 2.5: Let G be an integral edge sum graph without isolated vertices. Then G is a zero integral edge sum graph
( )
Z
2pΓ
orΓ
( )
Z
2p(
e
)
for some prime number ‘p’.Proof: Let
f
:
E
→
S
be an optimal integral edge function and F be its corresponding optimal integral edge sum function. Lete
=
uv
be such thatf
(
e
)
=
0
. Ife
1 is any other edge, thenf
(
e
1)
=
f
(
e
1)
+
f
(
e
)
∈
S
and therefore e ande
1 are incident on a vertex. That is, all the other edges are adjacent to e.If w is any other then w and v, then
F
(
w
)
=
F
(
w
)
+
f
(
0
)
∈
S
. Therefore ifw
1,
w
2,
....,
w
r are the vertices adjacent to w, thenww
1,ww
2, …,ww
r and e are adjacent on a vertex. The only possibility is that w is a pendent vertex adjacent on a vertex. The only possibility is that w is a pendant vertex adjacent to either u or v.Suppose
deg
u
>
1
. Letu
1,
u
2,
....,
u
m be the pendent vertices adjacent to u apart from v andv
1,
v
2,
...,
v
p−1 be the pendent vertices adjacent to v apart from u. ThenF
(
u
)
=
f
(
uu
1)
+
f
(
uu
2)
+
....
+
f
(
uu
m)
+
f
(
uv
)
∈
S
. Let)
(
)
(
u
f
e
1F
=
. Ife
1=
uu
i for some i (sayuu
1), thenF
(
u
)
=
f
(
uu
1)
=
f
(
uu
1)
+
f
(
uu
2)
+
....
+
f
(
uu
m)
.That is,
f
(
uu
2)
+
....
+
f
(
uu
m)
=
0
. HenceF
(
v
)
=
F
(
v
)
+
f
(
uu
2)
+
....
+
f
(
uu
m)
∈
S
. That is,m p
uu
uu
uu
vv
vv
vv
1,
2,
...,
−1,
1,
2,
...,
anduv
are incident on a vertex. That is positive only ifm
=
0
.If
e
1=
e
=
uv
, thenF
(
u
)
=
0
andF
(
u
)
+
F
(
v
)
∈
S
. Again this is a possibility only if v is not adjacent to any other than u. Here againm
=
0
.Let us consider the last case. Let
e
1=
vv
i for some i. assuminge
1=
vv
1, we get the following:1 2 1
( )
(
)
(
) ....
(
p)
F v
=
f vv
+
f vv
+
+
f vv
− =f uu
(
1)
+
f uu
(
2) ...
+ +
f uu
(
m)
+
f vv
(
2)
+
f vv
(
3) ...
+ +
f vv
(
p−1)
is an element of S. This is not possible ifp
−
1
≥
2
, where p is any prime greater than 2.Hence we have the following:
(i) e is adjacent to all other edges of
Γ
( )
Z
n .(ii) Any vertex other than u and v is a pendent vertex adjacent to either u or v.
(iii)If
deg
u
>
1
, thendeg
v
=
1
. Thus G is eitherΓ
( )
Z
2p orΓ
( )
Z
2p(
e
)
for somep
−
1
.In all the other integral edge sum graphs zero is not among the edge labels. In a similar way it is easily seen that zero sum of edge labels is also not possible in the nonzero integral edge sum graphs. That is, if f is an optimal integral edge
sum graph
Γ
( )
Z
2n , thenf
(
e
1)
+
f
(
e
2)
+
...
+
f
(
e
n)
is not zero for any sub collection{
e
1,
e
2,
....,
e
n}
of the edge setΓ
( )
Z
2n .The optimal edge function and its corresponding optimal edge sum function of an edge sum graph and the optimal integral edge function and its corresponding optimal integral edge sum of a nonzero integral edge sum graph act almost
similarly. We state the following observations concerning non zero integral edge sum of
Γ
( )
Z
n without proof, as the result and its proof are similar to that of edge sum graphs in each case.Let
ww
i=
l
i forl
≤
i
≤
m
. If there exists an edgee
=
uv
such thatf
(
l
1)
+
f
(
l
2)
+
...
+
f
(
l
m)
=
f
(
e
)
, then one of the following holds:(i)
{ }
u
,
v
forms aΓ
( )
Z
9 component inΓ
( )
Z
n .(ii)
{
u
,
v
,
w
}
is eitherΓ
( )
Z
4 orΓ
( )
Z
9 with one of u, v as a pendent vertex inΓ
( )
Z
n .Observation 2:
Γ
( )
Z
n be a nonzero integral edge sum graph with integral edge functionf
:
E
→
S
and the integral edge sum function F of f. let w be a non pendent vertex ande
=
uv
∈
E
be such thatF
(
w
)
=
f
(
e
)
. Then one of the following holds:(i)
{ }
u
,
v
forms aΓ
( )
Z
9 component inΓ
( )
Z
n .(ii)
{
u
,
v
,
w
}
is eitherΓ
( )
Z
9 with one of u, v as a pendent vertex inΓ
( )
Z
n .Remark: Let
f
:
E
→
S
be an optimal edge function and F be its corresponding optimal edge sum graphΓ
( )
Z
n .If u is a non pendent vertex ofΓ
( )
Z
n , then there exist vertices v, w such thatF
(
u
)
=
f
(
vw
)
. If vw is not aΓ
( )
Z
9component then either one of them is adjacent to u. Let v be the adjacent to u. Then
( )
(
)
(
)
( )
( )
F v
≥
f u v f vw
+
>
F u
>
f e
for every edge incident on u. Hence ifF
(
v
)
=
f
(
e
1)
, thene
1 is incident on v andF
(
u
)
=
f
(
e
2)
, thene
2 is not incident on u. But this is not the case in integral edge sum graphs. We will see it in proving complete graph inΓ
( )
Z
n is integral edge sum graph.Definition 2.6: Let
V
=
{
v
1,
v
2,
...,
v
p−1}
be the vertex set of( )
2p
Z
Γ
forp
≥
5
is any prime number. The integraledge function matrix
A
=
( )
a
i,j of orderp
−
1
of the integral edge function f of( )
2p
Z
Γ
is defined as)
(
,j i j i
f
v
v
a
=
ifi
≠
j
and is 0 ifi
=
j
. The integral edge function matrix is a symmetric matrix.Theorem 2.7:
Γ
( )
Z
p2 forp
≥
5
is an integral edge sum graph, where p is a prime number.Proof: Let
V
=
{
v
1,
v
2,
...,
v
p−1}
be the vertex set of( )
2p
Z
Γ
forp
≥
5
is any prime number and{
:
1
≤
≤
−
2
+
1
≤
≤
−
1
}
=
v
v
i
p
and
i
j
p
E
i j be the edge set ofΓ
( )
Z
p2 . For defining the integral edgefunction matrix we need the following definitions.
Let
B
=
( )
b
i,j be a(
p
−
5
)
×
(
p
−
1
)
matrix defined asb
0=
20
,b
i,j=
b
j,i for1
≤
i
,
j
≤
p
−
5
,{
}
∑
≤
≤
−
=
=
− −+1 1 1,
:
1
5
,
b
b
i
p
b
ii i i j andb
i,j=
∑
{
2
j−ib
i−1:
1
≤
i
≤
p
−
5
and
(
i
+
2
)
≤
j
≤
p
−
1
}
.Let
d
p−1=
[
(
b
1,p−1−
b
1,2) (
+
b
2,p−1−
b
2,3)
+
....
+
(
b
p−5,p−1−
b
p−5,p−4)
]
.(
) (
)
(
)
[
1, 2 1,2 2, 2 2,3 5, 2 5, 4]
2 − −
....
− − − −−
=
p−
+
p−
+
+
p p−
p pp
b
b
b
b
b
b
d
.(
) (
)
(
)
[
1, 3 1,2 2, 3 2,3 5, 3 5, 4]
3 − −
....
− − − −−
=
p−
+
p−
+
+
p p−
p pp
b
b
b
b
b
b
d
.(
) (
)
(
)
[
1, 4 1,2 2, 4 2,3 5, 4 5, 4]
4 − −
....
− − − −−
=
p−
+
p−
+
+
p p−
p pp
b
b
b
b
b
b
d
.Since
b
i,i+1=
b
i−1 for1
≤
i
≤
p
−
5
, we get that(
1,+
2,+
....
+
−5,)
−
( )
1,2+
(
2,3+
....
−5, −4)
=
j j p j p pj
b
b
b
b
b
b
d
.{
}
∑
{
}
∑
≤
≤
−
−
≤
≤
−
=
b
i,j:
1
i
p
5
b
i−1:
1
i
p
5
{
,:
1
≤
≤
−
5
}
−
−6=
∑
b
i ji
p
B
pWhere,
B
j=
∑
{
b
i:
1
≤
i
≤
j
}
forp
−
4
≤
j
≤
p
−
1
.Let
H
=
( )
h
i,j be a(
p
−
5
)
×
(
p
−
1
)
matrix defined ash
i,j=
h
0 for2
≤
j
≤
p
−
1
and{
}
∑
≤
≤
−
=
=
−1 −1,:
1
1
,
h
h
k
p
h
i j i i k for2
≤
i
≤
p
−
5
andi
+
1
≤
j
≤
p
−
1
whereh
j,i=
h
i,j for5
1
≤
i
≤
p
−
and1
≤
j
≤
p
−
5
. Letx
i=
b
i+
h
i for0
≤
i
≤
p
−
6
.The integral edge function matrix
A
=
( )
a
i,j of orderp
−
1
is defined as,j i j i j
i
b
h
a
,=
,+
, for1
≤
i
≤
p
−
5
andi
+
1
≤
j
≤
p
−
1
.{
}
∑
≤
≤
−
=
=
− −−
−4, 3
x
5a
5,:
1
j
p
1
a
p p p p jy
a
p−4,p−2=
−
a
x
a
p−4,p−1=
p−5+
{
}
∑
≤
≤
−
=
=
− −−
−3, 2
x
4a
4,:
1
j
p
1
a
p p p p jz
a
p−3,p−1=
−
{
}
∑
≤
≤
−
=
=
− −−
−2, 1
x
3a
3,:
1
j
p
1
a
p p p p jWhere a, z, y are defined as follows.
Let
X
i=
∑
{
x
j:
0
≤
j
≤
i
}
4
2
3
5−
4+
1+
3+
4+
=
X
p−x
p−d
p−d
p−d
p−y
4
2
−
5+
1−
2+
4+
=
y
X
p−d
p−d
p−d
p−z
and(
5)
12
2 33
45
2
a
=
y
−
X
p−+
d
p−−
d
p−−
d
p−−
d
p−We have,
d
j=
∑
{
b
i,j:
1
≤
i
≤
p
−
5
}
−
B
p−6 forp
−
4
≤
j
≤
p
−
1
(
)
{
}
∑
{
(
)
}
∑
+
≤
≤
−
−
+
≤
≤
−
=
b
i,jh
i,j:
1
i
p
5
b
i−1h
i,j:
1
i
p
5
{
}
∑
{
(
)
}
∑
≤
≤
−
−
+
≤
≤
−
=
a
i,j:
1
i
p
5
b
i−1h
i,j:
1
i
p
5
{
}
∑
{
}
∑
≤
≤
−
−
≤
≤
−
=
a
i,j:
1
i
p
5
x
i−1:
1
i
p
5
{
,:
1
≤
≤
−
5
}
−
−6=
∑
a
iji
p
X
pLet
S
=
{
a
,j:
1
≤
i
≤
p
−
2
and
i
+
1
≤
j
≤
p
−
1
}
.The edge function
f
:
E
→
S
is definedi
+
1
≤
j
≤
p
−
1
. The corresponding integral edge sum function F is as follows:( )
v
=
∑
{
f
( )
v
v
:
1
≤
j
≤
p
−
1
and
j
≠
i
}
for
1
≤
i
≤
p
−
3
F
i i j .{
}
∑
≤
≤
−
≠
=
a
i,j:
1
j
p
1
and
j
i
(
)
{
}
∑
+
≤
≤
−
=
b
i−1h
i,j:
1
j
p
1
i i
h
b
+
=
i
x
=
( )
v
−2=
∑
{
f
(
v
−2v
)
:
1
≤
j
≤
p
−
1
and
j
≠
p
−
2
}
F
p p j{
}
∑
≤
≤
−
≠
−
=
a
p−2,j:
1
j
p
1
and
j
p
2
{
−2,:
1
≤
≤
−
5
}
+
−2, −4+
−6−
+
−4+
−3=
∑
a
p jj
p
a
p pX
py
x
px
p5
−
=
x
p3 , 4 −
−
=
a
p p(
−4 −3)
( )
v
−1=
∑
{
f
(
v
−1v
)
:
1
≤
j
≤
p
−
2
}
F
p p j{
}
∑
≤
≤
−
=
a
p−1,j:
1
j
p
2
{
}
∑
−≤
≤
−
+
− −+
− −+
− −=
a
p 1,j:
1
j
p
5
a
p 1,p 4a
p 1,p 3a
p 1,p 25 3 5
6
1 − − − −
−
+
+
+
−
+
=
=
d
pX
px
pa
z
x
px
p3 , 4 −
−
=
a
p p(
−4 −3)
=
f
v
pv
pHence, F is into S. Let
S
1=
{
a
i,j:
1
≤
i
≤
p
−
5
and
i
+
1
≤
j
≤
p
−
1
}
and{
,:
4
2
1
1
}
2
=
a
p
−
≤
i
≤
p
−
and
i
+
≤
j
≤
p
−
S
i j . The elements ofS
1 satisfy the following properties:(i)
a
i j,=
b
i j,+
h
i j,(ii)
b
i i,+1=
∑
{
b
i−1,j:1
≤ ≤ −
i
p
5
}
(iii)
b
i j,>
(
b
1,2+
b
1,3+ +
...
b
1,p−1) (
+
b
2,3+
b
2,4+ +
...
b
2,p−1)
+
....
+
(
b
i−1,i+
b
i−1,i+1+
...
+
b
i−1,p−1) (
+
b
i,i+1+
b
i,i+2+
...
+
b
i,p−1)
(iv)
h
i j,=
h
0 for2
≤
j
≤
p
−
1
andh
i j,=
h
i−1=
∑
{
h
i−1,k:1
≤ ≤ −
k
p
1
}
for2
≤
j
≤
p
−
5
and1
1
≤
≤
−
+
j
p
i
.(v) All the elements of
S
1 are congruent to0
(
mod
20
)
. Hence, no element ofS
1 exceptx
i for6
1
≤
j
≤
p
−
is a sum of two or more elements of S.The elements of
S
2 satisfy the following properties:(i) ap−4 ,p−3 =xp−5 =
∑
{
ap−5,j: 1≤ ≤ −i p 1}
(ii) y=3Xp−5−2xp−5+dp−1+dp−3+dp−4+ ≡4 4 mod 20
(
)
.Hence,
a
p−3,p−2= − ≡
y
10 mod 20
(
)
.(iii)
a
≡
10 mod 20
(
)
and hencex
p−4,p−1=
x
p−5+
a
≡
10
(
mod
20
)
.(iv)
a
p−3,p−2=
x
p−4≡
6 mod 20
(
)
.(v)
z
≡
8 mod 20
(
)
and hencea
p−3,p−1=
−
z
≡
12
(
mod
20
)
. (vi)x
p−3≡
8 mod 20
(
)
.(vii)
x
p−5+ − =
a
z
2
.(viii)
x
p−4− =
y
d
p−1−
d
p−2+
2
.(ix)
x
0=
b
0+
h
0
>
{
d
p−1−
d
p−2+
2
}
=
x
p−4−
y
.Hence, no element of
S
2 exceptx
p−3,
x
p−4 andx
p−5 are a sum of two or more elements of S. Thus, we get that ifS
e
f
e
f
e
f
(
1)
+
(
2)
+
...
+
(
n)
∈
, thene
1,
e
2,
...,
e
n from the set of all edges incident on a vertex. Hence,( )
2p
Z
Γ
REFERENCES
1. D.F.Anderson, P.S.Livingston, The zero- divisor graph of a commutative ring, J.Algebra, 217, No.2, (1999), 434-447.
2. I.Beck, Colouring of Commutative Rings, J.Algebra, 116, (1988), 208-226.
3. J.Paulraj Joseph, D.S.T.Ramesh, S.Somasundaram, Edge sum graphs, IJOMAS, 18, No. 1, (2002), 71-78. 4. J.RaviSankar and S.Meena, Changing and Unchanging the Domination Number of a Commutative ring,
International Journal of Algebra, 6, No.27, (2012), 1343-1352.
5. J.RaviSankar and S.Meena, Connected Domination number of a commutative ring, International Journal of Mathematical Research, 5, No.1, (2012), 5-11.
6. J.RaviSankar and S.Meena, On Weak Domination in a Zero Divisor Graph, International Journal of Applied Mathematics, 26, No.1, (2013), 83-91.
Source of support: Nil, Conflict of interest: None Declared.
