Let be the zero divisor graph for the ring of the Gaussian integers modulo . Several properties of the line graph of , are studied. It is determined when 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 is given when is a power of a prime. On the other hand, several graph invariants for are also determined.
1. 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 , where the vertices of this graph are all elements in the ring and two vertices , are adjacent if and only if . Anderson and Livingston [2] modified the definition of zero divisor graphs by restricting the vertices to the nonzero zero divisors of the ring . Further study of zero divisor graphs by Anderson et al. [3] investigated several graph theoretic properties, such as the number of cliques in . They also gave some cases in which is planer. On the other hand, they answered the question when for some specified types of rings and . Akbari and Mohammadian [4] improved on those results. for rings which satisfy certain conditions are discussed by Anderson and Badawi [5]. The zero divisor graph of the ring of integers modulo 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 , , where they studied several graph properties and determined several graph invariants for . Further properties of the zero divisor graphs for the ring of Gaussian integers modulo are investigated in [12].
In this paper, we study the line graph of . 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 Eulerian, Hamiltonian, or planer. In Section 5, the chromatic and clique numbers of are found. While the diameter, the girth and the radius of are determined in Sections 6 and 7, respectively. Finally, the last two sections discuss the domination number of and as well as the independence and clique numbers of .
2. Preliminaries
The set of Gaussian integers is defined by .
A prime Gaussian integer is one of the following:(i) or ,(ii), where is a prime integer and ,(iii), , where , is a prime integer and .
It is clear that is a ring with addition and multiplications modulo . Throughout this paper, will be used to denote a prime integer which is congruent to 1 modulo 4, while will denote a prime integer which is congruent to 3 modulo 4. Since is finite, each element in is either a zero divisor or a unit. Also, since is a unique factorization domain, each integer can be uniquely factorized as where are Gaussian prime integers and are positive integers.
The zero divisor graph of a commutative ring denoted by , is the graph whose vertices is the set of all nonzero zero divisors of , and edge set . The line graph of a graph is defined to be the graph whose vertices are the edges of , with two vertices being adjacent if the corresponding edges share a vertex in . For , if , then this graph is one vertex, while if , then . Throughout this paper, all rings, , are commutative with unity.
For a connected graph , the distance, , between two vertices and is the minimum of the lengths of all paths of . The eccentricity of a vertex in is the maximum distance from to any vertex in . The radius of , , is the minimum eccentricity among the vertices of . The diameter of , , is the maximum eccentricity among the vertices of . The girth of , , is the length of a shortest cycle in . The center of is the set of all vertices of with eccentricity equal to the radius. If has a walk that traverses each edge exactly once goes through all vertices and ends at the starting vertex, then is called Eulerian. A graph is called Hamiltonian if there exists a cycle containing every vertex. The chromatic number of a graph , , is the minimum such that is -colorable (i.e., can be colored using different colors such that no two adjacent vertices have the same color). The clique number, , of a graph is the maximum order among the complete subgraphs of . A subset of the vertex set is said to be independent if no two vertices in this set are adjacent. The independence number of , , is the maximum cardinality of all independent sets in . A subset of the vertex set of a graph is a dominating set in if each vertex of , not in , is adjacent to at least one vertex of . The minimum cardinality of all dominating sets in , , is called the domination number of . Edge dominating sets are defined analogously. The minimum cardinality of all edge dominating sets in , , is called the edge domination number of . The minimum cardinality of all independent edge dominating sets, , is called the independence edge domination number of . The maximum vertex degree of a graph will be denoted by .
3. When Is Eulerian?
Now it is characterized when the line graph is Eulerian. Before proceeding, we prove the following lemma.
Lemma 3.1. (i) Every vertex of has even degree if and only if , or is a composite integer which is a product of distinct odd primes.
(ii) If , and , then has a vertex of odd degree and another of even degree.
(iii) Every vertex of has odd degree if and only if .
Proof. (i) Since the graph is Eulerian if and only if each vertex has an even degree by Theorem 29 of [11], the result holds.
(ii) Assume that is prime, and . Then we have three cases.
Case I (). Then deg and .Case II ( is an odd prime and ). By Theorem 23 of [11], has a vertex of degree , where and a vertex of degree , where .Case III ( and ). Since and divides , is odd and hence is even. Then by using part (i), the result holds.(iii) Let ,, and , where
Now if all are odd primes, then and if , then .
(⇐) Note that, . So, for every vertex in .
Since is Eulerian if and only if is even for every or is odd for every , [9], together with Lemma 3.1 and Theorem 26 of [11], the following theorem is obtained.
Theorem 3.2. (i) is Eulerian graph if and only if , or is a composite integer which is a product of distinct odd primes.
(ii) is Eulerian graph does not necessarily imply that is Eulerian.
4. When Is Hamiltonian or Planner?
First we determine which graphs, , are Hamiltonian. Before this paper comes to the light, a recent article by Abu Osba et al. [12] reached to similar results concerning Hamiltonian . However, we present our proof since it is simpler and shorter. The proof makes use of the following theorem.
Theorem 4.1 (see [4]). Let be a finite principal ideal ring, if is Hamiltonian, then it is either a complete graph or a complete bipartite graph.
Theorem 4.2. The graph is Hamiltonian if and only if or .
Proof. Since is a finite principal ideal ring, is a complete graph or a complete bipartite graph if is Hamiltonian. But the graph is complete if and only if , and it is complete bipartite if and only if or , where , [11]. On the other hand, a complete bipartite graph is Hamiltonian if and only if . So the result holds.
Note that is not Hamiltonian and hence the converse of Theorem 4.1 is not true.
Next, we move to the line graphs . Before proceeding, we present the following theorem.
Theorem 4.3.
(i) If is a graph of diameter at most 2 with , then is Hamiltonian, see [13].
(ii) The line graph of an Eulerian graph is both Hamiltonian and Eulerian, see [14].
If , , or , where , then . On the other hand, if or is a composite odd integer which is a product of distinct primes, then is Eulerian, [11]. Thus the following corollary is obtained.
Corollary 4.4.
(i) If , , or , where , then is Hamiltonian.
(ii) If is a composite odd integer which is a product of distinct primes, then is both Eulerian and Hamiltonian.
Now, we discuss planarity of the graph .
A graph 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 so that the line graph is planer.
Theorem 4.5 (see [15]). A nonempty graph has a planer line graph if and only if(i)G is planer,(ii),(iii)if , then is a cut vertex.
Recall that is planer if and only if or , [11]. But is not planer since . Therefore, we get the following theorem.
Theorem 4.6. The graph is never planer.
5. The Chromatic and Clique Numbers of
If is a finite ring, then , unless is complete graph of odd order, [4]. Note that, the only complete graph occurs when . However, in this case the order of the graph is which is even, so . Moreover, since the edge coloring of any graph leads to a vertex coloring of its line graph, we obtain . Clearly, . On the other hand, the line graph of has a complete subgraph of order . Thus . Observe that if or , then has a vertex which is adjacent to every other vertex in . While if , then . Thus . This leads to the following Theorem.
Theorem 5.1.
Finally, if , where and , then the clique number and the chromatic number for the graph is given by the following theorem.
Theorem 5.2. , where and , then
Proof.. The result follows by computing , since .
6. The Diameter of
Now, we will find the diameter of the line graph .
First, we will prove that when or .
Lemma 6.1. (i) If , then there are no , where are odd integers such that .
(ii) If then there are no where are relatively prime with , such that .
Proof.. (i) Assume that . Then and . Since are odd integers, , , , and for some . So . And + + + + , a contradiction.
(ii) Assume that . Then and . Since are relatively prime with , we have , , and , where . So
Multiplying (6.1) by and (6.2) by and adding gives . Then or . Since , . Therefore, , and hence , a contradiction.
So, we conclude the following.
Theorem 6.2. If or and , then .
Proof. (i) Suppose that . Then,(1) where are odd and or implies that ann,(2) where are odd and , then ann.
Moreover, if . Since .
(ii) Suppose that . Let and . Then ann. Moreover, , since implies that .
From Theorems 3.1 and 3.3 of [9], . In , where and , . So, .
Theorem 6.3. (i) If , where are two distinct primes and or , then .
(ii) If are two distinct primes and , then .
Proof. First note that diam, [9] and for with , .(i)Case I: If or where , then .Case II: If , then
(ii)Note that .
Theorem 6.4.
(i) If , where is prime and , then .
(ii) If , where , and , then .
(iii) If , where , , and , then .
(iv) If where are distinct primes and , then .
Proof. (i) Let and . Then .
(ii) Let . Then .
(iii) Let . Then .
(iv) Let . Then .
Theorem 6.5. (i) If are fields and , then .
(ii) If are finite rings and is not a field for some and , then .
(iii) If where , then .
Proof. (i) Let . Since are fields, or has exactly two components equal 0. W.L.O.G. let and . Since , or . Say , then . So, .
(ii) Suppose that is not a field. Let such that . Then .
(iii) Let , where and 0 otherwise, , where and 0 otherwise, , where and 0 otherwise and , where and 0 otherwise. Then .
Summarizing the above results, we get the following theorem.
Theorem 6.6. (i) if and only if or with .
(ii) otherwise.
7. The Girth and the Radius of
In this section, we give a complete characterization of the girth and the radius of .
Since for any commutative ring , is a tree if and only if or [9], is never a tree. On the other hand, if contains a cycle, then where equality holds only if , [9].
Consequently, the following result holds.
Theorem 7.1. .
Next, we prove that the radius of the line graph equals 2.
Since , [9] and for any graph , .
Lemma 7.2. If there exists a vertex with eccentricity 2, then
Proof. Note that, has no spanning star graph, since if such that and , then .
Theorem 7.3. If , , or , , then .
Proof. (1)If , then .(2)If ,, then for all .(3)If , then for all .
Theorem 7.4. If , where , or and , , then .
Proof. (1)If or , then for all .(2)If , then , , for all .
Summarizing the above results, we get the following.
Theorem 7.5. The radius of the line graph equals 2.
8. The Domination Number of
Pervious results concerning the domination number of 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 . 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 , but also gives a minimum dominating set of this graph. This dominating set, as we will see, reveals to have interesting properties.
Theorem 8.1 (see [16]). Let be a finite commutative ring with identity that is not an integral domain. If is not a star graph, then the domination number equals the number of distinct maximal ideals of .
Theorem 8.2. If , where and are distinct gaussian prime and are positive integers and or . Then , if is odd, and , if is even.
Proof. (I) If , then is the unique maximal ideal of .
If , then is the unique maximal ideal of .
If where , then and are the only distinct maximal ideals of .
If is odd then are the only maximal ideals of .
If is even then and where are the only distinct maximal ideals of . Finally, since is never a star graph [11], the result holds.
Proof. (II) We have two cases.
Case I: is odd. Then it is easy to see that is a dominating set of . To show that is a minimum dominating set, assume that is a minimum dominating set such that there is no , belongs to for some . Then . So, is a dominating set of , a contradiction.
Case II: is even. Then . Similar to case I, we can see that is a minimum dominating set of .
If a dominating set induces a complete graph, then, is called clique dominating set, the clique domination number is the cardinality of a minimum clique dominating set, and is denoted by , if every vertex in is adjacent to another vertex in , then is called total dominating set. The minimum cardinality of a total dominating set is called total domination number and is denoted by . For any graph , . Since the suggested dominating set, , for in the second proof of Theorem 8.2 induces a complete graph, then .
9. The Domination Number of
In this section we determine the domination number of when and is prime.
The study of the domination number of the line graph of leads to the study of edge or line domination number of , that is, . On the other hand, for any graph , , [17]. Further, if is the complete bipartite graph , then , thus we have the following.
Lemma 9.1. (i) .
(ii) , where .
Now, we study the domination number of the line graph of when is a power of a prime. The first theorem treats the case . Here we make use of the fact that , [12].
Theorem 9.2. For ,
Proof. For , let . Note that the sets form a partition to the vertices of . Let and . Then the set induces a complete subgraph of and the set form an independent set of it. And each vertex in is adjacent to each vertex in . has no other edges. Let be a dominating set of vertices for with minimum cardinality. Since, the set induces a complete subgraph of of order , then . On the other hand, since dominates all edges in the complete graph , also dominates every edge joining to , recall that forms an independent set and so .
The proof of Theorem 9.2. shows the set is an independent set with maximum cardinality in , while the set induces a complete subgraph with maximum order.
So, the following corollary is obtained.
Corollary 9.3. For , (i),(ii).
As another consequence to the proof of the preceding theorem, the following corollary, which gives the degree sequence for , is obtained.
Corollary 9.4. For , the graph has exactly vertices of degree if and vertices of degree if .
Proof. For each , where ,, so . And for each , where ,, so .
Furthermore, The proof of the above theorem shows that the eccentricity of is 1 and the eccentricity of any other vertex in is 2, since the vertex 2 is adjacent only to the vertex , and for any , , is a path of length 2. This aleads to the following corollary.
Corollary 9.5. The center of the graph is the set .
Next, we we find the domination number of the line graph when .
Lemma 9.6. (i) For , (1)If , then when , and when ,(2)If , then .
(ii) For , if , then .
Theorem 9.7. If , then if is even and if is odd.
Proof. Let , and be defined as given in Lemma 9.6. Clearly, the set induces a complete subgraph of with maximum order if is even and induces a complete subgraph of with maximum order if is odd. On other hand if , then form an independent set with maximum cardinality. Moreover, if a vertex belongs to the set , then is adjacent to every element in where and at the same time. has no other edges.
As a consequence of the proof of Theorem 9.7, we conclude the following.
Corollary 9.8. If , then (i) if is even and if is odd,(ii) if and if .
Corollary 9.9. Let , , and where . Then
Corollary 9.10. Let . Then(i)the eccentricity of each is 1 and the eccentricity of any other vertex is 2,(ii)the center of the graph is the set ,(iii)the radius of the graph equals 1,(iv)the diameter of the graph equals 2, for .
Finally, we find the domination number of the line graph when .
Recall that . Let, . Clearly, the sets and not both or 0, partition the vertices of .
Lemma 9.11. (i) For (1)if , then ,(2)if and , then ,
(ii) for :
(1)if , then ,(2)if , then ,
(iii) for :
If , and , then .
Theorem 9.12. Let and and be defined as given in Lemma 9.11, then if is even and if m is odd.
Proof. Using the same notations of Lemma 9.11. Note that the set induces a complete subgraph of , . Thus, any edge dominating set for must contain edges to dominate . If , the set induces a complete bipartite graph with bipartite sets and . This contributes edges in the dominating edge set for .
Edges joining vertices in to vertices in are covered by the same edge dominating sets for and . Moreover, vertices in and , where , are only adjacent to some vertices in and .
On the other hand, if , the set is an independent set. Fortunately, vertices in are only adjacent to vertices in . So, any edge dominating set for also dominates edges between and .
Now, for each , and , the set induces a complete bipartite graph with bipartite sets and . In order to dominate this collection of complete bipartite graphs induced by we need edges in the edge dominating set for . Fortunately, this dominating set with elements also dominates all edges in which are incident to any edge in this collection.
Finally, observe that if , then vertices in are only adjacent to some vertices in as well as in the collection of the complete bipartite graphs. The graph has no other edges.
The above proof shows that induces a complete graph in . In fact, is a complete subgraph with maximum order in case is even, while if is odd we can add one additional vertex of some , where either or , say , is while is greater than . On the other hand, the set is a maximum independent set of order , where .
Thus, using the same notation of Lemma 9.11 and the proof of the above theorem, we obtain the following corollary.
Corollary 9.13. If , then(i) if is even and if m is odd, for ,(ii), if , if , and , for .
Corollary 9.14. If , then(i) has eccentricity 3, while all other vertices has eccentricity 2,(ii)the center of the graph is the set , where both and ,(iii)the radius of the graph equals 2,(iv)the diameter of the graph equals 3.
Proof. (i) First, note that has no vertex of eccentricity 1, otherwise . Let and . If is adjacent to both and , then . So, , and hence, the eccentricity of each vertex in is 3. If are nonadjacent, then is adjacent to both vertices. Similarly, if are nonadjacent, then is adjacent to both vertices.