FACTOR GRAPH OF NON-COMMUTATIVE RING

Transcription

1 International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 9, Issue 6, November - December 2018, pp , Article ID: IJARET_09_06_019 Available online at ISSN Print: and ISSN Online: IAEME Publication FACTOR GRAPH OF NON-COMMUTATIVE RING Britto. A Assistant Professor, School of Maritime Studies, Vels Institute of Science, Technology & Advanced Studies (VISTAS), Thalambur, Chennai Gopinath T.B Assistant Professor, School of Maritime Studies, Vels Institute of Science, Technology & Advanced Studies (VISTAS), Thalambur, Chennai ABSTRACT In this paper we introduce and investigate the new type of graph called Factor Graph of non-commutative Ring with non-zero identity. We define the Factor Graph of Ring and for each commutative ring R we associate a simple graph F(R). We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of G then show that the entire Factor graphs are connected. Key Words: Factor Graph, Non-Commutative Ring, Simple Graph. Cite this Article: Britto. A and Gopinath T.B, Factor Graph of Non-Commutative Ring. International Journal of Advanced Research in Engineering and Technology, 9(6), 2018, pp INTRODUCTION Let R be a non-commutative Ring with identity 1 0. The Factor graph of R denote by F(R) is a simple graph whose vertices are labeled by the elements of R. Let a, b Z*(R), where Z*(R) be the set of all non-zero divisors of R (Z(R) - {0}) then there is an edge in the graph F(R) between the two vertices a and b if only if a is a proper factor of b (not the reverse). In this case we say that a and b are adjacent vertices. This paper we interplay between the algebraic properties of a Ring theory and the graph theoretic properties of its Factor graph. The Graphs of Commutative Ring G(R) was first introduced by I. Beck [1] when coloring a graph then it was continued and modified by D.F. Anderson et.al as Γ() the zero divisor graphs of commutative rings [2]. The zero divisor graph of non-commutative Ring was studied by S. Redmond [3]. In this paper we are going to introduce a new graph called Factor Graph and discuss the properties of the Factor Graph with non-commutative Ring and prove some theorems/ lemmas by different diagrams. Throughout this paper we denote R as a noncommutative Ring with unity and all non-commutative Ring are finite editor@iaeme.com

2 Britto. A and Gopinath T.B 2. DEFINITIONS We recall the definitions of Ring and Graph Theory with some examples to understand Factor graph of non-commutative Ring. A Ring is a set R with binary operations + and such that (R, +) is a monoid with identity element 0 and (s, ) is a monoid with identity element 1. In addition it is distributive with respect to operations + and. The non-commutative ring is a ring whose multiplication is not commutative i.e. there exists a and b in R with a b b a [5]. A graph is an ordered pair G = (V, E) comprising a set V of vertices, nodes or points together with a set E of edges, arcs or lines. In this paper the graph considered as undirected and simple. A graph is connected if there is a path between every pair of vertices of the graph F(R). The degree of the vertex is the number of vertices adjacent to the vertex. In contrast to the standard definition of Factor graph if any number is a Factor of itself; say r, then we allow an edge from r to itself in the Factor graph. Such an edge is called a loop. For any two vertices a and b in a graph F(R), the distance between a and b is denoted by d (a, b), is the length of the shortest path connecting a and b, if such a path exist, if no path exist between a and b then d(a, b) =. The diameter of the graph F(R) defined diam (F(R)) = sup {d(a, b): a and b are distinct vertices of F(G)}. A graph is complete if every vertex in the graph is adjacent to every other vertex in the graph and it is denoted by notation Kn. A cycle in a graph is a path of length at least 3 through distinct vertices which begins and ends at the same vertex. A cycle graph is an n-gon for some integer n 3. A graph is planar if it can be drawn in the plane with no crossing of edges. A star graph is a complete bipartite graph in which one of the partitioning subsets is a singleton set and it is denoted by K1,n. Length of shortest graph cycle in a graph is called a girth. Clique is a complete sub graph of F (G) largest possible size is referred as maximum clique [4]. 3. PROPERTIES OF F(R) To understand the Factor graph of non-commutative ring, the given examples which are useful to understand the theorems and propositions shown in fig.1. (a) And fig.1. (b). Figure.1. (a) Factor graphs of rings having integer modulo n ( Z ) editor@iaeme.com

3 Factor Graph of Non-Commutative Ring Figure.1. (b) Factor graphs of rings having cycle of length 3 and above. Theorem 1. Let R be a non-commutative ring. Then F(R) is connected and diam F(R) 1. Proof: Let x, y F(R), with x y. If x is a Factor of y, then d(x, y) = 1. Suppose that x is not a Factor of y. If xy is a Factor of x and y, then x xy y is a path of length two, and d(x, y) = 2. Suppose xy is not a Factor x but it is a Factor y. There exists an element a F(R) with a y such that a is a Factor of y. If a is a Factor of x, then x a y is a path of length two between x and y. If a is not a Factor of x, then x ax y is a path of length two between x and y. In either cases d(x, y) = 2. Thus we may suppose that neither x nor y is a Factor. Then there exist nonzero Factors a, b F(R) (not necessarily distinct) with a is a Factor of x and b is a Factor of y. If a is a Factor of b, then x a y is a path of length 2, and hence d(x, y) = 2. Thus we may assume a b. Consider the element ab. If a is a Factor of b, then x a b y is a path of length three, and hence d(x, y) = 3. If a is not a Factor of b, then x ab y is a path of length two and hence d(x, y) = 2. If we extend the process then we know the diameter of F(R) is greater than or equal to one. Theorem 2. If R be a non-commutative Ring then every Factor graph F(R) is Factorization. Proof: By the definition of Factorization Let R be a domain with 1, let a, b R and a 0. If there exist an element c R such that b = ac, we say that a divides b or a is a Factor of b and we denote this by a/b. From the definition c is also a Factor of b. So, in the Factor graph all the three Factors were connected. Hence this statement is proved. Theorem 3. Let R be a non-commutative Ring, then the graph F(R) has the cycle when n 9. Proof: It is true by the above example graphs of editor@iaeme.com

4 Britto. A and Gopinath T.B Theorem 4. Let R be a non-commutative Ring. Then F(R) is complete if and only if either 6. Proof: By theorem 1 it true., for n Theorem 5. Let R be a non-commutative ring, then the diameter of the Factor graph F(R) is the number of highest Factors of the elements of R. Proof: It is proved obviously by the example. Theorem 6. Let R[x] is a ring of Polynomial then F(R[x]) is connected with respect to its factors. Proof: Let ={ ,,,,, }. Let ()= =0 Let x = a be a root of f(x), then (x-a) is a factor of f(x). ()=( )(! +! + +! ) And Let"()=! +! + +! =0, then ()=( )"() Since (x-a) and "() are the factors of f(x). Now the Factor graph is It is a graph of F(R) and is connected. Let be an n-1 degree polynomial. Let x = b be a root of g(x), then (x-b) is a factor of g(x). Then "()=(!)(# +# $ $ + +# ) And let h()=# +# $ $ + +# be a polynomial of degree n-2. Now ()=( )(!)h() Hence the factor graph is editor@iaeme.com

5 Factor Graph of Non-Commutative Ring It is a graph of F(R) and is connected. Similarly we can define Factor graph of all factor of the Polynomial Ring R. Theorem 7. Let R[x] be non-commutative Ring of Polynomial and let complete graph for every n (Kn). Proof: Let = = = $ = Hence the Factor graph is &=(& '), then F(R[x]) is a It is a complete graph of 4 and it can be prove for every n. Theorem 8. Let R[x] is Ring of Polynomial then F(R[x]) is a regular graph of degree n editor@iaeme.com

6 Britto. A and Gopinath T.B Proof: It is proved from the above theorem. It is a complete graph of degree n-1. So it is a regular graph of degree n-1. Since every polynomial of x n have n-1 factors and all the factors were connected by an edge. Theorem 9 Let R[x] be Non-commutative Ring of Polynomial and let R[x} = (x ± a) n, for ( ) then F(R[x]) is a complete and regular graph. Proof: This theorem is true from the above two theorems. a) Corollary 1. If R is the ring in the set Z, with n 5, then F(R) contains a triangle. The girth of F(R) is greater than or equal to three. Proof: From the above example it is true. b) Corollary 2. If R is the ring in the set Z, with n 7, then F(R) contains a square. The girth of F(R) is greater than or equal to three. Proof: The result is true by the previous examples. c) Corollary 3. If R is the ring in the set Z, with n =2, then F(R) is an isolated vertex. Proof: The factor of 2 is itself. So there is one vertex which is 2. Corollary 4. If R is the ring in the set Z, F ( Z ) is a planar for all n 11. REFERENCES [1] I. Beck, Coloring of commutative rings J. Algebra, and VOL: 116, PP: , [2] D.F. Anderson, P.S. Livingston, The zero-divisor graph of commutative ring, J.Algebra VOL: 217, PP: , [3] S. Redmond, The zero-divisor graph of non-commutative ring. Int. J. of Comm. Rings 1(4): , [4] R. Diestel, Graph Theory, Springer-Verlag, New York, [5] T.Y. Lam, A First Course in Non-commutative Rings, Springer, New York, editor@iaeme.com