From the canonical representation, every
positive integer 
can
be written as 
, 
are
primes, 
is a
positive integer for every 
and 
. A
positive integer 
is
called the least common multiple of 
and 
, if is a
common multiple of and , and
also 
for
any another common multiple
of and . Also,
we denote as 
. Let
be
the set of all proper divisors of a positive integer 
,
i.e., 
. The
divisor function 
denotes
the cardinality of the set of all divisors of .
i.e., 
,
where 
. If 
is
called a common divisor of and
when
and 
. And
also is
called a greatest common divisor of and , if
is a
common divisor of and and
also if 
is
any other common divisor of and ,
then 
, we
write 
.
Also, we have 
. For
the positive integers 
and 
, if divides
the difference of and , we
denote that is
congruent to modulo
and
we define 
.
Otherwise, we denote that is
incongruent to modulo
and
we write 
. For
further definitions of number theory the reader may look at 2.
For the graph 
, the
two distinct vertices 
and 
are
said to be adjacent whenever there exist an edge between and . And
also we write 
. The
cardinality of the set 
is
called the order of . The
graph is
called connected if there exist a path between every pair of two distinct
vertices in ,
otherwise said to be disconnected. If there exist an edge between every pair of
two distinct vertices in the graph ,
then the graph is
called complete, it is denoted by 
for vertices
of the graph. A bipartite graph is a graph whose vertex set can be partitioned
into two sets 
and 
such
that each edge has one end in and
another end in . A
complete bipartite graph is a bipartite graph with partition 
, in
which each vertex of is
adjacent to each vertex of . The
eccentricity 
of a
vertex 
in a
graph is
the distance between and
a vertex farthest from in .
Radius of a graph is
the smallest
eccentricity among the vertices of , is
denoted by 
. Diameter
of a graph is
the greatest eccentricity
among the vertices of , is
denoted by 
. For
further definitions of graph theory the reader may see 3.
1.
ADJACENCY:
In this section we find the necessary
and sufficient condition for adjacency of vertices in the graph 
.
Lemma 1.1: If 
, 
is
prime, then
the graph does
not exist.
Proof:We know
that the ring 
consists
of no nonzero zero-divisors for everyprime . This
completes the proof.
Lemma 1.2: The
order of
the graph is 
.
Proof:From the
definition of the graph, we have
. We
know that 
.
This implies that 
.
Theorem 1.3: For two
distinct vertices and are
adjacent in the graph if
and only if the
least common multiple of and is
congruent to zero modulo .
Proof:We know that 
in .
Because 
.This
shows that 
if
and only if 
in .
We have the vertex is
adjacent to the vertex in
the graph if
and only if in ,
from the definition of the graph .This
follows the proof.
Example 1.4 explains Theorem 1.3
with Fig. 1.
Example 1.4: In the ring 
, the
vertex set of the graph 
is 
. And
also, 
is
adjacent to 
, is
adjacent to 
and is
not adjacent to in ,
since 
, 
and 
.
Fig. 1. 
2.
CONNECTEDNESS:
In this section we find the
connectedness of the intersection graph 
for
all characterizations of .The
set of
all proper
divisors of a positive integer can
be written as the disjoint union of two sets 
and 
as the
least common multiple of every element is incongruent to zero modulo with
every another element in and
congruent to zero modulo with
at least one another element in respectively.
i.e.,
and
.
Example 2.1:The set of
all proper divisors of 
is 
and
also the sets and are
and 
,
which clearly shows that the set can
be written as the disjoint union of and
.
Based on the sets and , the
set 
of
all nonzero zero-divisors of the ring can
also be written as the disjoint union of two sets 
and 
as
and
.
Example2.2:
The set of all nonzero zero-divisors of
the ring 
is
. And
also and are 
and 
.
From the definition of the intersection
graph , the
vertex set 
of the
set of all nonzero zero-divisors of the ring. So,
the vertex set of the graph can
also be written as the disjoint union of 
and 
.
Example 2.3:
Consider the intersection graph 
for
the ring .
Then the vertex set of the graph is 
. And
also 
, 
. The
graph is shown
in the Fig. 2.
Fig. 2. 
Result 2.4(Result 4.3 of 9):
(i)
If
and is
prime,
then 
and 
.
(ii)
If
, is
prime and 
, then
and 
.
(iii)
If
, are
primes and ,
then 
and 
.
(iv)
If
, are
primes, 
, is a
positive integer for every 
and 
. Then
both , , 
and 
are
non empty.
Result 2.5: Every vertex in
the
set is
not adjacent to remaining all the vertices in the graph .
Proof:Obviously, the
proof follows from the definitions of and with
the Theorem 1.3.
Theorem 2.6: The graph is
null if
and only if can
be written as power of a prime.
Proof:Weknow that , is
prime with if
and only if ,
from (ii) of Result 2.4.Hence the proof follows from Result 2.5.
Example 2.7:For the graph
, 
, 
. And
also the graph is
shown in the Fig. 3.
Fig.
3.
Theorem 2.8:If can
be written as a product of two distinct primes, then the
graph is
complete bipartite.
Proof: Let 
,
where 
are
primes. Then the total number of proper divisors of are 
,
which are 
and 
.Now,
the set of all vertices in can
be written as the disjoint union of two sets 
and 
.
Since 
,
then we have each vertex in the set 
is
adjacent to each vertex in the set 
.Andalso
we have 
,
then we have every vertex in the sets and are
not adjacent to remaining all the vertices in that same set.This implies that
the graph is
complete bipartite.
Fig.
1. illustrates the Theorem 2.8 with bipartition 
,
where 
and
.
Theorem 2.9: The graph is
connected, but neither complete and nor bipartite if and only if can
be written as a product of more than two distinct primes.
Proof:Assume that , where
are
primes and 
.Let
and
be
any two distinct arbitrary vertices in the graph.
(i)
If
is
adjacent to ,
then there is nothing to prove.
(ii)
If
is
not adjacent to ,
then the following possibilities are exists for every 
and 
.
i.e., if
,
then
and
also if 
,
then at least 
, 
.
If at least one 
, for
, 
,
then there exists a vertex 
such
that 
and 
are
congruent to zero modulo ,
where 
,
.
Thus, we have 
.
Otherwise, there exists two vertices and 
such
that , 
and 
are
congruent to zero modulo ,
where 
and
.
This shows that 
. Thus
the graph is
connected.
Since
, ,
then clearly there exits two distinct vertices and
such
that 
,
where 
and 
for 
,
and
also there exist a cycle 
, 
, 
, of
length 3. This proves that the graph is
neither complete nor bipartite by using Theorem 1.2 of 3.
Conversely,
assume the graph is
connected, but neither complete nor bipartite. Now we prove that with
.If 
,
then the graph complete
bipartite from Theorem 2.8.Thus we get a contradiction.This proof the theorem.
And also Fig. 4 explains the Theorem 2.9.
Fig. 4. 
Theorem 2.10: The
graph is
disconnected with one connected component if and only if can
be written as a product of more than one prime powers but not product of all
primes.
Proof:Let , , 
, 
and , then
both and are
non empty from (iv) of Result 2.4.By using the Result 2.5, we have the graph is
disconnected and also from the definition of , we
get one connected component with the vertex set as
same as in Theorem2.9.
Conversely assume that
the graph is
disconnected with one connected component. Now we show that can
be written as a product of more than one prime powers but not product of all
primes.
If possible assume 
, , 
with,
then we get a contradiction from the Theorems 2.6, 2.8, 2.9. This completes the
proof. And also Fig. 2 explains Theorem 2.10.
3.
RADIUS
AND DIAMETER:
In
this section we calculate the radius and diameter of the intersection graph 
for
all characterizations of.
Theorem
3.1:If
can
be written as a power of a prime, then 
.
Proof: Let ,prime
with .
Then from (ii) of Result 2.4 and Result 2.5, 
for
all
,
except 
.
Obviously the graph 
is
empty when .
This follows the proof.
Example3.2: For the graph , .
Then 
,
since 
. The
graph is
shown in Fig. 3.
Theorem 3.3: If can
be written as a product of two distinct primes.Then
(i)
and 
, if is
even
(ii)
, if is
odd
Proof:Let 
, 
are
primes, then from
Theorem2.8, the graph is
complete bipartite.
(i)
For
is
even, so that the graph becomes
a star graph. This completes the proof.
(ii) For is
odd, then each partition of the complete bipartite graph contains
more than one vertex. Thisclearly shows that 
for
all .Thus
the proof completes.
Example 3.4:
a.
For
,
then 
and 
.Because
and , 
and the
graph 
is
shown in Fig. 5.
b.
For
,
then 
.
Since , 
and
the graph 
is
shown in Fig. 6.
Fig.
5.
Fig. 6.
Theorem 3.5: If can
be written as a product of more than two distinct primes. Then 
and 
.
Proof. Let ,
where are
primes with 
.
Since the vertex set of the graph is 
We know that, every divisor in the
set of
all proper divisors of can
be written as 
, 
, 
. Let
and be
any two distinct arbitrary vertices in the graph.
(i)
If
a.
If
is
adjacent to ,
then 
b.
If
is
not adjacent to ,
then
, 
, 
and
also if ,
then at least , .
1.
If
at least one , for
and .
Then there exist a vertex such
that and are
congruent to zero modulo ,
where .
Thus,
we have and 
.
2.
Otherwise,
there exists two vertices and such
that 
, and are
congruent to zero modulo ,
where and .
This implies that and 
.
Sothe eccentricity of is 
.
(ii)
If
, 
a.
If
is
adjacent to ,
then 
b.
If
is
not adjacent to ,
then 
, and
also then there exists a vertex such
that and are
congruent to zero modulo ,
where .
Thus, we have and
hence .
So
the eccentricity of the vertex is .
This completes the proof.
Example 3.6:For 
,
then 
and 
.
Since, we have and 
,
where 
and 
. And
alsothe graph 
is
shown in Fig. 4.
Theorem 3.7: If can
be written as a product of more than one prime powers but not product of all
primes.Then 
.
Proof:From Theorem 2.10,
the
graph is
disconnected. Hence the proof follows.
Example3.8: For the graph , 
.
Then
.Because
the graph is disconnected
and shown in Fig. 2.
CONCLUSION:
In this paper we
conclude that two
distinct vertices and are
adjacent in the graph if
and only if the
least common multiple of and is
congruent to zero modulo .
Also the graph is
null if and only if can
be written as power of a prime and the graph is
complete bipartite when can
be written as product of two distinct primes. Further the radius and diameter
of the graph is
infinity when can
be written as power of a prime or product of two distinct primes.
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and
is defined as a simple undirected graph whose vertices are the set of all
nonzero zero-divisors of the ring 
