Download some domination parameters of zero divisor graphs

Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
SOME DOMINATION PARAMETERS OF ZERO DIVISOR GRAPHS
K.BUDADODDI
R. RAMAKRISHNA PRASAD
&
A.MALLIKARJUNA REDDY
Assistant Professor, Department of Mathematics,
Sri Krishnadevaraya University College of Engineering & Technology,
Ananthapuramu, Andhra Pradesh, India-515003.
Research Scholar, Department of Mathematics,
Sri Krishnadevaraya University, Ananthapuramu, Andhra Pradesh, India-515003.
Professor, Department of Mathematics,
Sri Krishnadevaraya University, Ananthapuramu, Andhra Pradesh, India-515003.
[email protected]
Article Info
ABSTRACT
Graph theory is one of the most developing branches
of mathematics with wide applications to various
branches of science and technology. A graph can be
used to represent almost any physical situation
Keyword:
Dominating
set,
Domination
number,
zero
divisors,
divisor graphs.
.
Zero
involving discrete objects and a relationship among
them. The origin of graph theory started with the
problem of Konigsberg bridge problem in 1735 by the
great mathematician Leonard Euler.
The theory of domination in graphs introduced by C.
Berge and O. Ore which is rapidly growing area of
research in graph theory today. The study of
domination in graphs originated around 1850 with the
problem of determining the minimum number of
queens that can be placed on an nxn chess board so as
to cover or dominate every square.
The concept of zero divisor graphs of a ring was
introduced by I. Beck in which the coloring of a
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Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
commutative ring is mainly discussed. In this work,
the set of vertices is taken to be R and two vertices x
and y are adjacent if their product should be zero. The
investigation of colourings of a commutative ring was
then continued by D. D. Anderson and M. Naseer. In
this paper, I investigate some domination parameters
of zero divisor graphs.
Copyright © 2015
Open Journal of Applied & Theoretical Mathematics
(OJATM)
All rights reserved.
1. INTRODUCTION
Let R be a commutative ring with unity
of R and
, Z(R) be the set of all zero-divisors
be the set of nonzero zero-divisors of R. The zero –divisor
graph of R denoted by (Z ( R) * ) is usually written (R) whose vertices are the
elements of the
and distinct vertices
are adjacent if and only if
. This definition defined by David F.Anderson and Philip S. Livingston in [3].
The main object of this unit is to study the inter relation of ring theoretic
properties of R with graph theoretic properties of (R) . This study helps illuminate the
structure of
. For
, define
if
or
. The relation
is
always reflexive and symmetric, but is usually not transitive. The zero divisor graph
(R) measures this lack of transitivity in the sense that
is transitive if and only if
(R) is complete.
The idea of a zero-divisor graph of a commutative ring was introduced by I. Beck
in [5], where he was mainly interested in colourings. This investigation of colourings of
a commutative ring was then continued by D.D.Anderson and M.Naseer in [4]. Their
definition was slightly different than the previous definition. They let all elements of R
be vertices and has distinct
and
are adjacent if and only if
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. They denote their
Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
zero-divisor graph of R by 0 ( R) . In 0 ( R) , the vertex 0 is adjacent to every other
vertex, but nonzero zero-divisor are adjacent only to 0. Note that (R) is a sub graph of
0 ( R) .
2. PROPERTIES OF ZERO DIVISOR GRAPH
1. (R) is the empty graph if and only if R is an integral domain.
2. Let A and B be integral domains and let R  A  B . Then (R) is a complete bipartite
graph with ( R)  A  B  2 .
3. Let R be a commutative ring then (R) is connected. Let R be a commutative ring.
Then (R) is complete if and only if either
or
, for all
.
4. Let R be finite commutative ring. If (R) is complete then either
local with char R= or
and ( R)  p n  1 where
In this work the commutative ring R is
is prime and
or R is
.
and zero-divisor graph (R) is ( Z n ) .
3. DOMINATION NUMBER OF ZERO DIVISOR GRAPH
We know that a dominating set of a graph G(V,E) is a subset D of V(G) such that
every vertex in V-D is adjacent to at least one vertex in D.
The minimum cardinality of a dominating set of G is called the domination
number of G and it is denoted by
.
In this section we study the domination number of zero divisor graphs.
Theorem 3.1: If n=2 where
is an odd prime, then the non-zero zero divisor graph is
star graph and it is also K 1, p 1 complete bipartite graph.
Proof: Let n=2 , where
graph is Z(R)*=V= {
is an odd prime. Then the vertex set of non-zero zero divisor
multiplication modulo
distinct even integers which is in V. The product of
which is not divisible by n where n=2 ,
is an odd prime.
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}. Let
be two
will become an even integer
Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
The product
vertex is
is divisible by n=2 , if one vertex is 2 or multiples of 2 and another
then only and
are adjacent.
Two distinct even integers
Let us take
and
are not adjacent.
is an even integer and
is a prime number in V. The product of
even and prime is an even integer and it is divisible by n where n=2 ,
is a prime and
.
Every even integer and prime number is adjacent.
is adjacent to 2,4,6,….2( -1)
It is clear that K P is a star graph. The graph K P is classified into two non-empty
subsets
 V. Where X={ } and Y={2,4,6,….2( -1)}. The vertex
and
X is adjacent
to all the vertices of Y.
K 1, p 1 is a complete bipartite graph.
=2 where
is a prime and
is a star graph and K 1, P 1 is a complete bipartite
graph.
Theorem 3.2: If =2 where
is an odd prime, then the domination number of non-
zero zero divisor graph is 1.
Proof: By theorem 3.1 ( Z n ) is a star graph. D= { } forms a minimum dominating set
with cardinality1. Then the domination number of non-zero zero divisor graph is 1,
when n=2 , where
is an odd prime.
( ( Z n ) ) = 1.
Theorem 3.3: If =
, where
is a prime and >3. Then the non-zero zero divisor
graph is K 2, P 1 complete bipartite graph.
Proof: Let =
6,…,
-3, ,
, where
is a prime number and >3.Then the vertex set V= Z(R)*= {3,
}. Decompose the vertex set V in to the following disjoint subsets.
1. The set S of non-zero multiples of 3 which are less than .
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Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
2. The set M of multiples of prime
which are less than .
The edge set E(G) defined by E(G)={(
)/
=0 (multiplication modulo n) 
Z(R)*}. Let us consider two distinct vertices
only multiples of 3. The product of
, where the vertices of S are
is divisible by 3 but not 3 . Hence two distinct
vertices of S are not adjacent. The product of
is divisible by , when one vertex is 3
or multiples of 3 and another vertex is prime or multiples of prime.
Now let us take
prime. The product of
and
, where the vertices of M are the multiples of
is divisible by
. Hence one vertex is in S and other in M are
adjacent.
The vertices of M are adjacent to all the vertices of S and vice-versa. The vertex set V
can be decomposed into the two disjoint subsets S and M. The edges of two vertices are
partition into the two sets one end in S and other end in M.
n=
where
is a prime and >3 is a K 2, P 1 complete bipartite graph.
Theorem 3.4: If
=
where
is a prime number and
>3. Then the domination
number of non-zero zero divisor graph is 2.
Proof: By theorem 3.3 ( Z n ) is a complete bipartite graph, if
=
where
is a prime
number and >3. The set D= { , } is a dominating set, since every vertex of V-D is
adjacent to at least one vertex of D. Continuing like this D= { ,
,3,6,…..
} is also
dominating set.
D= { , } is a minimum dominating set with cardinality 2.
The domination number of ( Z n ) is 2.
( Z n )
.
Theorem 3.5: If n=
, where
is a prime and >5. Then the domination number of
non-zero zero divisor graph is 2.
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Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
Proof: Let n=
15,….,
, where
, ,
,
,
is a prime and >5. Then the vertex set V=
= {5, 10,
}. Decompose the vertex set V into the following disjoint
subsets.
1. The set S of non-zero multiples of 5 which are less than .
2. The set M of multiples of prime
which are less than .
The edge set E(G) defined by E(G)={(
)/
=0 (multiplication modulo n) 
Z(R)*}. Let us consider two distinct vertices
multiples of 5. The product of
, where the vertices of S are
(mod n), 
is an integer. The product of
. So the distinct vertices of S are not adjacent.
Let
, and
divisible by
or
. The product of
but
is not divisible by
i.e.,
which is
(mod n). The distinct
vertices of M are not adjacent.
The distinct vertices of M are not adjacent.
Let us consider
and
where the vertices of S are multiples of 5 and
the vertices M are multiples of prime . The product of
is divisible by .
Every vertex in M are adjacent to all the vertices of S and vice-versa.
D= {(
)/
and
} is a dominating set because every vertex in V-D is adjacent
to at least one vertex in D. Continuing like this D={5, 10, 15,….5p-5, ,
,
,
} is also
dominating set.
D= {
( Z n )
} is a minimum dominating set with cardinality 2.
.
Theorem 3.6: If n=
, where
is a prime and >5, then the non-zero zero divisor
graph is K 4, P 1 complete bipartite graph.
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Proof: Let n=
15,….,5 -5, ,
, where
,
,
is a prime and
>5.Then vertex set V=
= {5, 10,
}. Decompose the vertex set V in to the following disjoint
subsets. By Theorem 3.5 ( Z n ) is a K 4, P 1 complete bipartite graph.
Theorem 3.7: If n=
, where
are primes and > . Then the domination number of
non-zero zero divisor graph is 2.
Proof: Let n=
{ ,
,
, where
,……, (
are primes and
> . Then the vertex set V=
=
), , 2 , 3 , …..,( -1) }. Decompose the vertex set V in to the
following disjoint subsets.
1. The set S of non-zero multiples of
2. The set M of multiples of
which are less than n.
which are less than n.
The edge set E(G) defined by E(G)={(
)/
=0 (multiplication modulo n) 
Z(R)*}.
Let us consider
is an integer.
where the vertices of S are multiples of
is divisible by , but
is not divisible by
. The product of
where n=
. Since
every integer can be written as product of primes but it is not in the form
. So the
distinct vertices of S are not adjacent.
Let us take
product of
is an integer.
, where the vertices of M are multiples of prime . The
is divisible by but
is not divisible by
where n=
.
The distinct vertices of M are not adjacent.
Let us consider
and
, where the vertices of S are multiples of prime
and the vertices of M are multiples of prime . The product
is divisible by
.
Every vertex in M is adjacent to all the vertices of S and vice-versa.
D= {(
)/
and
} is a dominating set, because every vertex in V-D is
adjacent to exactly one vertex in D. Continuing like this D={ ,
3 , …..,( -1) } is also dominating set.
D= {
} is a minimum dominating set with cardinality 2.
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,
,….., (
), , 2 ,
Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
( Z n )
.
Theorem 3.8: If n=
where
are primes and > . Then the non-zero zero divisor
graph is K p1,q1 Complete bipartite graph.
Proof: By theorem 3.7, ( Z n ) is a K p1,q1 complete bipartite graph.
Theorem 3.9: If
where
an odd prime, then the domination number of non-
zero zero divisor graph is 1.
Proof: Let
where
graph is V=
an odd prime. Then the vertex set of non-zero zero divisor
= { , 2 , 3 ,….., (
)}. The edge set E(G) defined by E(G)={(
=0 (multiplication modulo n) 
complete graph on
)/
Z(R)*}. When we draw the graph we get a
vertices. If graph is complete then a single vertex can
dominate remaining all the vertices.
D= { } is a dominating set with cardinality 1. It is clear that D= { } is a minimum
dominating set.
The domination number of ( Z n ) is 1, when
( Z n )
where
an odd prime.
.
Theorem 3.10: If
where
is an odd prime and
. Then the domination
number of non-zero zero divisor graph is 1.
Proof: Let
where
{ , 2 , 3 ,…,
,…....,
of ( Z n ) . Hence D= {
is an odd prime and
}. If
then
.Then the vertex set V=
=
is adjacent to every other vertex
} is a minimum dominating set. It is clear that D= {
} is a
minimum dominating set with cardinality 1.
The domination number of non-zero zero divisor graph is 1, when
an odd prime and
( ( Z n ) )
.
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where
is
Open Journal of Applied & Theoretical Mathematics (OJATM)
Vol. 2, No. 4, December 2016, pp. 496~504
ISSN: 2455-7102
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[2]
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rings” Comm.Algebra, 33, 2043-2050, 2005.
[3]
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434-447,1999.
[4]
Anderson.D.D and Naseer.M “Beck’s coloring of commutative ring” J. Algebra,
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[5]
Beck.I “coloring of commutative rings”, J. Algebra 116, 208-226, 1988.
[6]
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