1. Introduction and
Preliminaries
The concept of graph of a commutative ring , was introduced by Beck [1] in 1988. Beck considered all
the elements of the ring as the
vertices of the graph and two distinct vertices and are adjacent in the graph if and only
if , the additive
identity of the ring . This
definition of graph given by Beck was modified by Anderson and
Livingston [2] in 1999.
Anderson and Livingston considered only the non-zero zero divisors of
commutative ring to be the vertex
set of the graph known as zero divisor graph of denoted by in which two distinct vertices
and are adjacent in if and only if . In this paper we consider
the zero divisor graph of the reduced ring, , of direct product of finite fields. Let be the set of non-zero
zero-divisors of the ring and denote the graph with vertex
set as and edge set as .
The metric dimension of a graph is the smallest number of vertices
such that all vertices are uniquely determined by their distances to the
chosen vertices. For an ordered subset of vertices in a connected graph and a vertex of , the metric representation of with respect to is the The set
is a resolving set for if implies that for all pairs of vertices of . In other words, a set is called a resolving set,
if for each two distinct vertices there exists
such that , where
is the distance between the
vertices and . The minimum cardinality of a resolving
set for is called the metric
dimension of , and denoted by
.
The problem of Metric dimension was introduced by Slater [3] and also independently
introduced by Harary and Melter [4]. In [5-7]
graphs of order with metric
dimension and have been characterized. In [8], metric dimensions for many
particular classes of graphs have been determined.
The corona product of graphs
and is the graph obtained by taking one copy of
, called the center graph, copies of , called the outer graph, and making the
vertex of adjacent to every vertex of the copy of , where .
The Join graph of two
graphs and is the graph with vertex set and edge set
In this paper, we determine the Metric dimension of Corona product
and Join graph of zero divisor graphs of direct product of finite
fields.
2. Results
on Metric Dimension of Zero Divisor Graph of Direct Product of Finite
Fields
In [9, Proposition 6.2],
Raja, Pirzada & Redmond, proved that for any . They also
proved that [9, Theorem 6.3], .
In [10], it is
proved that if , and order
of each field is two, then .
Theorem 1. [10, Theorem 2.4] If with , then the metric dimension .
Consider the reduced rings and where are finite fields with .
In [11], the
following result is proved.
Theorem 2. [11] Let and where are finite fields with .
Then,
the diameter of Corona product of and is if ,
the diameter of Corona product of and is if or and or ,
the diameter of Corona product of and is if ,
the girth of Corona product of and is if ,
the girth of Corona product of and is if or .
3. Metric
Dimension of Corona Product of Zero Divisor Graphs of Direct Product of
Finite Fields
In this section, we determine the Metric dimension of Corona product
of zero divisor graphs of direct product of finite fields.
Theorem 3. If and where are finite fields with , then the Metric
dimension of Corona product of and is .
Proof. According to [9, Proposition 6.2], the metric dimension and . Let We first prove that is resolving set for the center graph
.
Let . Let
. Suppose contains number of and contains number of .
Case 1. .
Let . Then there
exists a position such that contains in the position and contains in the position. Then both and are adjacent to a vertex with a in the position and everywhere else. Also is not adjacent to . Suppose contains in the position . Then is adjacent to a vertex with a in the position and everywhere else. Also is adjacent to and is not adjacent to and . Therefore, . This is a contradiction.
Case 2. but and differ in atleast one position of . Suppose contains in the position and contains in the position. Then from case (i),
. This is a
contradiction. Thus, is a resolving set for . Let with . As proved above, similarly we can
prove that is resolving set for
the outer graph of copy of
. Let . Then, is
resolving set for the Corona product of and for , the metric
representation of with respect to
is the vector of length . Since, , therefore, the
Metric dimension of Corona product of and is . 
Theorem 4. If and where are finite fields with , then
the Metric dimension of Corona product of and is
Proof. Consider the reduced rings and where are finite fields with . Let
Then . Let .
According to [10, Theorem 2.4], is minimum resolving set for and . Now
consider the Corona product of and . There are copies of in the Corona product graph
and the
vertex of is adjacent to each and every
vertex of copy of . Let and . Then . Let .
According to [10, Theorem 2.4], is minimum resolving set for and . Let and Then and and
. We claim that
is minimum resolving set for the corona product graph .
For , the metric
representation of with respect to
is the vector of length .
Clearly, is distinct for
each , as the vector contains exactly one entry with the position of
in distinct co-ordinate
for each metric representation of . It is enough to show that for each , the metric representation is distinct.
For , if
, then clearly , since is in the copy of and is in the copy of in corona product . Now suppose
there exists distinct such that . In other words, .
Since the entries in and
are only and and , this implies and will differ in at least one position
containing . Suppose
contains in the co-ordinate position and contains in the co-ordinate position. Then is adjacent to a vertex , with a non-zero entry other
than in the position and in the remaining positions
(as each field contains at least three elements). This implies as . But since and both contain non-zero entry in the
position, this implies . This is a
contradiction to . Therefore, the
metric representation is
distinct for each . Thus,
is a resolving set for the corona
product graph .
Now consider . Suppose contains
in positions and non-zero entries
in remaining positions with at least one non-zero entry other than , and let with
in positions and in remaining positions. This
implies and contains in same co-ordinate positions
. This implies
the vertices adjacent(non-adjacent) to are also adjacent(non-adjacent) to
, implies implies
implies .
Thus, is not
a resolving set for each .
This implies is a minimal
resolving set. Thus, the metric dimension, .
Now there are exactly
vertices in , such that any two vertices with zero entries will differ in at least
one position containing . Since if and if , any two vertices in will either differ in the number of
entries or if the two
vertices contain same number of , then the two vertices will
differ in at least one position containing . Thus, if with when or with where ,
then there exists at least two vertices, say, such that both contains same number of and position of is also same in both and . This implies positions of non-zero
entries is also same, but since
and are distinct, they will
differ in at least one position containing non-zero entry. Since both contains in same co-ordinate positions, by
a similar argument above, we get, .
Let be the set obtained
from , by replacing the set with . Then . Therefore, if with
then there exists at least two vertices, say, such that,
implies . This implies
cannot be a resolving set with
implies implies
implies .
Hence, the metric dimension, 
4. Metric
Dimension of Join Graph of Zero Divisor Graphs of Direct Product of
Finite Fields
Theorem 5. If and where are finite fields with , then the Metric
dimension of Join graph of and is .
Proof. According to [9, Proposition 6.2], the metric dimension and . Let In the proof of Theorem 3, we
proved that, is resolving set
for . Let Similarly, is resolving set for .
Let . Then,
clearly is resolving set for the
Join graph of and and for , the metric representation
of with respect to is the vector of length . Hence, the Metric dimension of
Join graph of and is
. 
Theorem 6. If and where are finite fields with , then
the Metric dimension of Join graph of and is .
Proof. Let Then . Let .
According to [10, Theorem 2.4], is minimum resolving set for and . Let Then . Let . Then by
[10, Theorem 2.4], is minimum resolving set for and .
Consider the Join graph of and and let . For each is
distinct, and for all
in the Join graph of and . Similarly, for each is
distinct, and for all
in the Join graph of and . Thus, is resolving set for the
Join graph of and and for , the metric representation
of with respect to is the vector of length .
Now consider . Either or
. Without loss of
generality suppose .
Suppose contains in positions and non-zero entries
in remaining positions with at least one non-zero entry other than , and let with in positions and in remaining positions. This
implies and contains in same co-ordinate positions
. This implies
the vertices adjacent(non-adjacent) to are also adjacent(non-adjacent) to
, implies implies
implies .
Similarly, if then . Thus, is not a resolving
set for each . This implies
is a minimal resolving set. Thus,
the metric dimension is .
Now there are exactly
vertices in , such that
any two vertices with zero
entries will differ in at least one position containing . Since if and if , any two vertices in will either differ in the number of
entries or if the two
vertices contain same number of , then the two vertices will
differ in at least one position containing . Thus, if with where or with where , then there exists at least two
vertices, say,
such that both contains same
number of and position of
is also same in both and . This implies positions of non-zero
entries is also same, but since
and are distinct, they will
differ in at least one position containing non-zero entry. Since both contains in same co-ordinate positions, by
a similar argument above, we get, .
Let be the set obtained
from , by replacing
the set with . Then . Therefore, if with , then there exists at
least two vertices, say, , such that,
,
implies . This implies
cannot be a resolving set with
implies implies implies .
Hence, the metric dimension is . 
5. Conclusion
We have determined the Metric dimension of Corona product and Join
graph of zero divisor graphs of direct product of finite fields.
Author
Contributions
Both authors Dr. Nithya Sai Narayana and Subhash Mallinath Gaded
wrote the main manuscript text and reviewed the manuscript. Both authors
have contributed equally to the work. Both authors read and approved the
final manuscript.
Declaration of Competing
Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to
influence the work reported in this paper.