INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS VOL.
NO.
MARCH 2024
Connectivity of The Triple Idempotent Graph of Ring Zn Vika Yugi Kurniawan.
Bayu Purboutomo.
Sutrima.
Nughthoh Arfawi Kurdhi AbstractAiLet R be a commutative ring and I(R) denotes a set of all idempotent elements of R.
The triple idempotent graph of ring R, denoted by T I(R), is the undirected simple graph with vertex-set in ROe.
, .
Two distinct vertices u and v in T I(Zn ) are adjacent if and only if there exists w OO R Oe .
, .
where w = u and w = v such as uv OO / I(R), uw OO / I(R), vw OO / I(R) and uvw OO I(R).
this research, we study the connectivity of the triple idempotent graph of ring integer modulo n, denoted by T I(Zn ).
The result is that the triple idempotent graph of ring Zn is a connected graph if n prime and n Ou 7.
Index TermsAithe triple idempotent graph, ring of integers modulo n, connected graph
I NTRODUCTION
ET R be a commutative ring with unit element 1 = 0.
Beck .
introduced the concept of a zero-divisor graph that connect between ring theory and graph theory.
Anderson and Livingston modified zero-divisor graph, denotes by e(R), with vertices Z(R)O = Z(R) Oe .
and two distinct vertices x, y OO Z(R)O adjacent if and only if xy = 0.
There was shown that e(R) is a connected graph with gr.
(R)) OO .
, 4.
O}.
In other paper.
Akhtar and Lee .
, studied the connectivity of the zero divisor graph e(R) associated to a finite commutative ring R.
They investigated the conditions of ring R such that graph e(R) is a connected Later, many papers that investigated various kind of graphs associated with the ring, see .
, .
, .
, and .
Recently in .
Mohammad and Shuker introduced graph that is called idempotent divisor graph, denoted by JI(R), with the set of vertices RO = R Oe .
and two distinct vertices v1 and v2 adjacent if and only if v1 .
v2 = e, for some non-unit idempotent element e OO R.
e e2 = e = .
Let I(R) be a set of idempotent elements of ring R.
In this paper, the definition of the triple idempotent graph of a commutative ring R, denoted by T I(R), is the undirected simple graph with vertex-set R Oe .
, .
Two distinct vertices u and v are in T I(R) adjacent if and only if there exist w OO R Oe .
, .
where w = u and w = v such as uv OO / I(R), uw OO / I(R), vw OO / I(R) and uvw OO I(R).
We will investigate the properties that related to connectivity of the triple idempotent graph of ring integer modulo n, denoted by T I(Zn ).
Kurniawan.
Purboutomo.
Sutrima, and N.
Kurdhi are with the Universitas Sebelas Maret.
Surakarta, 57126 Indonesia e-mail:
vikayugi@staff.
Manuscript received October 30, 2023.
accepted January 30, 2024.
II.
P RELIMINARIES
According to Chartrand and Zhang .
, a graph G is a finite nonempty set V of objects is called vertices together with a possibly empty set E of 2-element subsets of V is called edges.
The number of vertices in a graph G is the order of G and the number of edges is the size of G.
A graph of size 0 is called an empty graph.
Two distinct vertices u and v said to be adjacent if there is an edge between u and v.
The degree of a vertex u in a graph G is the number of vertices in G that are adjacent to u.
If a path between two vertices of graph G can be found, then the graph G is connected.
If R is a ring.
Z(R) denotes the set of zero-divisors of R and I(R) denotes the set of idempotent elements of R.
Definition 1.
Graph triple idempotent of commutative ring R, denoted by T I(R), is the undirected simple graph with vertexset R Oe .
, .
, and two different vertices u and v are in T I(R) adjacent if and only if there exists w OO R Oe .
, .
where w = u and w = v such that u.
v OO / I(R), u.
w OO / I(R), v.
w OO / I(R) and w OO I(R), where I(R) is a set of all idempotent elements of R.
In the following, given an example of T I(Z9 ).
Example 2.
Let Z9 , with Z9 = .
E, 1E, 2E, 3E, 4E, 5E, 6E, 7E, 8E} and I(Z9 ) = .
E, 1E}.
By the Definition 1, the set of vertex V (T I(Z9 )) = .
E, 3E, 4E, 5E, 6E, 7E, 8E} and the set of edge E(T I(Z9 )) = .
2E,4E , e2E,8E , e4E,8E , e5E,7E , e5E,8E , e7E,8E }.
Graph T I(Z9 ) illustrated in the Figure 1.
Fig.
1: Graph T I(Z9 ) i.
R ESULT
In this section, the results of investigations regarding the conditions for connectivity of T I(Zn ) are given.
The following is a theorem regarding the condition for the T I(Zn ) to be an empty graph.
Theorem 3.
Let Zn .
If n O 6, then T I(Zn ) is an empty graph.
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS VOL.
NO.
MARCH 2024
Proof.
The proof given by 3 cases below.
For n = 3, 4.
Since |V (T I(Zn ))| < 3, then there are not found any adjacency such that for T I(Zn ) has no edge or T I(Zn ) is an empty graph.
For n = 5.
There are found V (T I(Z5 )) = .
E, 3E, 4E} and I(Z5 ) = .
E, 1E}.
Since |V (T I(Z5 ))| = 3 and Z5 is a commutative ring such that there is only one possible combination of vertices u,v,w i.
u = 2E, v = 3E, w = 4E.
As a result of uv = 1E, 1E OO I(Z5 ), then u, v, w are not adjacent.
Therefore T I(Z5 ) has no edge or T I(Z5 ) is an empty graph.
For n = 6.
There are found V (T I(Z6 )) = .
E, 3E, 4E, 5E} and I(Z6 ) = .
E, 1E, 3E, 4E}.
Since |V (T I(Z6 ))| = 4, same as before, then there are four possible combinations of vertices u,v,w.
First, for vertex u = 2E, v = 3E, w = 4E.
As result of uv = 0E 0E OO I(Z6 ), then u, v, w are not adjacent.
In the same way, for the others combination as well.
Therefore.
T I(Z6 ) has no edge or T I(Z6 ) is an empty graph.
So, it is proven that for Zn where n O 6, the T I(Zn ) is an empty graph.
n The illustrations of graph T I(Zn ) where n O 6 are shown in Figure 2.
In the following, a lemma is given regarding cases of vertices that are not adjacent to each other in T I(Zn ).
Lemma 5.
Let Zn where n is prime and n Ou 7.
For every u, v OO V (T I(Zn )), u not adjacent to v if uv = 1 or uv = x = 1 where x = uOe1 or x = vOe1 .
Proof.
Let u, v OO V (T I(Zn )) be distinct and arbitrary vertices.
Then, clearly that Zn where n prime are field, such that I(Zn ) = .
, .
where element 0 and 1 in Zn are related to 0E and 1E.
Since, the field has no zero divisor element, then adjacency condition can be reduced to uv = 1, vw = 1, uw = 1 and uvw = There will be showed 2 cases where u is not adjacent to v.
Case 1 uv = 1.
By Definition 1, if uv = 1, 1 OO I(Zn ), then it will be contradiction with one of adjacency conditions of T I(Zn ).
Therefore, u is not adjacent to v.
Case 2 uv = x = 1, where x = uOe1 or x = vOe1 .
For uv = uOe1 , if both side multiply with u, then u.
u = 1.
Seen that needed two elements of u in the left side so that triple vertices multiplication that involved u and v is equal By Definition 1, it will be contradiction with one of adjacency conditions where u = v = w respectively.
Therefore, u is not adjacent to v.
In the same ways for uv = vOe1 .
n As an illustration of the lemma 5, the following is an example of an explanation of the cases that two vertices is not adjacent to each other in Z11 .
T I(Z3 ) .
T I(Z4 ) Example 6.
Let Z11 .
The set of vertex element and the set of idempotent element.
V (T I(Z11 )) = .
E, 3E, 4E, 5E, 6E, 7E, 8E, 9E, 1E.
and I(T I(Z11 )) = .
E, 1E}.
We provide an example explanation by showing that any two vertices that are not adjacent in Z11 , can be included in one of the two cases in the Lemma 5 above.
Given below adjacency matrix of Z11 in the Table I.
TABLE I: Adjacency Matrix of T I(Z11 ) .
T I(Z5 ) .
T I(Z6 ) Fig.
2: Graph T I(Zn ) where n O 6 There is a lemma that related to divisibility properties for any non-zero element in a field F.
Lemma 4.
Let F be a field.
For every a, b OO F where a = 0 and b = 0 then .
b and .
Proof.
Let non zero element a, b OO F.
We will show that .
b and .
Because of F is a field, clearly there exist bOe1 such that a = a.
e = a.
bOe1 .
Using commutative and closed properties, then a.
bOe1 = a.
bOe1 .
b = c.
b with c = a.
bOe1 , c OO F.
Therefore, surely .
As the same way, for .
So, for every non zero element a, b OO F then .
b and .
n Seen that 2E is not adjacent to 3E.
This is because 2E.
3E = 6E = 2EOe1 such that include in case 2.
Then, vertex 2E is also not adjacent to 6E because 2E.
6E = 1E such that include in case 1.
Now, vertex 3E is not adjacent to 4E because 3E.
4E = 1E such that include in case Also, vertex 3E is not adjacent to 5E because 3E.
5E = 4E = 3EOe1 such that include in case 2.
In the same ways, for the others vertices that not adjacent each other in T I(Z11 ) and always can be included to one of the two cases in Lemma 5.
The T I(Z11 ) is showed in Figure 3.
The following result show that for Zn where n prime and n Ou 7.
T I(Zn ) is a connected graph.
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS VOL.
NO.
MARCH 2024
Fig.
3: Graph T I(Z11 ) Theorem 7.
Let Zn .
If n prime and n Ou 7, then T I(Zn ) is a connected graph.
Proof.
Let u, v OO V (T I(Zn )) be arbitrary.
Then, by Lemma 5 adjacency condition can be reduced to uv = 1, vw = 1, uw = 1 and uvw = 1.
If u and v are adjacent, then there is a direct path u Oe v.
If u and v are not adjacent, then by Lemma 5 there are divided to 2 cases.
Case 1.
If uv = 1.
Let w OO V (T I(Zn )).
There exist wOe1 OO V (T I(Zn )) where w.
wOe1 = 1.
Since u OO V (T I(Zn )), then by Lemma 4, wOe1 = u.
x such that w.
x = w.
wOe1 = 1.
Therefore, u adjacent to w.
In the same way for v, such that v adjacent to w.
Since u and v are adjacent to w, then there is a path uAewAev.
Case 2.
If uv = x = 1, where x = uOe1 or x = vOe1 .
Let uv = uOe1 .
Since uOe1 OO V (T I(Zn )), then by Lemma 4 there exist w OO V (T I(Zn )) such that uOe1 = w.
So, u.
y = u.
uOe1 = 1 such that u adjacent to w.
Then, also for vOe1 OO V (T I(Zn )) by Lemma 4, vOe1 can be divided by w such that vOe1 = w.
Therefore, v.
z = v.
vOe1 = 1, then v adjacent to w.
Since u adjacent to w and v adjacent to w then there is a path uAewAev.
With same ways, can be proved for uv = vOe1 .
Since there is always can be found a path between u and v, then T I(Zn ) where n prime and n Ou 7 is a connected graph.
n Fig.
4: T I(Z7 )
As shown in Figure 3.
T I(Z11 ) is a connected graph because it also satisfies Theorem 1.
In addition, we also give several other graphs T I(Zn ) where n is prime and n Ou 7 in Figure 5.
T I(Z13 ) .
T I(Z17 ) Given example that related to a connected graph of T I(Z7 ) as an illustration of Theorem 7.
Example 8.
Let Z7 .
There is obtained vertex set V (T I(Z7 )) = .
, 3, 4, 5, .
and idempotent set I(Z7 ) = .
, .
Now, let u, v OO V (T I(Z7 )) be distinct vertices, will be shown there always exist path between u and v.
For u = 2 and v = 3, there exist w = 6 such that u.
w = 1E OO I(Z7 ).
Therefore, there is a direct path between 2 Oe 3, 2 Oe 6 and 3 Oe 6.
For u = 2E and v = 4E, as result of 2E.
4E = 1E, 1E OO I(Z7 ) then vertex 2E and 4E are not adjacent.
So, there exist w = 6E such that wOe1 = 6E = 2E.
3E = 4E.
Therefore, 2E and 4E adjacent to 6E.
Can be found a path 2E Oe 6E Oe 4E, as the same ways for u = 3E and v = 5E.
For vertex u = 2E and v = 5E, u.
v = 3E where 3E is the inverse of vertex 5E.
There exist w = 6E such that vOe1 = 3E = 6E.
and uOe1 = 4E = 3E.
Then, clearly 2E and 5E adjacent to 6E such that there is a path 2E Oe 6E Oe 5E as the same ways for u = 3E and v = 4E.
Since there is always found a path between every two distinct vertices, then T I(Z7 ) is a connected graph.
Figure 4 shows graph T I(Z7 ).
T I(Z19 ) .
T I(Z23 ) Fig.
5: Graph T I(Zn ) where n = .
, 17, 19, .
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS VOL.
NO.
MARCH 2024
ACKNOWLEDGMENT