2. Definitions and Notations
Let G be a connected graph with vertex and edge sets and , respectively. A numerical quantity related to a graph that is invariant under graph automorphisms is topological index or topological descriptor. For a graph the degree of a vertex v is the number of edges incident with v and denoted by . The maximum degree of a graph G, denoted by , and the minimum degree of a graph, denoted by , are the maximum and minimum degree of its vertices. The sum of degree of all vertices u which are adjacent to vertex v is denoted by . The distance between the vertices u and v of G is denoted by and it is defined as the number of edges in a minimal path connecting them.
Connectivity descriptors are important among topological descriptors and used in various fields like chemistry, physics, and statistics. Let
, the distance
between
and
, be defined as the length of any shortest path in
G connecting
and
. In mathematical, eccentricity is defined as:
In 1997 eccentric connectivity index was introduced by Sharma [
10]. By using eccentric connectivity index, the mathematical modeling of biological activities of diverse nature is done. The general formula of eccentric connectivity index is defined as:
where
is the eccentricity of vertex
v in
. Some applications and mathematical properties of eccentric connectivity index can be found in [
11,
12,
13,
14]. The total eccentricity index is the sum of eccentricity of the all the vertex
v in
G. Total eccentricity index is introduced by Farooq and Malik [
15], which is defined as:
The first Zagreb index of a graph
G is studied in [
16] based on degree and the new version of the first Zagreb index based on eccentricity was introduced by Ghorbani and Hosseinzadeh [
17], as follows:
The eccentric connectivity polynomial is the polynomial version of the eccentric-connectivity index which was proposed by Alaeiyan, Mojarad, and Asadpour [
18] and some graph operations can be found in [
19]. The eccentric connectivity polynomial of a graph
G is defined as:
Gupta, Singh and Madan [
20] defined the augmented eccentric connectivity index of a graph
G as follows:
where
denotes the product of degrees of all vertices
u which are adjacent to vertex
v. Some interesting results on augmented eccentric connectivity index are discussed in [
21,
22]. Another very relevant and special eccentricity based topological index is connective eccentric index. The connective eccentric index was defined by Gupta, Singh, and Madan [
20] defined as follows:
Ediz [
23,
24] introduced the Ediz eccentric connectivity index and reverse eccentric connectivity index of graph
G, which is used in various branches of sciences, molecular science, and chemistry etc. The Ediz eccentric connectivity index and reverse eccentric connectivity index are defined as:
where
is the sum of degrees of all vertices,
u, adjacent to vertex
v,
is the eccentricity of
v.
Let
R be a commutative ring with identity and
is the set of all zero divisors of
R.
is said to be a zero divisor graph if
and
if and only if
Beck [
25] introduced the notion of zero divisor graph. Anderson and Livingston [
26] proved that
is always connected if
R is commutative. Anderson and Badawi [
27] introduced the total graph of
R as: There is an edge between any two distinct vertices
if and only if
. For a graph
G, the concept of graph parameters have always a high interest. Numerous authors briefly studied the zero-divisor and total graphs from commutative rings [
28,
29,
30,
31,
32]. Similar problems were investigated in [
33,
34].
Let , , and q be prime numbers, with and being zero divisor graph of the commutative rings . In this paper, we investigate the eccentric topological descriptors namely, eccentric connectivity index, total eccentric index, first Zagreb eccentricity index, connective eccentric index, Ediz eccentric index, eccentric connectivity polynomial, and augmented eccentric connectivity index of zero divisor graphs . Now onward, we use G as a zero divisor graph of the commutative rings .
3. Methods
In this paper, we adopted interdisciplinary methods by combining algorithmic approach for graph construction and outcome of algorithm are aligned with eccentric topological indices. For prime numbers with we consider the commutative ring with usual addition and multiplication. The zero divisor graph associated with ring R is defined as: For , , if and only if for and . Let , then . The elements of the set J are the non zero divisors of R. Also is a non zero divisor. Therefore, are the total number of non zero divisors of R and the total number of elements of R are Hence, are the total number of zero divisors. This implies that We can construct the zero divisor graph of commutative ring by the following algorithm:
Input: and q are three prime numbers.
Output: ordered pairs for zero divisor.
Algorithm 1 ZeroDivisorGraph () |
1: if 2: 3: for to 4: 5: for to q 6: 7: if OR 8: 9: createGraph () |
Algorithm 2 createGraph () |
1: for to 2: 3: for to q 4: 5: if AND 6: 7: if OR 8: 9: 10: 11: else 12: 13: 14: 15: if OR 16: 17: 18: 19: else 20: 21: 22: 23: if mod AND mod AND 24: 25: return |
Outcomes of above algorithm, the degree of each vertex can be depicted mathematically in the following cases:
Case 1: If and any then each such type of vertex is adjacent to the vertices for every . Hence the degree of each vertex is
Case 2: If and then each such type of vertex is adjacent to the vertices for every , and . Hence the degree of each vertex is Similarly, if and then the degree of each such type of vertices is
Case 3: If and then each such type of vertex is adjacent with only for every . Hence the degree of each vertex is
Case 4: If and then each such type of vertex is adjacent with only for every . Therefore, the degree of each vertex is Similarly, if and then degree of each such type of vertices is
From the above discussion and our convenance, let us partitioned the vertex set of
G based one their degrees as follows:
This shows that . It is easy to see that , , , , , and
4. Main Results
Let denote the degree of a vertex u in U and denotes the distance between the vertices of two sets U and V. In the following theorem, we determined the eccentricity of the vertices of G.
Theorem 1. Let G be the zero divisor graph of the commutative ring R, then the eccentricity of the vertices of G is 2 or 3.
Proof. From case 1, the vertices of the set are at distance 1 with the vertices of the sets , , & i.e., From Case 4, the vertices of the sets and are adjacent with the vertices of the sets and , respectively. This implies that . The distance between any two different vertices of the set is also 2. Therefore the eccentricity of the vertices of set is 2, i.e., . Similarly, it is easy to see that
As and . This implies that . This shows that . Similarly, it is easy to calculate that This completes the proof. □
Summarizing the above cases, partition of vertices and their cardinality and Theorem 1 in
Table 1.
In the following theorem, we determined the eccentric connectivity index of the graph G.
Theorem 2. Let , q be prime numbers, then eccentric connectivity index of graph G is
Proof. By using the degree of each vertex partition and corresponding their eccentricity from
Table 1 in the Equation (
2), we obtain:
After simplification, we get:
This completes the proof. □
The eccentricity of the vertices and their frequency is given in the
Table 1 of the graph
G, by putting these values and after simplification we obtain the following two corollaries.
Corollary 1. Let , q be prime numbers, then the total-eccentricity index of G is given by.
Corollary 2. Let , q be prime numbers, then the first Zagreb eccentricity index of G is given by
Theorem 3. Let , q be prime numbers, then the connective eccentric index of graph G is .
Proof. By using the values of degrees and their eccentricity in the Equation (
6), we obtain the following:
After simplification, we get
.
This completes the proof. □
Theorem 4. Let , q be prime numbers, then the Ediz eccentric connectivity index of graph G is .
Proof. is the sum of degrees of all vertices
u which are adjacent to vertex
v. Calculate the values of
with the help of
Table 1. The eccentricity of each vertex is also given in
Table 1. Putting these vales in Equation (
7), we obtain the following:
After simplification, we get
.
This completes the proof. □
Theorem 5. Let , q be prime numbers, then the eccentric connectivity polynomial of graph G is .
Proof. By using the degree of each vertex partition and their corresponding eccentricities from
Table 1 Equation (
4), we obtain:
After simplification, we get
.
This completes the proof. □
Theorem 6. Let , q be prime numbers, then augmented eccentric connectivity index of graph G is .
Proof. is the product of degrees of all vertices
u which are adjacent to vertex
v. Calculate the values of
with the help of
Table 1. The eccentricity of each vertex is also given in the
Table 1. Putting these vales in Equation (
5), we obtain the following:
After simplification, we get
This completes the proof. □
If
and
q are prime numbers with
, then Ahmad et al. [
35] determined the vertex-based eccentric topological indices of zero divisor graph of the commutative ring
as follows:
Theorem 7 ([
35])
. Let be prime numbers. If is the zero divisor graph of the commutative ring , then.