On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n

Let Γ Zn i be the zero divisor graph for the ring of the Gaussian integers modulo n. Several properties of the line graph of Γ Zn i , L Γ Zn i are studied. It is determined when L Γ Zn i is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of L Γ Zn i is given when n is a power of a prime. On the other hand, several graph invariants for Γ Zn i are also determined.


Introduction
The study of zero divisor graphs of commutative rings reveals interesting relations between ring theory and graph theory; algebraic tools help understand graphs properties and vise versa.In 1988, Beck 1 defined the concept of zero divisor graph of a commutative ring R, where the vertices of this graph are all elements in the ring and two vertices x, y are adjacent if and only if xy 0. Anderson and Livingston 2 modified the definition of zero divisor graphs by restricting the vertices to the nonzero zero divisors of the ring R. Further study of zero divisor graphs by Anderson et al. 3 investigated several graph theoretic properties, such as the number of cliques in Γ R .They also gave some cases in which Γ R is planer.On the other hand, they answered the question when Γ R 1 ∼ Γ R 2 for some specified types of rings R 1 and R 2 .Akbari and Mohammadian 4 improved on those results.Γ R for rings R which satisfy certain conditions are discussed by Anderson and Badawi 5 .The zero divisor graph of the ring of integers modulo n was extensively studied in 6-10 .
In 2008, Abu Osba et al. 11 introduced the zero divisor graphs for the ring of Gaussian integers modulo n, Γ Z n i , where they studied several graph properties and International Journal of Combinatorics determined several graph invariants for Γ Z n i .Further properties of the zero divisor graphs for the ring of Gaussian integers modulo n are investigated in 12 .In this paper, we study the line graph of Γ Z n i .We organized our work as follows: some basic definitions and terminology are given in Section 2. In Sections 3 and 4, we answer the question when is the line graph L Γ Z n i Eulerian, Hamiltonian, or planer.In Section 5, the chromatic and clique numbers of L Γ Z n i are found.While the diameter, the girth and the radius of L Γ Z n i are determined in Sections 6 and 7, respectively.Finally, the last two sections discuss the domination number of Γ Z n i and L Γ Z n i as well as the independence and clique numbers of Γ Z n i .

Preliminaries
The set of Gaussian integers is defined by Z i {a bi : a, b ∈ Z and i √ −1}.A prime Gaussian integer is one of the following: i 1 i or 1 − i , ii q, where q is a prime integer and q ≡ 3 mod4 , iii a bi, a − bi, where a 2 b 2 p, p is a prime integer and p ≡ 1 mod4 .
It is clear that Z n i is a ring with addition and multiplications modulo n.Throughout this paper, p will be used to denote a prime integer which is congruent to 1 modulo 4, while q will denote a prime integer which is congruent to 3 modulo 4. Since Z n i is finite, each element in Z n i is either a zero divisor or a unit.Also, since Z i is a unique factorization domain, each integer n can be uniquely factorized as n k j 1 π m j j where π j s are Gaussian prime integers and m j s are positive integers.
The zero divisor graph of a commutative ring R denoted by Γ R , is the graph whose vertices is the set of all nonzero zero divisors of R, and edge set E Γ R {xy : x, y ∈ V Γ R and xy 0}.The line graph L G of a graph G is defined to be the graph whose vertices are the edges of G, with two vertices being adjacent if the corresponding edges share a vertex in G.For Γ Z n i , if n 2, then this graph is one vertex, while if n q, then Γ Z n i K 0 .Throughout this paper, all rings, R, are commutative with unity.
For a connected graph G, the distance, d u, v , between two vertices u and v is the minimum of the lengths of all u − v paths of G.The eccentricity of a vertex v in G is the maximum distance from v to any vertex in G.The radius of G, rad G , is the minimum eccentricity among the vertices of G.The diameter of G, diam G , is the maximum eccentricity among the vertices of G.The girth of G, g G , is the length of a shortest cycle in G.The center of G is the set of all vertices of G with eccentricity equal to the radius.If G has a walk that traverses each edge exactly once goes through all vertices and ends at the starting vertex, then G is called Eulerian.A graph is called Hamiltonian if there exists a cycle containing every vertex.The chromatic number of a graph G, χ G , is the minimum k such that G is k-colorable i.e., can be colored using k different colors such that no two adjacent vertices have the same color .The clique number, ω G , of a graph G is the maximum order among the complete subgraphs of G.A subset D of the vertex set V G is said to be independent if no two vertices in this set are adjacent.The independence number of G, β G , is the maximum cardinality of all independent sets in G.A subset D of the vertex set V G of a graph G is a dominating set in G if each vertex of G, not in D, is adjacent to at least one vertex of D. The minimum cardinality of all dominating sets in G, γ G , is called the domination number of G. Edge dominating sets are defined analogously.The minimum cardinality of all edge dominating sets in G, γ G , is called the edge domination number of G.The minimum cardinality of all independent edge dominating sets, γi G , is called the independence edge domination number of G.The maximum vertex degree of a graph G will be denoted by Δ G .

When Is L Γ Z n i Eulerian?
Now it is characterized when the line graph L Γ Z n i is Eulerian.Before proceeding, we prove the following lemma., and it is complete bipartite if and only if n p or q 1 q 2 , where q 1 < q 2 , 11 .On the other hand, a complete bipartite graph K m,n is Hamiltonian if and only if m n.So the result holds.
Note that Γ Z q 1 q 2 i is not Hamiltonian and hence the converse of Theorem 4.1 is not true.
Next, we move to the line graphs L Γ Z n i .Before proceeding, we present the following theorem.[13].
ii The line graph of an Eulerian graph is both Hamiltonian and Eulerian, see [14].
If n p, 2 m , or q m , where m ≥ 2, then diam Γ Z n i ≤ 2. On the other hand, if n 2, p or n is a composite odd integer which is a product of distinct primes, then Γ Z n i is Eulerian, 11 .Thus the following corollary is obtained.
ii If n is a composite odd integer which is a product of distinct primes, then L Γ Z n i is both Eulerian and Hamiltonian.Now, we discuss planarity of the graph L Γ Z n i .A graph G is planar if it can be drawn in the plane without any edge crossing.The following theorem gives necessary and sufficient conditions on a graph G so that the line graph L G is planer.

Theorem 4.5 see 15 . A nonempty graph G has a planer line graph L G if and only if
Therefore, we get the following theorem.
Theorem 4.6.The graph L Γ Z n i is never planer.

The Chromatic and Clique Numbers of L Γ Z n i
If R is a finite ring, then χ Γ R Δ Γ R , unless Γ R is complete graph of odd order, 4 .Note that, the only complete graph Γ Z n i occurs when n q 2 .However, in this case the order of the graph is q 2 − 1 which is even, so χ Γ Z n i Δ Γ Z n i .Moreover, since the edge coloring of any graph leads to a vertex coloring of its line graph, we obtain This leads to the following theorem.

5.1
Finally, if n 2 m r j 1 p r j j l j 1 q j s j 1 q s j j , where s j ≥ 2 and m, r j ≥ 1, then the clique number and the chromatic number for the graph L Γ Z n i is given by the following theorem.
Theorem 5.2.n 2 m r j 1 p r j j l j 1 q j s j 1 q s j j , where m, r j ≥ 1 and s j ≥ 2, then Proof.The result follows by computing

The Diameter of L Γ Z n i
Now, we will find the diameter of the line graph L Γ Z n i .First, we will prove that diam L Γ Z n i 2 when n 2 m or n q m .Lemma 6.1.

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ii Assume that a bi c di ≡ 0 modq .Then ac − bd ≡ 0 modq and ad bc ≡ 0 modq .Since a, b, c, d are relatively prime with q, we have a qa . ii Suppose that n q m , m ≥ 2. Let x aq t bq k i and a, b ∈ U Z n .Then ann x {cq r dq s i : r, s ≥ m − min{t, k}}.Moreover, d a 1 q r 1 b 1 q s 1 i, c From Theorems 3.1 and 3. Proof.First note that diam L Γ R ≥ 2, 9 and for n n Case II: If n 2q or n q 1 q 2 , then x j , where x j 1 if j 1, 2 and 0 otherwise, y y j , where y j 1 if j 3, 4 and 0 otherwise, z z j , where z j 1 if j 2, 3 and 0 otherwise and w w j , where w j 1 if j 1, 4 and 0 otherwise.Then d x, y , z, w 3.
Summarizing the above results, we get the following theorem.
(ii) diam L Γ Z n i 3 otherwise.

The Girth and the Radius of L Γ Z n i
In this section, we give a complete characterization of the girth and the radius of L Γ Z n i .Since for any commutative ring R, L Γ R is a tree if and only if Γ R K 2 or K 1,2 9 , L Γ Z n i is never a tree.On the other hand, if L Γ R contains a cycle, then g L Γ R ≤ 4 where equality holds only if R Z 3 × Z 3 , 9 .
Consequently, the following result holds.International Journal of Combinatorics 2 If n q m ,m ≥ 2, then d q m−1 , q , x, y ≤ 2 for all x, y ∈ V L Γ Z n i .
3 If n p m , m ≥ 1, then d a bi m a − bi m−1 , a − bi m a bi m−1 , x, y ≤ 2 for all x, y ∈ V L Γ Z n i .
Theorem 7.4.If n r m t, where r 2, q, or p and m ≥ 1, g.c.d r, t 1, then rad L Γ Z n i 2. Proof.
Summarizing the above results, we get the following.
Theorem 7.5.The radius of the line graph L Γ Z n i equals 2.

The Domination Number of Γ Z n i
Pervious results concerning the domination number of Γ Z n i are very restricted; Abu Osba et al. 11 answered the question "when is the domination number 1 or 2?".Here we find the domination number of the graph Γ Z n i .Two independent proofs reflecting two different viewpoints are given, the first proof depends on ring theory.While the second proof is constructive in the sense that it does not only give the domination number of Γ Z n i , but also gives a minimum dominating set of this graph.This dominating set, as we will see, reveals to have interesting properties. . . .π m k k : 1 ≤ j ≤ k} is a dominating set of Γ Z n i .To show that D is a minimum dominating set, assume that D 1 is a minimum dominating set such that there is no x sP j , g.c.d s, π j 1 belongs to D 1 for some 1 ≤ j ≤ k.Then T j {π j , Case II: n is even.Then π 1 1 i , π 2 1 − i .Similar to case I, we can see that D If a dominating set D induces a complete graph, then, D is called clique dominating set, the clique domination number is the cardinality of a minimum clique dominating set, and is denoted by γ cl G , if every vertex in D is adjacent to another vertex in D, then D is called total dominating set.The minimum cardinality of a total dominating set is called total domination number and is denoted by γ t G .For any graph G, γ G ≤ γ t G ≤ γ cl G .Since the suggested dominating set, D, for Γ Z n i in the second proof of Theorem 8.2 induces a complete graph, then γ Γ Z n i γ t Γ Z n i γ cl Γ Z n i .

The Domination Number of L Γ Z n i
In this section we determine the domination number of L Γ Z n i when n t m and t is prime.
The study of the domination number of the line graph of G leads to the study of edge or line domination number of G, that is, γ L G γ G .On the other hand, for any graph G, γ i G γ G , 17 .Further, if G is the complete bipartite graph K r,s , then γ G min r, s , thus we have the following.
, where q 1 < q 2 .Now, we study the domination number of the line graph of Γ Z n i when n is a power of a prime.The first theorem treats the case n 2 m , m ≥ 2.Here we make use of the fact that Proof.For j 1, 2, . . ., 2m−1, let A j {α2 2m−j : α ∈ {1, 3, . . ., 2 j −1}}.Note that the sets A j form a partition to the vertices of Γ Z 2 2m .Let S m j 1 A j and T 2m−1 j m 1 A j .Then the set S induces a complete subgraph of Γ Z 2 2m and the set T form an independent set of it.And each vertex in A k is adjacent to each vertex in 2m−k j 1 A j .Γ Z 2 2m has no other edges.Let D ⊂ E Γ Z 2 2m be a dominating set of vertices for L Γ Z 2 2m with minimum cardinality.Since, the set S induces a complete subgraph of Γ Z 2 2m of order 2 m − 1, then γ L Γ Z 2 m i ≥ 1/2 2 m − 1 .On the other hand, since D dominates all edges in the complete graph S , D also dominates every edge joining S to T , recall that T forms an independent set and so γ L Γ Z 2 m i 1/2 2 m − 1 . 10

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The proof of Theorem 9.2.shows the set T is an independent set with maximum cardinality in Γ Z 2 m i , while the set S induces a complete subgraph with maximum order.
So, the following corollary is obtained.
As another consequence to the proof of the preceding theorem, the following corollary, which gives the degree sequence for Γ Z 2 m i , is obtained.Corollary 9.4.For j 1, 2, . . ., 2m−1, the graph Γ Z 2 m i has exactly 2 j−1 vertices of degree 2 2m−j −2 if 1 ≤ j ≤ m and 2 j−1 vertices of degree Furthermore, The proof of the above theorem shows that the eccentricity of 2 2m−1 is 1 and the eccentricity of any other vertex in Γ Z 2 2m is 2, since the vertex 2 is adjacent only to the vertex 2 2m−1 , and for any x ∈ V Γ Z 2 m i , 2-2 2m−1 -x, is a path of length 2. This leads to the following corollary.

Corollary 9.5. The center of the graph
Next, we we find the domination number of the line graph L Γ Z n i when n q m , m ≥ 2. Lemma 9.6.(i) For m ≥ 2, Proof.Let A kj , S, and T be defined as given in Lemma 9.6.Clearly, the set S induces a complete subgraph of Γ Z n i with maximum order if m is even and S ∪ {q m/2 } induces a complete subgraph of Γ Z n i with maximum order if m is odd.On other hand if m ≥ 3, then T form an independent set with maximum cardinality.Moreover, if a vertex v belongs to the set A rs , then v is adjacent to every element in A kj where m − min{r, s} ≤ k, j ≤ m and k, j / m at the same time.Γ Z n i has no other edges.
As a consequence of the proof of Theorem 9.7, we conclude the following.
Corollary 9.9.Let n q m , m ≥ 2, and v aq r bq s i where a, b ∈ U Z n .Then

9.2
Corollary 9.10.Let n q m , m ≥ 2. Then iii the radius of the graph Γ Z n i equals 1, iv the diameter of the graph Γ Z n i equals 2, for m ≥ 3.
Finally, we find the domination number of the line graph Clearly, the sets A kj , 0 ≤ k, j ≤ m and not both k, j m or 0, partition the vertices of Γ Z p m × Z p m .Lemma 9.11.(i) For m ≥ 2: Theorem 9.12.Let n p m , m ≥ 2 and s, l and b be defined as given in Lemma 9.11, then Proof.Using the same notations of Lemma 9.11.Note that the set S induces a complete subgraph of Γ Z n i , K s .Thus, any edge dominating set for Γ Z p m × Z p m must contain s/2 edges to dominate K s .If m ≥ 3, the set L L 1 ∪ L 2 induces a complete bipartite graph K l,l with bipartite sets L 1 and L 2 .This contributes l edges in the dominating edge set for Γ Z p m × Z p m .

International Journal of Combinatorics
Edges joining vertices in K l,l to vertices in K s are covered by the same edge dominating sets for K l,l and K s .Moreover, vertices in A k0 and A 0k , where 1 ≤ k ≤ m−1, are only adjacent to some vertices in K s and K l,l .
On the other hand, if m ≥ 3, the set T is an independent set.Fortunately, vertices in T are only adjacent to vertices in S. So, any edge dominating set for K s also dominates edges between S and T .Now, for each 1 ≤ k ≤ m/2 − 1, and m − k ≤ j ≤ m, the set A kj ∪ A jk induces a complete bipartite graph with bipartite sets A kj and A jk .In order to dominate this collection of complete bipartite graphs induced by A kj ∪ A jk we need b edges in the edge dominating set for Γ Z p m × Z p m .Fortunately, this dominating set with b elements also dominates all edges in E Γ Z p m × Z p m which are incident to any edge in this collection.
Finally, observe that if m ≥ 4, then vertices in W are only adjacent to some vertices in K s as well as in the collection of the complete bipartite graphs.The graph Γ Z p m × Z p m has no other edges.
The above proof shows that S induces a complete graph in Γ Z p m × Z p m .In fact, K s is a complete subgraph with maximum order in case m is even, while if m is odd we can add one additional vertex of some A kj , where either k or j, say k, is m/2 while j is greater than m/2 .On the other hand, the set T ∪ W ∪ m−1 k m/2 A k,0 ∪ A 0,k ∪ A m,0 is a maximum independent set of order t w r, where r | m−1 k m/2 A k,0 ∪A 0,k ∪A m,0 | p m−1 p −1 2p m/2 −1 .Thus, using the same notation of Lemma 9.11 and the proof of the above theorem, we obtain the following corollary.Proof.i First, note that Γ Z p m i has no vertex of eccentricity 1, otherwise γ Γ Z p m i 1.Let u, w, α, β ∈ U Z p m and 1 ≤ r, s ≤ m − 1.If x, y is adjacent to both up r , β and α, wp r , then x y 0. So, d up r , β , α, wp s  3, and hence, the eccentricity of each vertex in E is 3.If up r , y , wp s , x are nonadjacent, then p m−1 , 0 is adjacent to both vertices.Similarly, if x, up r , y, wp s are nonadjacent, then 0, p m−1 is adjacent to both vertices.
where a, b, c, d are odd integers such that a bi c di ≡ 0 mod4 .(ii) If n q m , m ≥ 2 then there are no a bi, c di ∈ Z n i where a, b, c, d are relatively prime with q, such that a bi c di ≡ 0 modq .Proof.i Assume that a bi c di ≡ 0 mod4 .Then ac − bd ≡ 0 mod4 and ad bc ≡ 0 mod 4 .Since a, b, c, d are odd integers, a 2a 1 1, b 2b 1 1, c 2c 1 1, and

Theorem 7 . 1 . 3 . 7 . 2 .
g L Γ Z n i 3.Next, we prove that the radius of the line graphL Γ Z n i equals 2. Since diam L Γ R ≤ 3, 9 and rad G ≤ diam G for any graph G, rad L Γ R ≤ Lemma If there exists a vertex v ∈ L Γ Z n i with eccentricity 2, then rad L Γ Z n i 2Proof.Note that, L Γ Z n i has no spanning star graph, since if a, b ∈ V Γ Z n i such that a / b and ab 0, then d a, b , ai, bi > 1.

Corollary 9 . 13 .Corollary 9 . 14 .
If n p m , then i ω Γ Z n i s if m is even and s 1 if m is odd, for m ≥ 2, ii β L Γ Z n i r, if m 2, β L Γ Z n i r t, if m 3, and β L Γ Z n i r t w, for m ≥ 3.If n p m , m ≥ 2, then i E {v ∈ V Γ Z n i: v u, w where either u or w ∈ U Z p m } has eccentricity 3, while all other vertices has eccentricity 2, ii the center of the graph Γ Z n i is the set C {v ∈ V Γ Z n i : v u, w , where both u and w ∈ Z Z p m } − { 0, 0 }, iii the radius of the graph Γ Z n i equals 2, iv the diameter of the graph Γ Z n i equals 3.
Case II t is an odd prime and m > 2 .By Theorem 23 of 11 , Γ Z n i has a vertex of degreet 2k−1 − 1, where 1 ≤ k < m/2and a vertex of degree t 2k − 2, where m/2 ≤ k < m.Case III t p a 2 b 2 and m 2 .Since deg a ib | p a − ib | − 1 and | p a − ib | divides |Z p 2 |, | p a − ib | is odd and hence deg a ib is even.Then by using part i , the result holds.

Hamiltonian or Planner?
9 , together with Lemma 3.1 and Theorem 26 of 11 , the following theorem is obtained.First we determine which graphs, Γ Z n i , are Hamiltonian.Before this paper comes to the light, a recent article by Abu Osba et al. 12 reached to similar results concerning International Journal of Combinatorics Hamiltonian Γ Z n i .However, we present our proof since it is simpler and shorter.The proof makes use of the following theorem.Since Z n i is a finite principal ideal ring, Γ Z n i is a complete graph or a complete bipartite graph if Γ Z n i is Hamiltonian.But the graph Γ Z n i is complete if and only if n q 2 Theorem 4.1 see 4 .Let R be a finite principal ideal ring, if Γ R is Hamiltonian, then it is either a complete graph or a complete bipartite graph.Theorem 4.2.The graph Γ Z n i is Hamiltonian if and only if n p or q 2 .Proof.
If n 2 m or q m and m ≥ 2, then diam L Γ Z n i 2.
Proof.i Suppose that n 2 m , m ≥ 2.Then, 1 x a2 t b2 k i where a, b are odd and t / k or t k ≥ m/2 implies that ann x {c2 r d2 s i : c and d are odd and r, s ≥ m − min{t, k}}, 2 x 2 t a bi where a, b are odd and t < m/2 , then ann x {c2 r d2 s i : c and d are odd and r, s , where p a 2 b 2 and m ≥ 2, d p, p m−1 , a ib m , a − ib m 3. So, diam L Γ Z p m i 3. (ii) If n st 2 are two distinct primes and s, t / p, then diam L Γ Z n i 2.
Theorem 6.3.(i) If n st, where s, t are two distinct primes and s / p or t / p, then diam L Γ Z n i 2.
Theorem 6.4.i If n sp 2 , where s is prime and p a 2 b 2 , then diam L Γ Z n i , m ≥ 1, l ≥ 2, and g.c.d p, t 1, then diam L Γ Z n i 3. iv If n s m t l where s, t are distinct primes and m, l ≥ 2, then diam L Γ Z n i 3.
If n 2 m , then 1 i is the unique maximal ideal of Z n i . 2 If n q m , then q is the unique maximal ideal of Z n i . 2 are the only distinct maximal ideals of Z n i .Finally, since Γ Z n i is never a star graph 11 , the result holds.Proof.II We have two cases.Case I: n is odd.Then it is easy to see that D {P j π m 1 1 π m 2 2 . . .π Theorem 8.1 see 16 .Let R be a finite commutative ring with identity that is not an integral domain.If Γ R is not a star graph, then the domination number equals the number of distinct maximal ideals of R. j , where k ≥ 1 and π j s are distinct gaussian prime and m j s are positive integers and n / 2 or q.Then γ Γ Z n i k, if n is odd, and γ Γ Z n i k − 1, if n is even.Proof.I 1 t where j / m j −1 j