Covering of Line Graph of Zero divisor Graph over Ring Zn

Let Zn be the commutative ring of residue classes modulo n, Γ(Zn) the zero divisor graph of Zn and L(Γ(Zn) be the line graph of Γ(Zn). We have studied the point covering number and independence number of L(Γ(Zn)), for some positive integer n. We have computed edge covering number for L(Γ(Zpq), and establish the relation among point covering, independence number and edge covering number of L(Γ(Zpq), where p and q are prime numbers.


Introduction
Let R be a commutative ring and Z(R) be its set of zero-divisors. We associate a graph Γ (R) to R with vertices Z(R) * = Z(R) − {0}, the set of non-zero zero-divisors of R and for distinct u, v ∈ Z(R) * , the vertices u and v are adjacent if and only if uv = 0. In this paper, the commutative ring R is Zn and zero divisor graph Γ (R) is Γ (Zn). The idea of a zero-divisor graph of a commutative ring was introduced by I. Beck in [Beack (1988)], where he was mainly interested in colorings. Now, these days so many mathematicians have been working on it as C. I. Aponte, P. S. johnson and N. A. Mims [Aponte, johnson and Mims (2005)], E. Emad AbdAlJawad and H. Al-Ezeh [Emad AbdAlJawad and Al-Ezeh (2008)], M. Ghanem and K. Nazzal [Ghanem and Nazzal (2012)].
The line graph L(Γ (Zn)) of the Γ (Zn) is defined to the graph whose set of vertices constitutes of the edges of Γ (Zn), where two vertices are adjacent if the corresponding edges have a common *Corresponding author: E-mail: sheelasuthar@gmail.com and om@iitp.ac.in vertex in Γ (Zn). The importance of line graphs stems from the fact that the line graph transforms the adjacency relations on edges to adjacency relations on vertices. For example, the chromatic index of a graph leads to the chromatic number of its line graph. We illustrate an example of a zero divisor graph and its line graph, Figure 1 shows Γ (Z10) and Figure 2 shows L(Γ (Z10)). Here , we study some properties such as point covering, independence point covering and edge covering of L(Γ (Zn)). Also, we determine relation between point covering and edge covering. Moreover, it is noted that if n is prime, then zero is the only zero divisor of Zn. Thus, Γ (Zn) is an empty graph. Also, Γ (Z4) has single vertex because 2 is the only non-zero zero divisor. In these cases L(Γ (Zn)) is an empty graph which will not be studied in the sequel. Definition 1.1. Point covering set:-Two points of a graph are covering each other if they are adjacent. A subset S of vertex set V (Γ (Zn)) of vertices in a graph L(Γ (Zn)) is a point covering set if every vertex v ∈ V (L(Γ (Zn))) is an element of S or adjacent to an element of S. The point covering number of graph is denoted by α00(L(Γ (Zn))), defined as the cardinality of minimum point covering set of L(Γ (Zn)). Definition 1.2. independent set:-An independent set in a graph is a subset of vertex set V (L(Γ (Zn))) of L(Γ (Zn)) such that no two vertices of subset are adjacent. The independence number of L(Γ (Zn)), denoted by α 00 L((Γ (Zn))), is defined as the cardinality of a minimum independent set of L(Γ (Zn)). Definition 1.3. Minimum Edge cover:-An edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. A minimum edge cover is an edge cover having the smallest possible number of edges for a given graph. The size of a minimum edge cover of the graph L(Γ(Zn)) is known as the edge cover number of graph L(Γ(Zn)) and is denoted by ρ(L(Γ(Zn))).

Results
Theorem 2.1. Let L(Γ(Zn)) be a line graph of Γ (Zn), where n = 2p, and p is an odd prime number. Then point covering and independence number of L(Γ(Zn)), both are one.
Proof. When n = 2p, then Γ(Zn) is a star graph. So there is a common vertex which is adjacent to all other vertices and that vertex is also called center of the graph. When we draw the line graph of Γ(Zn), for n = 2p, and let v1 be the common vertex of Γ(Zn) which is the end point of every edge of Γ(Zn). Then v1 appears in every vertex of the line graph. [v1, ui] ∈ V (L(Γ(Zn))), where {i = 2, 2.2, ..., 2(p − 1), p = v1} forms a complete line graph of Γ(Zn) and here, [v1, u1] is adjacent with all other vertices of line graph. In other words, we can say that single vertex cover all other vertices of line graph of zero divisor graph Γ(Zn). Thus, point cover is one and that vertex is also use for independence number. Proof. Γ (Zn) is a complete bi-partite graph, when n = 3p. Then, there are two idependent set of vertices, one set has p − 1 elements which are multiple of 3 and second set has (3 − 1) elements multiple of p. In the zero divisor graph Γ (Zn), there are two vertices which are adjacent to all other vertices but not to each other. These two vertices of Γ (Zn) appears in line graph as the end points of edges. Let [u1,v1] and [u2, v2] ∈ V (L(Γ (Zn))). Then they are not adjacent to each other but [u1, v1] u2, v2] are two vertices covers all vertices of line graph of Γ (Zn). So, independence number is two. Again, [u1, v1] and [u2, v1] cover all vertices of line graph of Γ (Zn), which are adjacent to each other. Thus, point covering is also two. 3. When p = 5, there are six vertices in L(Γ (Z25)), then point covering is two and independence number is also two.
Theorem 2.5. If n = p 3 , then point covering and independence number of L(Γ (Zn)) is one, where p = 2 or 3.
Proof. In zero divisor graph of Zn, where n = p 3 ,(Γ (Zn)) is complete 3-parptite graph. There are p − 1 vertices which are adjacent to each other and adjacent with all multiples of p. These (p − 1) elements has minimum eccentricity and form centers of Γ (Zn). But in the line graph of Γ (Zn), p − 1 2 elements are centers. These p − 1 2 elements are adjacent to each other and also adjacent with multiple elements of p. Therefore, a single vertex which is also a center, that cover all veritces of line graph and minimum in numbers. Thus, point covering is one and the same vertex which covers all vertices is also give independence number.
Hence from above two examples, when n = p 3 , where p = 2 or 3. point covering and independence number is one. When p = 2 and p = 3, then point covering and independence number are p − 1 2 .
Theorem 2.7. Let p be a prime number and n = 2p, for p > 2 and n = 3p, for p > 3. Then edge covering of L(Γ (Zn)) is one.
Proof. Case 1:-When n = 2p, the line graph of Γ (Zn) is a complete graph. Therefore, every vertex is adjacent to each other. Let [v1,u1] and [v1, u2] ∈ V (L(Γ (Zn))). Then there is an edge between [v1, u1] and [v1, u2] and these are end points of that edge. There is v1 point appears in every vertex in line graph of Γ (Zn). Since we know for n = 2p, Γ (Zn) is a star graph, so let v1 be the center of Γ (Zn) which covers all vertices. Therefore, edge of vertices [v1, u1] and [v1, u2] covers all the vertices which is also minimum in number. Hence, edge covering is one.
where n = pq, p < q and p, q are prime numbers.
Theorem 2.9. Edge covering is less than the point covering and independence number in L(Γ(Zn)), where n = pq, p > 2, p and q are distinct prime numbers.
Proof. If n = 2p, then point covering and edge covering is one.(From Theorem (1) and Theorem (6) (case-1)) If n = pq, and p > 2, then from Theorem (2) to Theorem (7), shown that edge covering is always less than to point covering and independence number.
It is also easily seen that point covering and independence numbers are equal.
In this paper, we studied the point covering number and independence number associated with line graph of zero divisor graph over commutative ring Zn. Some details are given below. a In the section (1), a brief historical background and definitions of line graph related to Zn has been discussed. Also, importance of line graph is given.
b In the section (2), we have studied the point covering number and independence number of line graph of Zn for n is product of primes. Also, we discussed the edge covering number of line graph over Zpq. Furthermore, we have established the relation between point covering number, independence number and edge covering number and concluded that edge covering number is less than the point covering number and independence number in L(Γ(Zpq)) where n = pq . .