ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
The Nilpotent Cayley Graph of the Residue Class Ring ( , ⊕, ⊙)
Tippaluri Nagalakshumma1, Jangiti Devendra2 and Levaku Madhavi3*
1Research Scholar, Department of Applied Mathematics, Yogi Vemana University, Kadapa-516005, A.P., INDIA.
2Research Scholar, Department of Applied Mathematics, Yogi Vemana University, Kadapa-516005, A.P., INDIA.
3*Assistant Professor, Department of Applied Mathematics, Yogi Vemana University, Kadapa-516005, A.P., INDIA.
email: [email protected], [email protected], [email protected]
(Received on: June 1, 2019)
ABSTRACT
In this paper the notion of the nilpotent Cayley graph ( , ), where ( , ⊕, ⊙) is the ring of residue classes modulo , ≥ 1, an integer and is the set of nonzero nilpotent elements, is introduced and it is shown that ( , ) can be decomposed into components, for all values of and a method of enumeration of triangles is present.
AMS Subject Classification (2010): 05C07, 05C25, 05C30, 05C38, 05C40,16N40, 20F65.
Keywords: Nilpotent elements, Symmetric set, Cayley graph,Nilpotent Cayley graph.
1. INTRODUCTION
Graphs constructed by using the algebraic concepts have become a rich source of research activity, providing abundance of applications in science and technology. Nathanson19 introduced the concept of congruences in number theory into graph theory and initiated the study of number theoretic graphs. In this way many researchers have studied the graphs associated with various concepts in algebra, to mention some of them, arithmetic functions in number theory5,6,8,9,17,18
and zero-divisors and nil-potent elements in ring theory1,2,4,10,11,20
. The notion of a Cayley graph is introduced to study, whether it is possible to find a graph Γ, associated with a group ( , . ) such that the automorphism group of the graph Γ is
isomorphic to the group ( , . )14, which was affirmatively answered by Frucht12. A detailed presentation of these facts can be found in Prop. 13.9, pp 447-448 of 21. The Cayley graph ( , ) associated with a group ( , . ) and its symmetric subset (which contains the inverse
of every element in it) is the graph, whose vertex set is and the edge set
= {( , ): , ∈ and either ∈ , or, ∈ }. This graph is | | −regular and has
| || | edges (Theorem 1.4.5 15). Madhavi15 has introduced arithmetic Cayley graphs associated with number theoretic functions, namely, the Euler totient function ( ), the divisor function ( ), ≥ 1, an integer and quadratic residues modulo a prime and studied their basic properties and cycle structures. Later Madhavi et al.,16,17,18 also studied some of the domination parameters of these graphs.
In recent times Chen10, Nikmehr and Khojasteh20 and Dhiren Kumar Basnet et al.,11 have studied nilpotent graphs associated with a finite commutative ring and the × matrix ring ( ). Let be a ring with unity, ( ), the set of all nilpotent elements of and let ( )∗= { ∈ , ≠ 0: ∈ ( ), for some ∈ , ∈ ≠ 0}. According to Chen10 and Nikmehr and Khojasteh20, the nilpotent graph Γ ( ) has vertex set ( )∗ and , ∈ ( )∗ are adjacent, if and only if, , ∈ ( ). They studied the properties of diameter and girth of these graphs for a finite ring and the matrix ring ( ). However, Dhiren Kumar Basnet et al.,11 studied a variant form of nilpotent graph Γ ( ), whose vertex set is ( )∗ and , ∈
( )∗ are adjacent, if and only if, + ∈ ( ).
In this paper we introduce another class of nilpotent graphs associated with the set of nilpotent elements in a ring ( , +, . ), which is the Cayley graph of the group ( , +) and its symmetric subset , with special reference to the ring ( ,⊕,⊙) of residue classes modulo ,
≥ 1, an integer. For concepts and terminology that are used in this paper the reader is referred to7 for graph theory,13 for algebra and3 for number theory.
2. BASIC PROPERTIES OF THE GRAPH ( , )
Definition 2.1: Let ( , +, . ) be a ring. An element ∈ , ≠ 0, is called a nilpotent element in ( , +, . ), if there exists a positive integer such that = 0.
Lemma 2.2: The set of nilpotent elements of a ring ( , +, . ) is a symmetric subset of the group ( , +).
Proof: Let ( , +, . ) be a ring and let be the set of nilpotent elements in the ring ( , +, . ).
For any ∈ , there exists a positive integer such that = 0. Let – be the inverse element of in the group ( , +). Then
(− ) = , if is even
− , if is odd.
So (− ) = 0 and – is also a nilpotent element in the ring ( , +, . ), which shows, that
− ∈ and is a symmetric subset of group ( , +). ∎
Lemma 2.3: If ( , +, . ) is a commutative ring and, if is a nilpotent element in ( , +, . ), then for every ∈ , is also a nilpotent element in ( , +, . ).
Proof: Let ( , +, . ) be a commutative ring and let be a nilpotent element in the ring ( , +, . ). Then there exists a positive integer such that = 0. Let ∈ . Since the ring ( , +, . ) is commutative and = 0, we have ( ) = = 0 = 0. So is also a
nilpotent element in ( , +, . ). ∎
For an integer ≥ 1, let us investigate the nature and the number of nilpotent elements in the ring ( , ⊕, ⊙), the ring of residue classes modulo .
Lemma 2.4: If = … , where < < ⋯ < are primes, ≥ 1, and 1 ≤ ≤ , are integers, then for 1 ≤ ≤ and 1 ≤ ≤ , the element … ∈
, is a nilpotent element in the ring ( , ⊕, ⊙).
Proof: Let = … , 1 ≤ ≤ , 1 ≤ ≤ , be an element in the ring ( , ⊕, ⊙) and let = max{ , , … , }. Then
= ( … ) = …
= ( … ) …
= …
= 0 ⊙ … = 0,
since = 0. That is, = 0, for the integer > 0, so that is a nilpotent element in the ring
( , ⊕, ⊙). ∎
The following corollary is immediate from the Lemma 2.4.
Corollary 2.5: In the ring ( , ⊕, ⊙), where = ∏ , < < ⋯ < , are primes and ≥ 1, 1 ≤ ≤ , are integers, the element , where = … is a nilpotent element in ( , ⊕, ⊙) and this is the smallest among all the nilpotent elements in the ring
( , ⊕, ⊙). ∎
Remark 2.6: One can easily see that ( ) is ∏ and (∏ ) is in the group ( , ⊕).
Lemma 2.7: Let = ∏ , where < < ⋯ < , are primes, ≥ 1 and 1 ≤ ≤ , are integers. The set of non-zero nilpotent elements in the ring ( , ⊕, ⊙), is given by
= ⊙ 1, ⊙ 2, … , ⊙ ∏ − 1 ,
where = … .
Proof: For = … , consider the subset
ℕ = ⊙ 0, ⊙ 1, ⊙ 2, . . ., ⊙ ∏ − 1
of the ring ( , ⊕, ⊙). By the Corollary 2.5, = … is the smallest nilpotent element in ( , ⊕, ⊙). Since ∏ < , we have 1, 2, … , ∏ − 1 ∈ . So by the Lemma 2.3, the elements ⊙ 0, ⊙ 1, ⊙ 2, … , ⊙ ∏ − 1 are all nilpotent elements in the ring ( , ⊕, ⊙). Further they are all distinct. We have
= {0, 1, 2 , 3, 4 , … , − 2, − 1}
= {0, 1, … , ∏ − 1 , ∏ , ∏ + 1, …,
(∏ − 1) + , … , (∏ − 1) + (n − ∏ ) }.
Also by the Lemma 2.3, gives all the nilpotent elements in the ring ( , ⊕, ⊙).
Now = { ⊙ 0, ⊙ 1, ⊙ 2, … , ⊙ ∏ + 1, …,
⊙ ∏ + , … , ⊙ ∏ − 1 + ( − ∏ ) }.
But, for any , 0 ≤ ≤ − ∏ ,
⊙ ∏ + = ⊙ ∏ ⊕ ⊙ ̅ = + .
Since + > , + can not be a nilpotent element in the ring( , ⊕, ⊙). Further,
⊙ 0 = 0, is not a nilpotent element in the ring ( , ⊕, ⊙). Thus the set
= ⊙ 1, ⊙ 2, … , ⊙ ∏ − 1
contains all the nonzero nilpotent elements in ( , ⊕, ⊙). ∎ From the Lemma 2.7, the following corollary is immediate.
Corollary 2.8: The number ( ) of the nilpotent elements in the ring ( , ⊕, ⊙), where
= ∏ , < < ⋯ < , are primes and ≥ 1, 1 ≤ ≤ , are integers, is given by
( ) = ∏ − 1. ∎
In the Lemma 2.2, it is observed that the set of nilpotent elements in the ring ( , +, . ) is a symmetric subset of the group ( , +). So one can think about the Cayley graph associated with the group ( , ⊕) and its symmetric subset of nilpotent elements in the ring ( , ⊕, ⊙). This is formally given in the following definition.
Definition 2.9: The Cayley graph associated with the group ( , ⊕) and its symmetric subset of nonzero nilpotent elements in the ring ( , ⊕, ⊙) is the graph, whose vertex set is
= {0, 1, 2, 3, … , − 1} and the edge set = {( , ): , ∈ and either − ∈ , or, − ∈ }. This is referred to as the Nilpotent Cayley graph associated with the ring ( , ⊕, ⊙) and it is denoted by ( , ).
Lemma 2.10: The graph ( , ) is (∏ − 1) − regular and the number of edges in ( , ) is given by (∏ − 1).
Proof: By the Theorem 1.4.5,[15] . The graph ( , ) is ( ) − regular and the total number of edges in ( , ) is | | ( ) . That is, ( , ) is (∏ − 1) − regular and
it contains (∏ − 1) edges. ∎
Lemma 2.11: For an integer = … , where < < ⋯ < are primes, the graph ( , ) contains only vertices.
Proof: Let = … , where < < ⋯ < are primes. Then the only nilpotent element in ( Z , ⊕, ⨀) is and it is the zero element of ( Z , ⊕, ⨀), since = 0. So the edge set is empty and the graph has only vertices. ∎
Example 2.12: The graphs ( , ) , ( , ), ( , ) and ( , ) are given below:
Figure 1: ( , ) Figure 2: ( , ) Figure 3: ( , ) Figure 4: ( , )
3. THE STRUCTURE OF COMPONENTS OF THE GRAPH ( , )
In this section it is shown that for any integer ≥ 1, the nilpotent Cayley graph is not connected. Further it has an interesting property that, it is decomposed into disjoint components, each of which is a complete graph. Let = ∏ , where < < ⋯ < , are primes and ≥ 1, 1 ≤ ≤ are integers and let = … .
Remark 3.1: The following decomposition of vertex set of ( , ) as subsets , , , … , comes handy to establish these facts.
= 0 , 2 , … , , … , − 1 ,
= 1, + 1, 2 + 1, … , + 1, … , − 1 + 1 ,
⋮
= − 1, + − 1, … , + − 1, … , − 1 + ( − 1) .
Observe that each , 0 ≤ ≤ − 1, contains vertices.
Lemma 3.2: For 0 ≤ ≤ − 1, each contains ( )+ 1 distinct vertices of ( Z , ).
Proof: For 0 ≤ ≤ − 1, consider
= , + , 2 + , 3 + , … , + , … , + , … , ( − ) + .
If possible, let + = + , for ≠ , 0 ≤ < ≤ ∏ − 1. Then ( − ) = 0, or, ( − ) = 0. Now, 0 ≤ < ≤ ∏ − 1 < ∏ , implies that, 0 < −
< ∏ . By the Remark 2.6, ( )= ∏ , so that ̅ ≠ 0, for any positive integer
< ∏ . So − < ∏ implies that ( − ) ≠ 0, and this leads to a contradiction. Hence our assumption that
+ = + , for ≠ , , 0 ≤ < ≤ ∏ − 1, is wrong and each contains distinct vertices of ( , ). Further
| | = = ∏ = ( ) + 1. ∎
Lemma 3.3: For 0 ≤ ≤ − 1, each is a complete subgraph of ( , ).
Proof: For 0 ≤ ≤ − 1, we have
= , + , 2 + , 3 + , … , + , + , … , ( − ) + .
We shall show that every pair of distinct vertices in are adjacent, so that is a complete subgraph of ( , ). So let + , + ∈ , for i ≠ j, 0 ≤ < ≤
∏ − 1. To show that there is an edge between + and + , one has to show that + − + ∈ . Now, + − + = ( − ) and ( − ) is a nilpotent element in ( Z , ⊕, ⨀), since is a nilpotent element in ( Z , ⊕, ⨀). So + −
+ ∈ , so that there is an edge between every pair of distinct vertices of , showing
that is a complete subgraph of ( , ). ∎
Lemma 3.4: For 0 ≤ , ≤ − 1, ≠ , ∩ = ∅.
Proof: If possible, assume that ∩ ≠ ∅, for 0 ≤ < ≤ − 1, ≠ . Then, there exists a vertex such that ∈ and ∈ . Then = + and = + for some
, , 0 ≤ , ≤ ∏ − 1. Without loss of generality one may assume that < . Then we have,
+ = = + , or, [ ( − )⨀ ] ⊕ − = 0.
Taking the product of this with ∏ , we get
[ ( − ) ⨀ ]⨀ ∏ ⊕ − ⨀ ∏ = 0.
Since ⨀ ∏ = = 0, this gives
− ∏ = 0, or, ( − ) ∏ = 0.
Now 0 ≤ < ≤ − 1, implies that 0 < − ≤ − 1 < . By the Remark 2.6,
∏ = in the group ( Z , ⊕), so that ∏ ≠ 0 for every integer , 0 < < . So ( − ) ∏ = 0, with − < leads to a contradiction. Hence
and must be disjoint. ∎
Theorem 3.5: The graph ( Z , ) is a union of disjoint connected components of ( Z , ), each of which is a complete subgraph of ( Z , ).
Proof: Let = ∏ , where < < ⋯ < , are primes, ≥ 1, 1 ≤ ≤ are integers and let = … . We have seen that the vertex set of ( Z , ) can be decomposed into subsets of , , , … , of vertices as given in Remark 3.1. By the Lemma 3.3, it follows that for every 0 ≤ ≤ − 1, is a complete subgraph of ( , )
containing ( ) + 1 vertices, and by the Lemma 3.4, and are disjoint for 0 ≤ , ≤ − 1, ≠ .
We shall now show that there is no edge between any vertex of and any vertex of , for 0 ≤ , ≤ − 1, ≠ , so that the nilpotent graph ( Z , ) is decomposed into disjoint union of complete components. To see this, let + ∈ and + ∈ , for
≠ . Then + − + = [( − )⨀] ⊕ − . Since 0 ≤ , ≤ − 1, ≠ , it
follows that | − | < , so that | − | is not a multiple of . Hence [( ̅ − ̅)⨀ ] + − is not a multiple of and hence [( ̅ − ̅)⨀ ] + − is not a nilpotent element in the ring ( Z , ⊕, ⨀), so that there is no edge between + and + . Hence, the graph ( Z , ) contains disjoint components, and each component is complete. ∎
Corollary 3.6: If is a prime and > 1 an integer, then the graph Z , is a disjoint union of components of Z , , each of which is a complete subgraph of Z , . Proof: Let = , > 1 be an integer. Then the set of nilpotent elements of the ring
Z , ⊕, ⨀ is given by = { , , , … , }, and be the smallest nilpotent element in Z , ⊕, ⨀ . By the Theorem 3.5, Z , is the disjoint union of the following complete components of Z , .
= 0, ̅, 2 , 3 , … , , … , ( − 1) ,
= 1, + 1, 2 + 1, 3 + 1, … , + 1, … , ( − 1) + 1 , = 2, + 2, 2 + 2, 3 + 2, … , + 2, … , ( − 1) + 2 ,
⋮
= ( − 1), + ( − 1), … , + ( − 1) + … , ( − 1) + ( − 1) . ∎ Example 3.7: The graphs ( , ), ( , ) and ( , ) and their components are given below:
The graph ( , ) The components of ( , )
The graph ( , ) The components of ( , )
The graph ( , ) The components of ( , )
4. ENUMERATION OF TRIANGLES IN THE GRAPH ( , )
The number of triangles in ( , ) is determined in this section. The following theorems give the number of triangles in ( , ) depending on the nature of .
Theorem 4.1: If = ∏ , where < < ⋯ < , are primes, then the graph ( , ) has no triangles.
Proof: By the Lemma 2.11, the graph ( , ) contains only vertices and no edges. So the
number of triangles in ( , ) is zero. ∎
Theorem 4.2: Let = ∏ , where < < ⋯ < , are primes, ≥ 1, 1 ≤ ≤ are integers. Then the number ( , ) of triangles of the graph ( , ) is given by
( , ) = ( ) ( ( ) − 1) ,
where ( ) is the number of nonzero nilpotent elements in the ring ( Z , ⊕, ⨀).
Proof: By the Theorem 3.5, the graph ( , ) has disjoint complete components, where
= … . By the Lemma 3.2, each complete component has ( ) + 1 number of vertices and it is ( )− regular. So, each of the components has ( ( ) + 1) number of triangles. Thus the total number of triangles ( , ) in ( , ) is given by
( , ) = ( ( ) + 1) = [ ( ) ][ ( )][ ( ) ]
.
But [ ( )+ 1] = , so that ( , ) = ( ) ( ( ) ). ∎
The following corollary gives a working formula for the number of triangles of ( , ), where is a product of prime powers.
Corollary 4.3: Let = ∏ , where < < ⋯ < , are primes, ≥ 1, 1 ≤ ≤ are integers. Then the number ( , ) of triangles of the graph ( , ) is given by
[(∏ − 1)(∏ − 2)].
Proof: Since ( ) = (∏ − 1), the number ( , ) of triangles in the graph ( , ) is given by
( , ) = {(∏ − 1)[(∏ − 1) − 1]}
= [(∏ − 1)(∏ − 2)]. ∎
Corollary 4.4: If = 2 … , where 2 < < < ⋯ < are primes, then the graph ( , ) has no triangles.
Proof: Let = 2 … , where 2 < < < ⋯ < are primes. By the Corollary 4.3, the number ( , ) of triangles in the graph ( , ) is given by
( , ) = {[(2 … ) − 1][(2 … ) − 2]} = (1)(0) = 0 ∎
Example 4.5: Consider the graph ( , ). Since 15 = 3 × 5, by the Theorem 4.1, the number of triangles in ( , ) is zero.
Example 4.6: Consider the graph ( , ). Since 36 = 2 × 3 , by the Corollary 4.3, the number ( , ) of triangles is given by
( , ) = [(2 3 ) − 1][(2 3 ) − 2] = 120.
Example 4.7: Consider the graph ( , ). Since 12 = 2 × 3, by the Corollary 4.4, the number of triangles in ( , ) is given by
( , ) = [(2 3 ) − 1][(2 3 ) − 2] = 0.
5. ACKNOWLEDGEMENTS
The authors express their sincere thanks to Prof. L. Nagamuni Reddy for his suggestions during the preparation of this paper.
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