BIPARTITE GRAPH ASSOCIATED WITH ELEMENTS AND COSETS OF SUBRINGS OF FINITE RINGS

ABSTRACT


INTRODUCTION
The study of algebraic structures and graph theory has been considerable attention over the past several years. There are many important notions of such interplay, for instance, see [1]- [6].
Let Γ be a graph with vertex set (Γ) and edge set (Γ). For any edge , where and are in (Γ), we call and as the end points of . We say that Γ is connected if there is a path between every pair of vertices of Γ. The length of the smallest cycle contained in a graph Γ is called the girth of Γ. The distance between and in graph Γ is the length of the shortest path between and and denoted ( , ). The length of the longest path between two distinct vertices of connected graph Γ is called the diameter of Γ and it is denoted by diam(Γ). A bipartite graph is a graph whose vertices can be partitioned into two disjoint sets and such that every edge connects a vertex in and a vertex in . In the case that every vertex in is adjacent to every vertex in , we call the graph as a complete bipartite graph. A cycle that meets every vertex in a graph exactly once is called a Hamiltonian cycle and a graph that includes a Hamiltonian cycle is called a Hamiltonian graph. A planar graph is a graph that can be drawn in the plane without crossings but possibly at the end points. By a vertex cover, we mean a set of some vertices of a graph that contains at least one of the end points of every edge in the graph. Moreover, a vertex cover having the smallest possible number of vertices for a given graph is known as a minimum vertex cover of Γ, denoted (Γ). A matching of a graph Γ is a set of edges of Γ having no common end points. A matching of Γ is said to be maximum if Γ has no matching ′ with | ′| > | |.
Throughout this article, denotes any finite ring. For any subring of and for any ∈ , the sets = { | ∈ } and = { | ∈ } are called as a left coset and a right coset, respectively. The set of all subrings of is denoted by . For further definitions and theorems of graph theory, group theory, and matrices over commutative theory, we refer to [7]- [14], respectively. This article concerns on the bipartite graph associated with elements and cosets of subrings of that is motivated by [5]. In Section 2, we introduce the bipartite graph associated with elements and cosets of subrings of and we investigate some basic properties of the graph including connectivity, diameter, girth, and planarity. We also study the hamiltonicity property of this graph. Moreover, we present some relations between graph theory and matrices over ℤ through this graph and give some conjectures about minimum vertex cover and maximum matching. We close the result and discussion section by giving the diameter of the bipartite graph associated to quaternion ring for some cases with the definition of quaternion ring refers to [15].

RESULTS AND DISCUSSION
In this section, firstly, we give definition of a bipartite graph associated to elements and cosets of subrings of finite rings and some basic properties of the graph. Definition 1. Let be a finite ring. The bipartite graph associated to element and cosets of subrings of is a simple undirected graph Γ( ) with vertex set (Γ( )) = ∪ where is the set of all subrings of and two vertices ∈ and ∈ are adjacent if and only if = .

Proof.
We are going to prove that for every two arbitrary vertices, there exists a path of length at most 3 between them. We consider the following cases.
(i) For every 1 , 2 , ∈ , it is obvious that 1 and 2 have common neighbours {0} in . Hence, we have a path 1 − {0} − 2 of length 2. (ii) Similar to case (i), two arbitrary vertices 1 and 2 in have common neighbors 0 in and we have path 1 − 0 − 2 of length 2.
(iii) For every vertex ∈ and ∈ , by the above two cases, we have a path − {0} − 0 − 2 of length 3.
Recall that for any ring , the set ( ) = { ∈ |(∀ ∈ ) = } is called the center of . In the following theorem give a sufficient condition for the graph ( ) has girth 4, related to the center of . Proof. First of all, note that if for any bipartite graph has no odd cycle, then the girth of Γ( ) cannot be 3.
The following theorem gives a necessary condition for the graph ( ) to be Hamiltonian. Proof. Let Γ( ) be a hamiltonian graph. Then, we have a cycle that meets all vertices of Γ( ). Let the cycle begin from a vertex 1 ∈ . Then there exists 1 ∈ such that 1 − 1 , and also we will have 2 ∈ such that 1 − 1 − 2 . If we do this until | | steps, then we have | | ∈ and | | ∈ such that 1 − 1 − 2 − ⋯ − | | − | | − 1 . Since Γ( ) is Hamiltonian, then the cycle must meet all vertices in and . Therefore, we conclude that | | = | |.
In the following theorem we give a property of Γ( ) for a particular . Thus, Γ(ℤ ) it is complete bipartite graph which is planar as shown in Figure 1. Let be a commutative ring with unity and let ( ) be the ring of square matrices of size × over . From the theory of matrices over rings, we have following theorem.
Furthermore, in [10] Grigore Calugȃreanu has determined the subrings generated by some units matrices. Let be a commutative ring with unity. Let = {1, 2, . . . , } and let ⊆ × . Then ⊆ ( ) is defined as the set of matrices generated by { ∶ ( , ) ∈ } where is a unit matrix. In the following lemma it is given the necessary and sufficient condition for to be subring.
The minimum cover of the graph 2 (ℤ ) is conjectured as the following.
Theorem 11. (Kőnig's Theorem) Let ( , ) be a bipartite graph. The size of a maximum matching in equals the size of a minimum vertex cover of .
According to Conjecture 2 and Kőnig's Theorem then the size of maximum matching can be conjectured as given below.

Conjecture 3. T he size of maximum matching in 2 (ℤ ) is | ( 2 (ℤ )) |.
Now, we will discuss the bipartite graph associated to quaternion ring. Prior, we give the definition of the quaternion ring.

Definition 2. [15]
Let ℂ and ℝ denote the fields of the complex and real numbers, respectively. Let ℚ be a four dimensional vector space over ℝ with an ordered basis, denoted by , , and . A real quaternion, simply called quaternion is a vector = 0 + 1 + 2 + 3 ∈ ℚ with real coefficients 0 , 1 , 2 , 3 .
Real numbers and complex numbers can be thought of as quaternions in the natural way. Thus 0 + 1 + 2 + 3 can simply be written as 0 + 1 + 2 + 3 . Moreover, Adkin in [11] defined ℍ = ℚ(−1, −1; ℝ) as the quaternion ring with addition and a multiplication on ℍ are given in the following definition.

CONCLUSIONS
According to the description above we know that the bipartite graph associated to elements and cosets of subrings of any ring has no isolated vertex and has a diameter 2 if is nonzero and commutative. But however, the complete information on the structure of the graphs in general is not yet obtained. This could be interesting for future works.