The Prime Graphs () and () over a Ring
PINKU SARKAR, KUNTALA PATRA
Department of mathematics, Gauhati university, Guwahati, 781014, assam, INDIA
Abstract: - Let G be a simple graph. L( G) is the Laplacian matrix of G. We de fine a simple undirected graph
() whose vertices are all the elements of the ring R and two distinct vertices a, are adjacent if and only if
.0.00. Also, we define a si mple undirected
graph () whose vertices are all the elements of the ring R and two distinct vertices , are adjacent if and
only if .0.00. In this paper we discuss degree of the vertices (), () for
where, is the group of integer modulo . Also, discuss planarity of the graph (), () for
. Here we introduced Laplacian of the graphs (), () for and where, is
prime and we find their girth, algebraic connectivity, clique number and discuss Eulerian property.
Key-Words: - Laplacian matrix, prime graph, planarity, girth.
Received: July 7, 2022. Revised: January 7, 2023. Accepted: February 6, 2023. Published: March 2, 2023.
1 Introduction
In 1973 Fiedler worked o n Algebraic connectivit y
of graphs, [ 6]. In 1994 Merris introduced m any
properties of the Laplacian matrix of graphs, [11]. In
1995 He discussed r elations between the spectrum
of the Laplacian and properties of graphs, [12]. The
first branch of algebraic graph theor y involves the
study of graphs in connection with linear algebra. In
particular, it studies the spectru m of the adjacency
matrix or the Laplacian matrix of a graph. The
Laplacian matrix of a graph G which is denoted by
L(G) is si mply the m atrix DGAG where,
is degree matrix and AG is adjacency matrix
of the graph G whose ,- entry is equal to 1 if
vertices , are adjacent and 0 otherwise. In this
paper we introduce the Laplacian matrix of ,
forand.Also,we
discusssomepropertiesofthegraphs.Kishor F.
Pawar and Sandeep S. Joshi defined a sim ple
undirected graph () whose vertices are all the
elements of the ring R and two distinct vertices ,
are adjacent if and only if .0.
0, [4]. He also
defined the graph () whose vertices are all the
elements of the ring R and two distinct vertices ,
are adjacent if and only if .0.
0 is a zero-div isor (including zero) of t he
ring R, [15]. Satyanarayana defined the prime graph
whose vertices are all ele ments of the ring R and
two distinct vertices a, are adjacent if and only if
00. This grap h is den oted by
[14]. In this paper m odified the adjacency
condition of these graphs we introduce two new
simple undirected graphs one is (), whose
vertices are all the elements of the ring R and two
distinct vertices , are adjacent if and only if
.0.00
and other is ()
whose vertices are all the elements of the ring R and
two distinct vertices , are adjacent if and only if
.0.00.
2 Preliminary Definitions
Definition 2.1: Let be a graph of vertices. The
Laplacian matrix of the graph is denoted by
is defined as where is
the degree matrix of the graph and is the
adjacency matrix of . Then -th entry of the
Laplacian matrix
are given by
=
1
0
Definition 2.2: let R be a ring. A non-zero element
of R is called a zero-div isor if there is a non-zero
element in R such that .0 or .0. The
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set of zero-divisors in a ring R is denot ed by ,
[4].
Definition 2.3: The ele ments which are not zero-
divisors are called units. The set of all units in a ring
R is denoted by U(R), [4].
Definition 2.4: The num ber of edges incident to a
vertex is called the degree of the vertex v, and it is
denoted by .
Definition 2.5: Let G be a graph, The minimum
number of vertices(lines) whose removal makes the
graph G disconnected is called vertex-connectivi ty
(line-connectivity) of the graph G. Vertex
connectivity of G is denoted by .
Definition 2.6: The girth of a graph G is the shortest
cycle in the graph G, which is denoted by . If
the graph G contains no cy cle, then the girth of th e
graph G is equal to infinity.
Definition 2.7: The second smallest eigenvalue of
L(G) is called algebraic connectivity of G. algebraic
connectivity is denoted by . If L(G) has
eigenvalues λλ⋯λ0(where n
|VG|), then λ is called algebraic connectivity,
[5].
Definition 2.8: A connected graph G is called
Eulerian if there exists a walk with no repeated
edges which includes all edges of the graph G.
Definition 2.9: A graph that can be d rawn in the
plane without any edge crossing is called a planar
graph. If any graph contains a non-planar subgraph,
then the graph is non-planar.
Definition 2.10: In a graph G the maximal complete
subgraph is called a clique. The number of vertices
in a clique is called clique number.
Definition 2.11: For a ring R a sim ple undirected
graph G=(V,E) is said to be prime graph of R which
is denoted by () if all elements of are taken
as vertices of the graph and two distinct vertices a
and b are adjacent if either (i) .0.0
or (ii), [4].
Definition 2.12: Let R be a ring. A graph G= (V, E)
is said to be the prim e graph of R (denoted b y
) if vertices of the graph G are all elements of
the ring R and two distinct vertices ar e adjacent if
and only if 00, [14].
Definition 2.13: For a ring R a sim ple undirected
graph G= (V, E) is said to be graph Г if all the
non-zero elements of R as vertices an d two distinct
vertices a and b are adjacent if and only if either
.0.0is a zero- divisor
(including zero), [ 12].
Theorem 2.14: Algebraic connectivity of a graph
is if and only if
.
Theorem 2.15: A connected graph G i s Eulerian if
and only if all the vertices of the graph G are of
even degree.
Theorem 2.16: Every non-zero vertices of the graph
are unit ele ment of and degu
ϕp1 for any odd prime
Theorem 2.17: Every planar graph has a vertex of
degree at most five.
3 Main Results
Definition 3.1: A sim ple undirected graph ()
whose vertices are all the elements of the ring R and
two distinct vertices , are adjacent if and only if
.0.00
.
Definition 3.2: A sim ple undirected graph ()
whose vertices are all the elements of the ring R and
two distinct vertices , are adjacent if and only if
.0.00.
Definition 3.3: Degree, Planarity, Eulerian
property of
Theorem 3.3.1: The graph is planar if and
only if 2,3,4 and 6
Proof: For 2, the graph is a complete
graph , so it is planar. The graph is a
complete graph which is also a planar graph. For
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4, the graph is a com plete graph ,
so it is a planar graph.
Fig. 1: Graph
In the above graph structures, we can see the graph
can be drawn with no edge crossing. So,
the graph is a planar graph.
If is prime, then the graph is a complete
graph . If 3 ( is prime) the graph
always has a subgraph , So the gra ph is not
planar. If 2 where 3, a subgraph
induced by ,,,,0 forms a co mplete
graph where, is an even element and is an
odd element of . So, the graph is not planar.
For where 1, A subgraph induced b y
,, ,,0 forms a com plete
graph . So, the graph for where
1 is not planar.
If
.
.
…
where, ,,⋯
are
distinct prime and ∈ for 1,2,…, then the
subgraph induced by ,,
,
,0
forms a com plete subgraph where,
.
.
…
.
Hence the result.
Theorem 3.3.2: Degree of any unit vertex of the
graph is n. Where; n.
Proof: Let be a unit ele ment of . There
is n number of unit elements. For ev ery unit
element, there exists an element such that is
a unit ele ment. Also, 2 is a unit in
, but the gr aph is sim ple, so is not
adjacent with itself.
So, the number of is n1. ⋯1
.00 ⇒ ~0.
0 is already count in (1)
Also, there is a verte x for which
0 ⇒ ~.
So, the num ber of adjace nt vertices w ith any unit
vertex is n11.
Therefore, the degree of any unit vertex of the graph
is n.
Theorem 3.3.3: Degree of any zero-di visor of the
graph is 1.
Proof: zero vertex is adjacent with every vertex
(because, 0.0). So, the degree of zero vertex is
1.
Non-zero zero-divisors of are multiples of .
Any two n on-zero zero-divisors , are adjacent
because both are multiples of , so . is multiple
of which is 0 in .
Let be a unit element and be any zero-divisor of
. Since, is not m ultiple of , so is
unit. Therefore, any zero-divisor is adjacent with
every unit element.
Hence, any zero-divisor of is adjacent with
every element of .
Degree of any zero-divisor of the graph is
1.
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Theorem 3.3.4: If is odd prime, then the graphs
and are Eulerian.
Proof: Degree of any unit vertex of the graph
is 1 which is even for
any prime . Also, the degree of any zero-divisor of
the graph is 1, which is even for any
odd prime . Therefore, the graph is
Eulerian if is odd prime.
The graph is a co mplete graph . So, the
degree of every vertex is 1, which is even for
any odd prime . Hence, the graph is
Eulerian if is odd prime.
Theorem 3.3.5: If is a non-zero zero divisor and
is any unit element of . then
a) degree of in the graph . is
pq, where is multiple of .
b) degree of in the graph . is
pq, where is multiple of .
c) degree of in the graph . is
pq1.
Where , are distinct prime.
Proof: Any non-zero zero di visors of . are
multiples of and multiples of .
a) Let be a non-zero zero divisor whic h is
multiple of .
For any unit ele ment there exists ∈ . such
that , there a re pq number of
which are adjacent with .
is adjac ent with such zero-divisors which are
multiples of . Because, .≡0 where
is a multiple of . There are 1 numbers of zero-
divisors which are multiples of .
Also, it is adjacent with zero vertex.
Therefore, the degree of is pq1
1pq.
b) Let be a no n-zero zero divisor which is
multiple of .
For any unit ele ment there exists ∈ . such
that , there a re pq number of
which are adjacent with .
is adjac ent with such zero-divisors which are
multiples of . Because, .≡0 where
is a multiple of . There are 1 numbers of zero-
divisors which are multiples of .
Also, it is adjacent with zero vertex.
Therefore, the degree of is pq1
1pq.
c) Let be a unit element of ..
For any unit ele ment there exist s ∈ . such
that , also, is a unit element in ..
So, there are pq1 number of which
are adjacent with . Therefore, the degree of in
the graph . is pq1.
Where , are distinct prime.
Example 3.3.5.1: In the graph , non-
zero zero-divisors are 3, 5, 6, 9, 10, 12. And
unit elements are 1, 2, 4, 7, 8, 11, 13, 14.
Degrees of 3, 6, 9, 12 a re 1538
311.
Degrees of 5, 10, 15 ar e 1558
513.
Degrees of 1, 2, 4, 7, 8, 11, 13, 14 are
151817.
3.4 Laplacian and Algebraic Connectivity of
Theorem 3.4.1: Laplacian of is
1
1
Proof: 0
,1
,2
,⋯1
are vertices of .
Here 0
is adja cent with all other vertices, because,
.0
0
where, is any vertex of the graph.
Therefore, 0
1.
Let is a non-zero element of .
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If we use t he adjacency condition ‘ .
0
or
.0
’
We get ~ 0
⋯ (1)
If we use the adja cency condition ‘
’.
We get ~
where,
is any element of
[
is non-zero for
. In every
non-zero element is a unit so
is a unit for
.] ⋯ (2)
If we use adjacency condition
0
We get ~
⋯ (3)
From (1) (2) and (3) is adjacent with all elements
of except itself (because the graph is simple).
Therefore, the degree of is 1. [ from (1),
(2) and (3)]
Hence, the d egree of ever y vertex of the graph is
1.
So, the graph forms a complete graph .
By definition of Laplacian
1
1
Theorem 3.4.2: If w here p is a
prime, then algebraic connectivity .
Proof: The graph forms a complete graph
. We know that ( ) = if and onl y if
.
Therefore, algebraic connectivity of is
.
3.5 Laplacian of
Theorem 3.5.1: Laplacian of the graph
is
11
121, is zero
divisor
11, is unit element
10
,and
,0
where a,
are non-zero elements of
or
a,0
and 0
,
where a,
are non-zero elements of
or
,0
and b
,
where b
, are non-zero elements of and b
or
0
,aand b
, where
b
, are non-zero elements of and c
or
,0
and ,0
where, is non-zero elements of
or
0
,aand 0
,
where, is non-zero element of
Or
,
and ,d
where, a,
,c, are non-zero elements of and
c and d
or
,
and
,
where b
, are non-zero elements of
0 otherwise
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Where, is an odd prime.
Proof:
All the elements of are consider ed as
vertices of the graph (). Zero divisor s
of are of the form ,0
and 0
,
,Where
,
are ele ments of .Unit elements of
are of the for m ,
where ,
are non-ze ro
elements of .
0
,0
~,
∀,
∈ Since,0
,0
.,
0
,0
∀,
∈.
Clearly, the vertex 0
,0
is adjacent with all other
vertices except itself [since the graph is simple].
∴0
,0
=1. ⋯1
For ,0
∈ , where is n on-zero
element of ;
,0
~0
,
where
is any element of .
[Since, ,0
.0
,
0
,0
]
There are numbers of 0
,
in which are
adjacent with ,0
, where
is any element of .
⋯
,0
~
, where
, are non-zero el ements of
and
.
[Because, ,0
,
, is unit
element of since for2 2 is
non-zero and
is non-zero for
]
There are 1 number of elem ents in
are of the form ,. Where , are non-zero
elements of . And for no n-zero
, ther e are
1 no. of
, in .
So, number of
, in the graph which are adjacent
with ,0
is 11 ⋯
Also, ,0
~,0
[Since ,0
,0
0
,0
] ⋯
∴ From , and
Number of adjacent vertices in () with
the vertex ,0
is
11112
⋯2
For any unit element ,
of where ,
are non-zero elements of ;
,
is not adjacent with ,. Where, is
any element of and
,
[ Since, ,,,
=
0
,
is not unit element of and not
zero element for
]
⋯
,
is not adjac ent with ,
. Where,
is any element of and ,
[ since ,
,
,0
is not unit element of and not zero element
for ]
⋯
From there are 1 number of , in
. Where, is any element of and
,
From there are 1 number of ,
in .Where, is any element of and
,
And also, ,
is non-adjacent with itself (since,
the graph is simple).
In () total number of non-adjacent
vertices with the vertex ,
is 1
1121
Therefore, in () total num ber of
adjacent vertices with the vertex ,
is
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211
So, degree of all units of in
is 1
By definition of Laplacian,
Laplacian of the graph is
11
121, is zero
divisor
11, is unit element
10
,and
,0
where a,
are non-zero elements of
or
a,0
and 0
,
where a,
are non-zero elements of
or
,0
and b
,
where b
, are non-zero elements of and b
or
0
,aand b
, where
b
, are non-zero elements of and c
or
,0
and ,0
where, is non-zero element of
or
0
,aand 0
,
where, is non-zero element of
Or
,
and ,d
where, a,
,c, are non-zero elements of and
c and d
or
,
and
,
where b
, are non-zero elements of
0 otherwise
Where, is an odd prime.
Theorem 3.5.2: The graph is
Eulerian, for any odd prime .
Proof: 0
,0
=1. For any odd prime
, 0
,0
=1 is even. Degree of an y zero-
divisor of is 12, which is even
for any odd prime . Degree of any uni t element is
1, which is also even for any odd prime.
Since is connected and all vertices of
are of e ven degree. Therefore
is Eulerian, where p is any od d
prime.
Theorem 3.5.3: Girth of the graph
is 3, Where p is any prime.
Proof: For an y odd prime , unit ele ments
,
, ,
and zero element 0
,0
make
a cycle of length 3 in . For 2 zero
element and 1
,0
, 0
,1
make a cycle of length 3.
Therefore, girth of is 3, where p is
any prime.
Theorem 3.5.4 is not a planar
graph, where p is any odd prime.
Proof: For any odd prime the graph
contains a subgraph with the vertices 1
,0
,
2
,0
, 0
,1
, 0
,2
, and 0
,0
, which is not a
planar. Therefore, the graph is not a
planar graph for any odd prime .
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Fig. 2:
3.6: Degree, planarity, girth of :
Theorem 3.6.1: Let
.
.
…
where,
,,⋯
are distinct prime and ∈ for
1,2,…,; and be any non-zero vertex of .
The degree of any vertex
of the graph
is gcd,1. Where; 0 on
.
If ≡0 then degree of
gcd,1.
If is even, then
The degree of
in is
1 if
is even
if
is odd
If then degree of the vertex 1
If gcd,1, then degree of is 2.
Proof: Let
.
.
…
where,
,,⋯
are distinct primes and ∈ for
1,2,…,. Also, let (0) be any vertex of
and gcd,
.
.
…
.
If is m ultiple of
.
.
…
then . is multiple of . Therefore .0 in .
So, the vertex is adjacent with .
Number of is the number of m ultiples of
.
.
…
between 0 to .
Number of multiples of
.
.
…
between 0 to is
.
.
…
.
.
.
…
.
.
…
.
.
…
gcd,.
Number of which satisfies the adjacency condition
.0 is gcd,.
Also, is adjacent with (using the adjacency
condition 0).
So, the degree of is gcd,1.
If ≡0n then .0. So, a satisfi es
the adjacency condition .0. But, the vert ex
is not adjacent to itself because the graph is a simple
graph. So, Number of which satisfies the
adjacency condition .0 is gcd,1.
Also, is adjacent with (using the adjacency
condition 0). But, .0 in .
Therefore, the degree of is gcd,1.
If
is even and 0, 2, 4, ⋯,
,⋯,
2. Then the a djacency condition .0.
0 holds. But,
(since, the graph is a sim ple
graph). If we use the adja cency condition
0, then
is a djacent to itself. But the graph is
simple graph. So, the number of adjacent vertices
with
is
1.
If
is odd an d 0, 2, 4, ⋯,
1,
1 ,⋯,
2. Then t he adjacency conditi on .
0.0 holds. If we use the adjacency
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condition 0, then
is adj acent to itsel f.
But the graph is a sim ple graph. So, the number of
adjacent vertices with
is
.
Let be any vertex of the graph such that
and gcd,
.
.
…
Since, , if is m ultiple of then is
adjacent with because, here .0 on .
Number of Number of multiples of in
gcd,. Also, is adjacent with
because,
0on.But,
1whichismultipleofa.
sincethegraphisasimplegraph.So,thedegree
of is gcd,.
If gcd,1, then is unit element of ,
which is adjacent with and zero ele ment.
Therefore, the degree of is 2.
Theorem 3.6.2: The girth of the graph is,
∞ if 2,3,4,5 or
is a prime
3 otherwise
Proof: The graph is a com plete graph ,
So girth is infinite. , are union of
two copies of and uni on of three cop ies of
with common vertex zero respectively . So, the girth
of , is infinite. is a union
of four copi es of with co mmon vertex zero.
Therefore, the girth of is infinite.
If 5 and is not prim e then in the graph
always exists a cy cle of length three with the zero
vertex and two non-zero zero divisors of which
are adjacent. Therefore, the girth is 3.
Theorem 3.6.3: The graph is Eulerian if
and only if is odd.
Proof: If is odd then the degree of zero vertex of
the graph 1 which is an even, and the degree
of any non-zero vertex is either gcd,1 or
gcd,1. Since, is odd, gcd, is odd. So,
gcd,1 and gcd,1 are even.
Therefore, the graph is Eulerian.
If n is even t he degree of zero vertex of the graph
1 which is odd. So, the graph is not Eulerian if
n is even.
Hence, the graph is Eulerian if and onl y if
is odd.
Theorem 3.6.4: The graph is not planar if
,3. Where, , are two distinct primes.
Proof: Zero-divisors of are multiples of
and multiples of . Set of non-zero zero-divisors
form a co mplete bipartite graph ,, whose
one set of vertices is a set of multiples of and the
other is a set of m ultiples of . In there
are 1 number of m ultiples of and 1
number of multiples of . So, if ,3 then the
graph has a subgraph , which is not planar.
Therefore, the graph is not a planar graph,
if ,3.
Theorem 3.6.5: The graph is not planar if
5.
Proof: In th e graph any zero-divisor is
multiple of . If , two distinct zero-divisors then
. has a factor . So, .0 in . Therefore,
any two zero-divisors are adjacent in the graph. For
5, the number of zero-divisors is 5 and zero-
divisors form a co mplete graph. So, if 5 the
graph has a complete subgraph . Hence the result.
3.7. Laplacian of
Theorem 3.7.1: Laplacian of is
11
21
10
and ,
or
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and 0
,
or
and
or
and
0
Otherwise
where is any non-zero vertex of and is
any odd prime.
Proof: 0
,1
,2
,⋯1
are vertices of .
Here the vertex 0
is adjacent with all other vertices.
Therefore, 0
1.
If is non-zero element of , then is adjacent
with 0
and
[since
0
, using
adjacency condition of the graph]
Therefore, the degree of any non-zero vertex is 2
By definition of Laplacian
11
21
10
and
Or
and 0
or
and
Or
and
0
Otherwise
Where, is any non-zero vertex of and
is any odd prime.
Theorem 3.7.2: If
where p is a
prime 3then 1.
Proof: 0
,1
,2
,⋯1
are vertices of .
The graph and are complete
graphs and respectively. For 3 the graph
is union of
copies of in which zero
vertex is co mmon. For deletion of zero vertex the
graph will be disconnected. So, vertex connectivit y
1. We kn ow that if
then
. Therefore, 1.
Example:
Fig.3:
3.8. Laplacian of
Theorem 3.8.1: Laplacian of is
11
11, is zero divisor
21, is unit element
10
,and
,0
where is non-zero and
is any element of
Or
a,0
and 0
,
where is non-zero and
is any element of
or
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,0
and ,0
where is non-zero element of
or
0
,aand 0
,
where is non-zero element of
or
,
and
,
0 otherwise
Proof: All the elements of are considered
as vertices of the graph . Zero divisors
of are of the form ,0
and 0
,
, Where
,
are elements of .
Units of are of the form ,
where ,
are non-zero elements of .
0
,0
.,
0
,0
∀,
∈ [from 1st
adjacency condition of ()]
Clearly, zero ele ment 0
,0
is adjacent wi th all
other vertices
∴0
,0
=1
Now for ,0
∈ , where is any non-
zero element of .
,0
~0
,
where
is any element of
[ using adjacency condition of
()
,0
.0
,
0
,0
]
There are number of
in .
,0
~,0
where is any non-zero
element of .
[using 2nd adjacency condition of
()
,0
,0
0
,0
]
∴ Number of adjacent ve rtices with t he vertex
,0
is 1
Degree of ,0
in is 1
Similarly, Degree of 0
, in is
1
Therefore, Degree of all non-zero zero divisors of
in is 2
Now for any unit element ,
of where,
,
are non-zero elements of .
,
~0
,0
,
~,
[Since, ,
,
0
,0
]
∴ The num ber of adjacent vertices with the vertex
,
is 112
The degree of ,
in is 2
Degree of all units of in is
2
By definition of Laplacian
11
11, is non-zero zero
divisor
21, is unit element
10
,and
,0
where is non-zero and
is any element of
Or
a,0
and 0
,
where is non-zero and
is any element of
or
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DOI: 10.37394/23206.2023.22.22
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Volume 22, 2023
,0
and ,0
where is non-zero element of
or
0
,aand 0
,
where is non-zero element of
or
,
and
,
where ,
are non-zero elements of
0 otherwise
Theorem 3.8.2: is Eulerian, where p
is any odd prime.
Proof: 0
,0
=1 , for any odd prim e
1 is even. Degree of any zero-divisor of
is 1, which is even for any odd prime.
Degree of any unit element is 2, wh ich is even.
Since is connected and all vertices of
are of e ven degree. Therefore
is Eulerian, where p is any od d
prime.
Theorem 3.8.3: Girth of is 3,
Where p is any prime.
Proof: For any odd prime, zero element and unit
elements ,
and ,
make a cycle of
length 3 in . For 2, zero element
and 1
,0
, 0
,1
makes a cy cle of length 3.
Therefore, girth of is 3, where p is
any prime.
Theorem 3.8.4: is not a planar
graph, where p is any odd prime.
Proof: The graph is union of
and with a common vertex 0
,0
.So the graph is
a planar for 2. If is an y odd prime then the
graph contains a sub graph with
the vertices ,0
, 0
,,
,0
, 0
,
,
and 0
,0
, which is not a planar. Th erefore
is not a planar graph f or any odd
prime .
Theorem 3.8.5: Clique number of is
5, Where p is any odd prime.
Proof: For any zero divisor ,0
there exists a
complete subgraph with vertices ,0
, 0
,
,
,0
, 0
,
, and 0
,0
. Similarly, for
any zero divisor of the form 0
, there exists a
complete subgraph with vertices ,0
, 0
,
,
,0
, 0
,
, and 0
,0
. For any unit
element ,
there exists a complete graph
with vertices ,
, 0
,0
,
,
.
Therefore, is the m aximal complete subgraph in
. Hence, Clique num ber of
is 5, Where p is any odd prime.
Example:
Fig. 4:
4 Conclusion
The graph is planar only for 2,3,4
and 6. Any zero-divisors of the graph is connected
with every vertex of the graph except itself. The
degree of the unit element of the graph is
. The gra ph , and
are Eulerian for any odd prime . On
the other hand; the graph is Eulerian if and
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DOI: 10.37394/23206.2023.22.22
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Volume 22, 2023
only if is odd. The grap h is also
Eulerian for any odd prim e . Algebraic
connectivity of the graph is but t he
algebraic connectivity of the is less than or
equal to 1. The graphs ,
are not planar for any odd prime . In the graph
there does not exist any c ycle for
2,3,4 and 5 or is prime.
5 Future Scope
We can work on the graphs (), () for any
non-commutative finite ring . Chromatic number
and dominating number of the graphs c an be found
out. Also, we can study Laplacian en ergy of the
graphs (), ().
Acknowledgements:
We gratefully thank the refere es for their
suggestions and valuable comments. This work is
financially supported by the Council of Scientific
and Industrial R esearch (File no:
09/059(0068)/2019-EMR-I.
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Pinku Sarkar, Kuntala Patra
E-ISSN: 2224-2880
183
Volume 22, 2023
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This work is financially supported by the Council of
Scientific and Industrial Research (File no: 09/059(0068)
/2019-EMR-I.
