For positive integers
, we consider
to be a zero divisor graph of commutative ring
whose vertex set is
and edge set is
Then
and
This section is devoted on the discussion of eccentric topological indices based on the edges of zero divisor graphs for rings
with
and
, where
and
q are prime numbers.
5.1. Case 1:
Let be a zero divisor graph of commutative ring , where are prime numbers with , in this subsection we discuss the eccentric topological indices for .
Theorem 1. Let be the zero divisor graph of commutative ring , where are prime numbers with Then and .
Proof. Let . Then we have following cases:
If and for any with then each of this type is connected to the vertices of type . Hence the degree of each such vertex is Similarly if and , then the degree of each such vertex is
If and then each such vertex is connected to the vertices of types and for every , , and with Hence the degree of each such vertex is
If and then each such vertex is connected to the vertices of type and for every , , and Therefore the degree of each such vertex is Similarly if and then the degree of each such vertices is
If and then each such vertex is connected to vertex for every . Hence the degree of each such vertex is Similarly if and then the degree of each such vertex is
If and then each such vertex is connected to the vertices of type and for every . Therefore the degree of each such vertex is Similarly, if and then the degree of each such vertex is
If and , and then each such vertex is connected to the vertex for every . Hence the degree of each such vertex is
The total number of vertices of
from the above cases is:
Hence,
The total number of edges of
is obtained by using hand shaking lemma as:
After simplification we get,
. □
From the proof of Theorem 1 and our convenance, we partition the vertex set of
as follows:
This shows that
. It is observed that
,
,
,
,
,
,
,
and
Let the degree of a vertex
u in set
A be denoted by
and the distance between two vertices sets
X and
Y be
Then following theorem determines the eccentricity of the vertices in graph
.
Theorem 2. If and q are prime numbers such that and the associated zero divisor graph to the commutative ring then the eccentricity of the vertices of corresponding graph is either 2 or 3.
Proof. It is easy to see from the proof of Theorem 1 that,
Hence the eccentricity of the corresponding vertices of set
is 3 i.e.,
. Similarly, we can see that
for
and
Therefore,
.
As and for This shows . Similarly, it is easy to calculate for and This implies . □
Lemma 1. For any three prime numbers and q such that and the associated zero divisor graph of commutative ring we have, Proof. The graph
contains
vertices and
edges. Let us divide the edges of
into partition sets according to the eccentricity of its end vertices.
This means that the set
contains the edges incident with one vertex of eccentricity
r and the other vertex of eccentricity
s. From the Theorem 1 and Theorem 2, we have
,
,
and
. Then
□
Now we present values of the eccentric topological indices of the graph which are edge-based.
Theorem 3. If and q are prime numbers such that and the associated zero divisor graph to the commutative ring then
the First Zagreb eccentricity index of is:
the third Zagreb eccentricity index is:
the geometric-arithmetic eccentricity index is:
the atom-bond connectivity eccentricity index is:
the fourth type of eccentric harmonic index is:
Proof. For the first Zagreb eccentricity index
of
we obtain
. So
,
and
Thus by Lemma 1,
For the third Zagreb eccentricity index
of
we obtain
. So
and
So by Lemma 1,
For the geometric-arithmetic eccentricity index
of
, we get
. So
and
Therefore,
For the atom-bond connectivity eccentricity index
of
, we obtain
. Thus
and
Therefore,
For the fourth type of eccentric harmonic index
of
, we get
, therefore
and
Thus,
□
5.2. Case 2:
In this subsection, we discuss the eccentric topological indices for zero divisor graph defined on commutative ring where are prime numbers.
Theorem 4. Let p and q be prime numbers and be the zero divisor graph of commutative ring . Then and .
Proof. For any two prime numbers
p and
we consider the ring
with usual operations. The associated zero divisor graph
on ring
R is defined as: Any
if and only if the elements
or
with
Therefore, a non zero element
is not a zero divisor in the ring
R if and only if
u is not a multiple of
p and
v is not a multiple of
As the number of elements in
which are not a multiple of
p are
and the number of elements in
which are not a multiple of
q are
Also, the element
is not a zero divisor in
Thus the total number of non zero divisors in the ring
R is
This implies
For any
we discuss the degree of each vertex as follows:
If and v is not a multiple of q, then vertex of this form is connected to each vertex where Hence each vertex has degree By symmetry each vertex such that u is not a multiple of p has degree
If and , then vertex of this form is connected to each vertex where and with Hence the each vertex of this type has degree By symmetry the degree of each vertex such that is
If u is not a multiple of p and , then each vertex is connected to each vertex Hence each vertex of this type has degree By symmetry the degree of each vertex if and v is not a multiple of q is
If and , then each vertex of this form is connected to each vertex where and with Hence the degree of each vertex is
From the above discussion, it is easy to see that the zero divisor graph contains vertices of degree , vertices of degree , vertices of degree , vertices of degree , vertices of degree , vertices of degree and vertices of degree . By using the hand shaking lemma the number of edges of are □
Let denotes the set of containing the vertices of degree i. From the proof of Theorem 4, we get and . The following theorem determines the eccentricity of the vertices in graph of .
Theorem 5. Let p and q be two prime numbers and be the zero divisor graph of commutative ring Then the eccentricity of a vertex of is either 2 or 3.
Proof. Similar proof of Theorem 2, we get and . □
Lemma 2. Let p and q be two prime numbers and be the zero divisor graph of commutative ring Then Proof. The graph
contains
vertices and
edges. Let us divide the edges of
into partition sets with respect to the degree of endpoints of each edge as:
This means that the set
contains the edges incident with one vertex of eccentricity
r and the other vertex of eccentricity
s. From the Theorem 4 and Theorem 5, we get
,
,
and
. Then
□
Now we present values of eccentric topological indices based on edges of graph .
Theorem 6. Let p and q be two prime numbers and be the zero divisor graph of commutative ring Then the First Zagreb eccentricity index of is:the third Zagreb eccentricity index is:the geometric-arithmetic eccentricity index is:the atom-bond connectivity eccentricity index is:the fourth type of eccentric harmonic index is: Proof. For the first Zagreb eccentricity index
of
we obtain
. So
,
and
Thus by Lemma 2,
For the third Zagreb eccentricity index
of
we obtain
. So
and
So by Lemma 2,
For the geometric-arithmetic eccentricity index
of
, we get
. So
and
Therefore,
For the atom-bond connectivity eccentricity index
of
, we obtain
. Thus
and
Therefore,
For the fourth type of eccentric harmonic index
of
, we get
, therefore
and
Thus,
□