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Neutrosophic graphs of finite groups.

1 Introduction

Most of the real world problems in the fields of philosophy, physics, statistics, finance, robotics, design theory, coding theory, knot theory, engineering, and information science contain subtle uncertainty and inconsistent, which causes complexity and difficulty in solving these problems. Conventional methods failed to handle and estimate uncertainty in the real world problems with near tendency of the exact value. The determinacy of uncertainty in the real world problems have been great challenge for the scientific community, technological people, and quality control of products in the industry for several years. However, different models or methods were presented systematically to estimate the uncertainty of the problems by various incorporated computational systems and algebraic systems. To estimate the uncertainty in any system of the real world problems, first attempt was made by the Lotfi A Zadesh [1] with help of Fuzzy set theory in 1965. Fuzzy set theory is very powerful technique to deal and describe the behavior of the systems but it is very difficult to define exactly. Fuzzy set theory helps us to reduce the errors of failures in modeling and different fields of life. In order to define system exactly, by using Fuzzy set theory many authors were modified, developed and generalized the basic theories of classical algebra and modern algebra. Along with Fuzzy set theory there are other different theories have been study the properties of uncertainties in the real world problems, such as probability theory. intuitionistic Fuzzy set theory, rough set theory, paradoxist set theory [2-5], Finally, all above theories contributed to explained uncertainty and inconsistency up to certain extent in real world problems. None of the above theories were not studied the properties of indeterminacy of the real world problems in our daily life. To analyze and determine the existence of indeterminacy in various real world problems, the author Smarandache [6] introduced philosophical theory such as Neutrosophic theory in 1990.

Neutrosophic theory is a specific branch of philosophy, which investigates percentage of Truthfulness, falsehood and neutrality of the real world problem. It is a generalization of Fuzzy set theory and intuitionistic Fuzzy set theory. This theory is considered as complete representation of a mathematical model of a real world problem. Consequently, if uncertainty is involved in a problem we use Fuzzy set theory, and if indertminancy is involved in a problem we essential Neutrosophic theory.

Kandasamy and Smarandache [7] introduced the philosophical algebraic structures, in particular, Neutrosophic algebraic structures with illustrations and examples in 2006 and initiated the new way for the emergence of a new class of structures, namely, Neutrosophic groupoids, Neutrosophic groups, Neutrosophic rings etc. According to these authors, the Neutrosophic algebraic structures N(I) was a nice composition of indeterminate I and the elements of a given algebraic structure (N, x). In particular, the new algebraic structure (N(I), x) is called Neutrosophic algebraic structure which is generated by N and I.

In [8], Agboola and others have studied some properties of Neutrosophic group and subgroup. Neutrosophic group denoted by (N(G), x) and defined by N(G) = (d [union] I), where G is a group with respect to multiplication. These authors also shown that all Neutrosophic groups generated by the Neutrosophic element I and any group isomorphic to Klein 4-group are Lagrange Neutrosophic groups.

Recent research in Neutrosophic algebra has concerned developing a graphical representation of the elements of a given finite Neutrosophic set, and then graph theoretically developing and analyzing the depiction to research Neutrosophic algebraic conclusions about the finite Neutrosophic set. The most well-known of these models is the Neutrosophic graph of Neutrosophic set, first it was introduced by Kandasamy and Smarandache [9].

Recently, the authors Kandasamy and Smarandache in [9-10] have introduced Neutrosophic graphs, Neutrosophic edge graphs and Neutrosophic vertex graphs, respectively. If the edge values are from the set (G [union] I) they will termed as Neutrosophic graphs, and a Neutrosophic graph is a graph in which at least one edge is indeterminacy. Let V(G) be the set of all vertices of G. If the edge set E(G), where at least one of the edges of G is an indeterminate one. Then we call such graphs as a Neutrosophic edge graphs. Further, a Neutrosophic vertex graph [G.sub.N] is a graph G with finite non empty set [V.sub.N] = [V.sub.N] (G) of p--points where at least one of the point in [V.sub.N] (G) is indeterminate vertex. Here [V.sub.N](G) = V(G) + N, where V(G) are vertices of the graph G and N the non empty set of vertices which are indeterminate.

In the present paper, indeterminacy of the real world problems are expressed as mathematical model in the form of new algebraic structure (GI, x), and its properties are studied in second section, where G is finite group with respect to multiplication and I indeterminacy of the real world problems.

In the third section, to find the relation between G, I and N (G) we introduced Neutrosophic graph Ne(G, I) of the Neutrosophic group (N(G), x), by studying its important concrete properties of these graphs.

In the fourth section, we introduced basic Neutrosophic triangles in the graph Ne(G, I) and obtained a formula for enumerating basic Neutrosophic triangles in Ne(G, I) to understand the internal mutual relations between the elements in G, I and N (G).

In the last section, all finite isomorphic groups G and G' such that N(G) [congruent to] N(G') and Ne(G, I) [congruent to] Ne(G', I') are characterized with examples.

Throughout this paper, all groups are assumed to be finite multiplicative groups with identity e. Let N(G) be a Neutrosophic group generated by G and I. For classical theorems and notations in algebra and Neutrosophic algebra, the interest reader is refereed to [11] and [8].

Let X be a graph with vertex set V(X) and edge set E(X). The cardinality of V(X) and E(X) are denoted by [absolute value of V(X)] and [absolute value of (X)], which are order and size of X, respectively. If X is connected, then there exist a path between any two vertices in X. We denote by [K.sub.n] the complete graph of order n. Let u [member of] V(X).

Then degree of u, deg(u) in X is the number of edges incident at u. If deg(u) = 1 then the vertex u is called pendent. The girth of X is the length of smallest cycle in X. The girth of X is infinite if X has no cycle. Let d(x, y) be the length of the shortest path from two vertices x and y in X, and the diameter of X denoted by

Diam(X) = max{d(x, y):x, j [member of] V(X)}. For further details about graph theory the reader should see [12].

2 Basic Properties of Neutrosophic set and GI

This section will present a few basic concepts of Neutrosophic set and Neutrosophic group that will then be used repeatedly in further sections, and it will introduce a convenient notations. A few illustrations and examples will appear in later sections.

Neutrosophic set is a mathematical tool for handling real world problems involving imprecise, inconsistent data and indeterminacy; also it generalizes the concept of the classic set, fuzzy set, rough set etc. According to authors Vasantha Kandasamy and Smarandache, the Neutrosophic set is a nice composition of an algebraic set and indeterminate element of the real world problem.

Let N be a non-empty set and I be an indeterminate. Then the set N(I) = (N [union] I} is called a Neutrosophic set generated by N and I. If '*' is usual multiplication in N, then I has the following axioms.

1. 0 x I = 0

2. I x I = I = I x 1

3. [I.sup.2] = I

4. a x I = I x a, for every a [member of] N.

5. [I.sup.-1] does not exist.

For the definition, notation and basic properties of Neutrosophic group, we refer the reader to Agbool [8], As treated in [8], we shall denote the finite Neutrosophic group by N(G) for a group G.

Definition 2.1 Let G be any finite group with respect to multiplication. Then the set GI defined as GI = {gI: g [member of] G} = {Ig: g [member of] G}.

Definition 2.2 If a map f from a finite nonempty set S into a finite nonempty set S' is both one-one and onto then there exist a map g from S' into S that is also one-one and onto. In this case we say that the two sets are equivalent, and, abstractly speaking, these sets can be regarded as the same cardinality. We write S ~ S' whenever there is a one-one map of a set S onto S'.

Two finite rings R and R' are equivalent if there is a one-one correspondence between R and R'. We write R ~R'.

Definition 2.3 Let G be any finite group with respect to multiplication and let N(G) = (G [union] I}. Then (N(G), x) is called a Neutrosophic group generated by G and I under the binary operation '*' on G. The Neutrosophic group N(G) has the following properties.

1. N(G) is not a group.

2. G [subset] N(G).

3. GI [subset] N(G).

4. N(G) is a specific composition of G and I.

Lemma 2.4 Let G be any finite group with respect to multiplication and [I.sup.2] = I . Then G ~ GI. In particular, [absolute value of G] = [absolute value of GI].

Proof. For any finite group G, we have G [not equal to] GI and GI [subset not equal to] G. Now define a map f: G [right arrow] GI by the relation f (a) = a1 for every a [member of] I. Let a, b [member of] G.

Then

a = b [??] a - b = 0 [??] (a - b)I = 0I [??] aI = bI [??] f(a) = f(b). This shows that f is a well defined one-one function. Further, we have

Range(f) = {f(a) [member of] GI: a [member of] G}

= {aI [member of] GI: a [member of] G} = GI.

This show that for every aI [member of] GI at least one a [member of] G such that f (a) = aI.

Therefore, f: G [right arrow] GI is one-one correspondence and consequently a bijective function. Hence G ~ GI. Lemma 2.5 Let G be any finite group with respect to multiplication and let N(G) = (G [union] I}. Then the order of N(G) is 2[absolute value of G].

Proof: We have GI = {gI: g [member of] G}. Obviously, GI [subset not equal to] G and G [subset not equal to] GI but GI [subset] N(G). It is clear that N(G) is the disjoint union of G and GI. That is,

N(G) = G [union] GI and G [intersection] GI = [empty set].

Therefore, [absolute value of N(G)] = [absolute value of G] + [absolute value of G] = 2 [absolute value of G], since [absolute value of G] = [absolute value of GI].

Lemma 2.6 The set GI is not Neutrosophic group with respect to multiplication of group G.

Proof: It is obvious, since GI [not equal to] (G [union]).

Lemma 2.7 The elements in GI satisfies the following properties,

1. e x gI = gI

2. [(gI).sup.2] = [g.sup.2]I

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all positive integers n.

4. [{gI).sup.-1] does not exist, since [I.sup.-1] does not exist.

5. gI = g'I [??] g = g'.

Proof: Directly follows from the results of the group (N(G), x).

Theorem 2.8 The structure {GI, x) is a monoid under the operation(aI)(bI) = abI for all a, b in the group (G, x) and [I.sup.2] = I.

Proof: We know that GI = {gI: g [member of] G}.

Let aI, bI and cI be any three elements in GI. Then the binary operation (aI)(bI) = abI in (GI, x) satisfies the following axioms.

1. abI [member of] GI [??] (aI)(bI) [member of] GI.

2. [{aI){bI)]{cI) = [{ab)I\{cI) = [(ab)c)]I = [a(bc))]I = aI[(bI)(cI)]

3. Let e be the identity element in (G, x). Then

eI = I = Ie and I(aI) = [aI.sup.2] = aI = (aI)I.

Remark 2.9 The structure {GI, x) is never a group because [I.sup.-1] does not exist.

Here we obtain lower bounds and upper bounds of the order of the Neutrosophic group N(G). Moreover, these bounds are sharp.

Theorem 2.10 Let G be a finite group with respect to multiplication. Then,

1 [less than or equal to] [absolute value of G] [less than or equal to] n [??] 2 [less than or equal to] [absolute value of N(G)] [less than or equal to] 2n.

Proof. We have,

[absolute value of G] = 1 [??] G = {e} [??] N(G) = G[union]GI = {e, I} [??] [absolute value of N(G)] = 2. This is one extreme of the required inequality. For other extreme, by the Lemma [2.4],

[absolute value of G] > 1 [??] [absolute value of GI] > 1

[??] [absolute value of G] + [absolute value of GI] > 2 and [absolute value of G] + [absolute value of GI] is not odd [??][absolute value of G] + [absolute value of GI] is even. [??][absolute value of N (G)] = [absolute value of G] + [absolute value of GI] = 2n.

Hence, the theorem.

3 Basic Properties of Neutrosophic Graph

In this section, our aim is to introduce the notion and definition of Neutrosophic graph of finite Neutrosophic group with respect to multiplication and study on its basic and specific properties such as connectedness, completeness, bipartite, order, size, number of pendent vertices, girth and diameter.

Definition 3.1 A graph Ne (G, I) associated with Neutrosophic group (N(G), x) is undirected simple graph whose vertex set is N(G) and two vertices x and y in N(G) if and only if xy is either X or y. Theorem 3.2 For any group (G, x), the Neutrosophic graph Ne (G, I) is connected.

Proof: Let e be the identity element in G. Then e [member of] N(G), since G [subset] N(G). Further, xe = x, for every x [not equal to] e in N(G). It is clear that the vertex e is adjacent to all other vertices of the graph Ne(G, I).

Hence Ne(G, I) is connected.

Theorem 3.3 Let [absolute value of G] > 1. Then the graph has at least one cycle of length 3.

Proof: Since [absolute value of G] > 1 implies that [absolute value of N(G)] [greater than or equal to] 4. So there is at least one vertex gI of N(G) such that gI is adjacent to the vertices e and I in Ne(G, I), since eI = I, I(gI) = [gI.sup.2] = gI and (gI)e = geI = gI. Hence we have the cycle e - I - gI - e of length 3, where g [not equal to] e.

Example 3.4 Since

N([G.sub.10]) = {2, 4, 6, 8, 2I, 4I, 6I, 8I} is the Neutrosophic group of the group [G.sub.10] = (2, 4, 6, 8} with respect to multiplication modulo 10, where e = 6. The Neutrosophic graph Ne([G.sub.10], I) contains three cycles of length 3, which are listed below.

[C.sub.1]: 6 - I - 2I - 6, [C.sub.2]: 6 - I - 4I - 6, [C.sub.3] = 6 - I- 8I - 8.

Theorem 3.5 The Neutrosophic graph Ne{G, I) is complete if and only if [absolute value of G] = 1.

Proof: Necessity. Suppose that Ne (G, I) is complete. If possible assume that [absolute value of G] > 1, then [absolute value of N(G)] [greater than or equal to] 4. So without loss of generality we may assume that [absolute value of N(G)] = 4 and clearly the vertices e, g, I, gI [member of] V{Ne(G, I)). Therefore the vertex g is not adjacent to the vertex I in Ne(G, I), since gI [not equal to] g or I for each g [not equal to] e in G, this contradicts our assumption that Ne(G, I) is complete. It follows that [absolute value of N(G)] cannot be four. Further, if [absolute value of N(G)] > 4, then obviously we arrive a contradiction. So our assumption is wrong, and hence [absolute value of G] = 1.

Sufficient. Suppose that [absolute value of G] = 1. Then, trivially [absolute value of (G)] = 2. Therefore, Ne(G, I) [congruent to] [K.sub.2], since eI = I. Hence, Ne{G, I) is a complete graph.

Recall that [absolute value of V(Ne(G, I))] is the order and [absolute value of E{Ne(G, I))] is the size of the Neutrosophic graph Ne{G, I). But,

[absolute value of V{Ne(G, I))] = [absolute value of N(G)] = 2[absolute value of G] and the following theorem shows that the size of Ne(G, I).

Theorem 3.6 The size of Neutrosophic graph Ne(G, I)is 3[absolute value of G]-2.

Proof: By the definition of Neutrosophic graph, Ne(G, I) contains 2[([absolute value of G]-1).sup.2] non adjacent pairs. But the number of combinations of any two distinct pairs from N(G) is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence the total number of adjacent pairs in Ne(G, I) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.7 [11] The size of a simple complete graph of order n is 1/2n(n - 1).

Corollary 3.8 The Neutrosophic graph Ne{G, I), [absolute value of G] > 1 is never complete.

Proof: Suppose on contrary that Ne{G, I), [absolute value of G] > 1 is complete. Then, by the Theorem [3.7], the total number of edges in NE(G, I) is 1/2 (2[absolute value of G](2[absolute value of G]-1))= [absolute value of G](2[absolute value of G]-1), but in view of Theorem [3.6], we arrived a contradiction to the completeness of Ne{G, I).

Theorem 3.9 The graph Ne(G, I) has exactly [absolute value of G] - 1 pendent vertices.

Proof: Since N(G) = G[union]GI and G[intersection]GI = [empty set]. Let X [member of] N(G). Then either x [member of] G or x [member of] GI. Now consider the following cases on GI and G, respectively.

Case 1. If x [member of] GI, then X = gI for g [member of] G. But xI = (gI)I = [gI.sup.2] = gI = x and ex = egI = gI = X. This implies that the vertex x is adjacent to both the vertices e and 1 in N(G). Hence deg(x) [not equal to] 1 for every x [member of] GI.

Case 2. If x [member of] G, then ex = x, for every X [not equal to] e and egI = gI, for every gI [member of] GI. Therefore deg(e) = [absolute value of] N(G)] - 1 [not equal to] 1. Now show that deg(x) = 1, for every X [not equal to] e in G. Suppose, deg(x) > 1, for every X [not equal to] e in G. Then there exist another vertex y [not equal to] e in G such that either xy = X or y, this is not possible in G, because G is a finite multiplication group. Thus deg(x) = 1, for X [not equal to] e in G.

From case (1) and (2), we found the degree of each non identity vertex in G is 1. This shows that each and every non identity element in G is a pendent vertex in Ne(G, I). Hence, the total number of pendent vertices in Ne(G, I) is [absolute value of G] - 1.

The following result shows that Ne(G, I) is never a traversal graph.

Corollary 3.10 Let [absolute value of G] > 1. Then Ne(G, I) is never Eulerian and never Hamiltonian.

Proof. It is obvious from the Theorem [3.9].

Theorem 3.11 [11] A simple graph is bipartite if and only it does not have any odd cycle.

Theorem 3.12 The Neutrosophic graph Ne(G, I), [absolute value of G] > 1 is never bipartite.

Proof. Assume that [absolute value of G] > 1. Suppose, Ne(G, I) is a bipartite graph. Then there exist a bipartition (G, GI), since N(G) = G [union] GI and G [intersection] GI = [empty set]. But e [member of] G and I [member of] GI, where e [not equal to] I. So there exist at least one vertex gI in Ne(G, I) such that e - I - gI - e is an odd cycle of length 3 because eI = I, I(gI) = gl and (gI)e = gI.

This violates the condition of the Theorem [3.11].

Hence Ne(G, I) is not a bipartite graph.

Theorem 3.13 The girth of a Neutrosophic graph is 3. Proof. In view of Theorem [3.3], for [absolute value of G] > 1, we always have a cycle e - I - gI - e of length 3, for each g [not equal to] e in G, which is smallest in Ne(G, I).

This completes the proof.

Remark 3.14 Let G be a finite group with respect to multiplication. Then gir (Ne{G, I)) = [infinity] if [absolute value of G] = 1, since Ne (G, I) is acyclic graph if and only if [absolute value of G] = 1.

Theorem 3.15 Diam(Ne(G, I)) [less than or equal to] 2.

Proof. Let G be a finite group with respect to multiplication. Then we consider the following two cases.

Case 1 Suppose [absolute value of G] = 1. The graph Ne(G, I) [congruent to] [K.sub.2]. It follows that Ne(G, I) is complete, so diam (Ne(G, I)) = 1.

Case 2 Suppose [absolute value of G] > 1. Then the vertex e is adjacent to every vertex of Ne(G, I). However the vertex a1 is not adjacent to bI for all a [not equal to] b in G, so d(aI, bI) > 1. But in Ne(G, I), there always exist a path aI - I - bI, since (aI)I = aI and I(bI) = bI, which gives d(aI, bI) = 2, for every aI, bI [member of] N{G).

Hence, both the cases conclude that:

Diam (Ne(G, I)) [less than or equal to] 2.

4 Enumeration of basic Neutrosophic triangles in Ne (G, I)

Since Ne(G, I) is triangle free graph for [absolute value G] = 1, we will consider [absolute value of G] > 1 in this section.

Let us denote a triangle by (x, y, z) in Ne(G, I) with vertices X, y and z. Without loss of generality we may assume that our triangles (e, I, gI) have vertices e, I and gI, where g [not equal to] e in G. These triangles are called basic Neutrosophic triangles in Ne(G, I), which are defined as follows.

Definition 4.1 A triangle in the graph Ne(G, I) is said to be basic Neutrosophic if it has the common vertices e and I. The set of all basic Neutrosophic triangles in Ne(G, I) denoted by

[T.sub.eI] = {(e, I, gI): g [not equal to] e in G}.

A triangle (x, y, z) in Ne (G, I) is called non-basic Neutrosophic if (x, y, z) [not member of] [T.sub.eI].

The following short table illustrates some finite Neutrosophic graphs and their total number of basic Neutrosophic triangles.
Ne(G, I)              Ne                     Ne
                      ([Z.sup.*.sub.p], I)   ([C.sub.n], I)

[absolute value of    p - 2                  n - 1
[T.sub.eI]]

Ne(G, I)              Ne                Ne
                      ([G.sup.2p], I]   ([V.sub.4], I)

[absolute value of    p - 2             3
[T.sub.eI]]


where [Z.sup.*.sub.p] = [Z.sub.p] - {10} is a group with respect to multiplication modulo p, a prime, [C.sub.n] = {1, g, [g.sup.2],..., [g.sup.n-1]: = 1} is a cyclic group generated by g with respect to multiplication, [G.sub.2p] = {0, 2, 4,...,2(p - 1)} is a group with respect to multiplication modulo 2p and [V.sub.4] = {e, a, b, c: [a.sup.2] = [b.sup.2] = [c.sup.2] = e} is a Klein 4-group.

Before we continue, it is important to note that the multiplicative identity e may differ from group to group. However, for simplicity sake we will continue to notate that e = 1, and we leave it to reader to understand from context of the group for e.

The following results give information about enumeration of basic and non-basic Neutrosophic triangles in the graph Ne (G, I).

First we begin a lemma, which gives a formula for enumerating the number of Neutrosophic triangles in Ne(G, I) corresponding to fixed elements e and I in the Neutrosophic set N(G).

This is useful for finding the total number of non-basic Neutrosophic triangles in Ne(G, I). Theorem 4.2 Let [absolute value of G] > 1. Then the total number of basic Neutrosophic triangles in Ne(G, I) is [absolute value of [T.sub.eI]] = [absolute value of G] - 1. Proof. Since N(G) = G[union]GI and G[intersection]GI = [empty set]. It is clear that e [not equal to] I. For any aI [member of] GI, the traid (e, I, aI) [member of] [T.sub.eI] [??] (e, I), (e, aI), and (I, aI) are edges in Ne(G, I) [??] eI = I, e(aI) = aI, I(aI) = aI [??] I, aI [member of] GI, where a [not equal to] e in G.

That is, for fixed vertices e, I and for each aI [member of] GI, the traid (e, I, aI) exists in Ne(G, I). Further, for any vertex a [member of] G, the vertices e, I and a does not form a triangle in NE(G, I) because (I, a) is not an edge in Ne(G, I), since aI [not equal to] a or I for all a [not equal to] e. So that the total number of triangles having common verities e and I in Ne(G, I) is

[absolute value of [T.sub.eI]] = [absolute value of N(G)] - ([absolute value of G] + 1) = 2 [absolute value of G] - ([absolute value of G] + 1) = [absolute value of G] - 1.

Theorem 4.3 The total number of non-basic Neutrosophic triangles in Ne(G, I) is zero.

Proof. Suppose that two vertices either X, y or y, z or z, x are not equal to e and I.

Then the traid (x, y, z) is a non-basic triangle in Ne(G, I) [??] (x, y, z) [not member of] [T.sub.eI] [not member of] xy = x, yz = y and zx-z [not member of] either xyzx = x or yzxy = y or zxyz = z.

This is not possible in the Neutrosophic group N(G). Thus there is no any non-basic triangle in the graph Ne (G, I), and hence the total number of non-basic Neutrosophic triangles in Ne(G, I) is zero.

In view of Theorems [3.9] and [4.2], the following theorem is obvious.

Theorem 4.4 The total number of pendent vertices and basic Neutrosophic triangles in Ne(G, I) is same, which is equal to [absolute value of G] - 1.

5 Isomorphic properties of Neutrosophic groups and graphs

In this section we consider important concepts known as isomorphism of groups and Neutrosophic groups. But the notion of isomorphism is common to all aspects of modern algebra [14] and Neutrosophic algebra. An isomorphism of groups and Neutrosophic groups are maps which preserves operations and structures. More precisely we have the following definitions which we make for finite groups and Neutrosophic finite groups.

Definition.5.1 Two finite groups G and G' are said to be isomorphic if there is a one-one correspondence f: G [right arrow] G' such that f (ab) = f(a) f(b) for all a, b [member of] G and we write G [congruent to] G'.

Now we proceed on to define isomorphism of finite Neutrosophic groups with distinct indeterminate, which can be defined over distinct groups with same binary operation. We can establish two main results.

1. Two groups are isomorphic and their Neutrosophic groups are also isomorphic.

2. If two Neutrosophic groups are isomorphic, then their Neutrosophic graphs are also isomorphic.

Definition 5.2 Let (G, x) and (G', x) be two finite groups and let I [not equal to] I' be two indeterminates of two distinct real world problems. The Neutrosophic groups N(G) =(<G[union]I>, x) and N(G') =(<G'[union]I'>, x) are isomorphic if there exist a group isomorphism [empty set] from G onto G' such that [empty set](I) = I' and we write N(G) [congruent to] N(G').

Definition 5.3 [13] If there is a one-one mapping a [left and right arrow] a' of the elements of a group G onto those a group G' and if a [left and right arrow] a' and b [left and right arrow]b' implies ab [left and right arrow] a'b', then we say that G and G' are isomorphic and write G [congruent to] G'. If we put a' = f (a) and b' = f(b) for a, b [member of] G, then f:G [right arrow] G' is a bijection satisfying f(ab) = a'b' = f(a)f(b).

Lemma 5.4 G [congruent to] G' [??] N(G) [congruent to] N(G').

Proof. Necessity. Suppose G [congruent to] G'. Then there exist a group isomorphism [empty set] from G onto G' such that [empty set](a) = a' for every a [member of] G and a' [member of] G'. By the definition [12], the relation says that [empty set] sends ab onto a'b', where a' = [empty set](a) and b' = [empty set](b) are the elements of G' one-one corresponding to the elements a, b in G. We will prove that N(G) [congruent to] N(G'). For this we define a map f: N(G) [right arrow] N(G') by the relation f(G) = G'. f(I) = I' and f(GI) = G'I'.

Suppose x, y [member of] N(G).

Then either X, y [member of] G or X, y [member of] GI. Now consider the following two cases.

Case 1 Suppose x, y [member of] G.

Then X [left and right arrow] x' and y [left and right arrow] y'.

Trivially, f(x) = x' = [empty set]{x), for every X [member of] G and x' [member of] G', since G [congruent to] G'. Thus. N(G) [congruent to] N(G').

Case 2 Suppose X, y [member of] GI.

Then X = al and y = bI for a, b [member of] G. Obviously, f is one-one correspondence between N(G) and N(G'). since G [congruent to] G' and f (I) = I'. Further, f(xy) = f{(aI)(bI)) = f(abI) =a'b'I', since f(GI) = G'I' = (a'I')(b'I') = f(aI)f(bI) = f(x)f(y).

Thus f is a Neutrosophic group isomorphism from N(G) onto N(G'), and hence N(G) [congruent to] N(G').

Sufficiency. It is similar to necessity, because (G[union]I) [congruent to] (G' [union] I') implies that G [congruent to] G' and

GI [congruent to] G'I' under the mapping a [left and right arrow] a' and aI [left and right arrow] a'I', respectively.

Theorem 5.5 If G [congruent to] G', then Ne(G, I) [congruent to] Ne{G', I'), where I [not equal to] I'.

But converse is not true.

Proof. Suppose N(G) =(G[union]I) and N(G') = (G' [union] I') be two different Neutrosophic groups generated by G, I and G'. I', respectively.

Let [empty set] be an isomorphism from G onto G'. Then [empty set] is one-one correspondence between the graphs Ne{G, I) and Ne(G', I') under the relation [empty set](x) = x' for every x [member of] N(G) and x' [member of] N(G'). Further to show that [empty set] preserves the adjacency. For this let x and y be any two vertices of the graph Ne(G, I), then x, y [member of] N(G). This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, G and G' are adjacent in Ne(G', I'). similarly, [empty set] maps non-adjacent vertices to non-adjacent vertices. Thus, [empty set] is a Neutrosophic graph isomorphism from Ne(G, I) onto Ne(G', I'), that is, Ne(G, I) [congruent to] Ne(G', I').

The converse of the Theorem [5.5] is not true, in general. Let G = [V.sub.4] and let G' = [Z.sup.*.sub.5]. Clearly, Ne(G, I) [congruent to] Ne(G', I'), but [V.sub.4] is not isomorphic to [Z.sup.*.sub.5].

This is illustrated in the following figure.

Acknowledgments

The authors express their sincere thanks to Prof. L. Nagamuni Reddy and Prof. S. Vijaya Kumar Varma for his suggestions during the preparation of this paper. My sincere thanks also goes to Dr. B. Jaya Prakash Reddy.

References

[1] L.A. Zadeh, Fuzzy sets, Inforni and Control. 8 (1965), 338-353.

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[3] K. Atanassov. More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989), 37-46.

[4] Z. Pawlak: Rough sets, International Journal of Computer and Information Sciences, 11(1982) 341-356.

[5] Florentin Smarandache. A unifying field in logics. Neutrosophy: Neutrosophic probability, set and logic. Americal research press, Rehoboth (1999).

[6] Florentin Smarandache, Unmatter, mss. Archives Valcea (1980).

[7] W.B. Vasantha Kandasamy, Florentin Smarandache. Basic Neutrosophic Algebraic Structures and their Applications to Fuzzy and Neutrosophic Models, Hexis, Church Rock (2004).

[8] A.A.A. Agboola, A.O. Akwu, Y.T. Oyebo, Neutrosophic Groups and Neutrosophic Subgroups, Int. J. of Math. Comb. 3 (2012)1-9.

[9] W. B. Vasantha Kandasamy, Ilanthenral K, Florentin Smarandache. Neutrosophic Graphs: A New Dimension to Graph Theory. EuropaNova (2015).

[10] W.B. Vasantha Kandasamy, F. Smarandache, Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures, Hexis, Phoenix, Arizona (2006).

[11] David Joyner. Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical. Toys. 2nd edition, JHU Press (2008).

[12] I. Vitaly, I. Voloshin. Introduction to graph theory, Nova Science Publishers. Inc. New York (2009).

[13] Kiyosi Ito.: Encyclopaedic Dictionary of Mathematics, Volume 1. The MIT Press. 2nd edition (2000).

[14] Linda Gilbert. Elements of Modern Algebra. 8th edition Cengage Learning, (2014).

Received: January 23, 2017. Accepted: February 13,2017.

T.Chalapathi (1) and R. V M S S Kiran Kumar (2)

(1) Assistant Professor, Department of Mathematics, Sree Vidyanikethan Eng.College Tirapati,-517502, Andhra Pradesh, India.

Email: chalapathi.tekuri@gmail.com

(2) Research Scholar, Department of Mathematics, S.V.University, Tirapati,-517502, Andhra Pradesh, India. Email: kksaisiva@gmail.com

Caption: Figure Neutrosophic graphs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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