NEW CONCEPTS OF PRODUCT INTERVAL-VALUED FUZZY GRAPH

Abstract

In this paper, we introduce product interval-valued fuzzy graphs and prove several results which are analogous to interval-valued fuzzy graphs. We conclude by giving properties for a product interval-valued fuzzy graph.

1. Introduction

In 1965, Zadeh [22] introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. The theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, engineering, statics, graph theory, computer networks, decision making and automata theory. In 1975, Rosenfeld [8] introduced the concept of fuzzy graphs, and proposed another elaborated definition, including fuzzy vertex and fuzzy edges, and several fuzzy analogs of graph theoretic concepts such as paths, cycles, connectedness and etc. Zadeh [22] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy set [23] in which the values of the membership degrees are intervals of numbers instead of the number. Interval-valued fuzzy set provid a more adequate description of uncertainty than traditional fuzzy sets. It is therefore important to use interval-valued fuzzy sets in applications, such as fuzzy control. The first definition of interval-valued fuzzy graph was proposed by Akram and Dudek [1]. Rashmanlou et al. [9,10,11,12,13,14] studied bipolar fuzzy graphs, balanced interval-valued fuzzy graph, complete interval-valued fuzzy graphs and some properties of highly irregular interval-valued fuzzy graphs. Samanta and Pal [17,18,19,20,21] defined fuzzy tolerance graphs, fuzzy threshold graphs, fuzzy planar graphs, fuzzy k-competition graphs and p-competition fuzzy graphs and irregular bipolar fuzzy graphs. In this paper we develop the concept of product interval-valued fuzzy graphs of interval-valued fuzzy graphs, further investigate properties of product interval-valued fuzzy graphs. The definitions that we used in this paper are standard. For other notations, the readers are referred to [2,3,4,5,6,7,15,16].

 

2. Preliminaries

Definition 2.1. The interval-valued fuzzy set A in V is defined by A = {(x, [μA-(x), μA+(x)]) : x ∈ V}, where μA-(x) and μA+(x) are fuzzy subsets of V such that μA-(x) ≤ μA+(x) ∀x ∈ V.

For any two interval-valued sets and in V we define:

Definition 2.2. By an interval-valued fuzzy graph of a graph G* = (V, E) we mean a pair G = (A, B), where A = [μA-, μA+] is an interval-valued fuzzy set on V and B = [μB-, μB+] is an interval-valued fuzzy set on E, such that

Definition 2.3. Let G = (A, B) be an interval-valued fuzzy graph of a graph G* = (V, E). If μB-(xy) ≤ μA-(x) × μA-(y) and μB+(xy) ≤ μA+(x) × μA+(y), for all x, y ∈ V, then the interval-valued fuzzy graph G is called product interval-valued fuzzy graph of G*.

Remark 2.1. If G = (A, B) is a product interval-valued fuzzy graph, then since μA-(x) and μA-(y) are less than or equal to 1, it follows that μB-(xy) ≤ μA-(x) × μA-(y) ≤ μA-(x) ∧ μA-(y) and μB+(xy) ≤ μA+(x) × μA+(y) ≤ μA+(x) ∧ μA+(y), for all x, y ∈ V.

Thus every product interval-valued fuzzy graph is an interval-valued fuzzy graph.

Definition 2.4. A product interval-valued fuzzy graph G = (A, B) is said to be complete if μB-(xy) = μA-(x) × μA-(y) and μB+(xy) = μA+(x) × μA+(y), for all x, y ∈ V.

Proposition 2.5. Let G = (A, B) be a complete product interval-valued fuzzy graph where μA- and μA+ are normal. Then and for all x, y ∈ V in which for all positive integer and

Proof. We prove by method of induction on n. Let n ≤ 2, x, y ∈ V. We have

Since μA-(z)2 ≤ 1, for all z, (μA-(z) ≤ 1), Hence

Since μA- is normal, μA-(t) = 1, for some t ∈ V. Then

Therefore

Now from (1) and (2), we get Also

Since μA+(z)2 ≤ 1, for all z, [μA+(z) ≤1], Hence,

Since μA+ is normal, μA+(t) = 1, for some t ∈ V. Then

Hence,

From (3) and (4) we get Let and We will prove that and We have

Similarly, we get □

Definition 2.6. The complement of a product interval-valued fuzzy graph G = (A, B) is an interval-valued fuzzy graph Gc = (Ac, Bc) where Ac = A = [μA-, μA+] and is defined by

Remark 2.2. The complement of a product interval-valued fuzzy graph is denoted by Gc. It follows that G is a product interval-valued fuzzy graph. Throughout this paper suppose that G1 = (A1, B1) and G2 = (A2, B2) are product interval-valued fuzzy graph of respectively.

Definition 2.7. Let G1 = (A1, B1) and G2 = (A2, B2) be interval-valued fuzzy graph. The union G1 ∪ G2 = (A1 ∪ A2, B1 ∪ B2) is defined as follows:

Proposition 2.8. The union of two product interval-valued fuzzy graphs is a product interval-valued fuzzy graph.

Proof. Let G1 = (A1, B1) and G2 = (A2, B2) be product interval-valued fuzzy graph. We prove that G1 ∪ G2 is a product interval-valued fuzzy graph of the graph Let xy ∈ E1 ∩ E2. Then

If xy ∈ E1 and xy ∉ E2 then

Similarly if xy ∈ E2 and xy ∈ E1, then we get □

Proposition 2.9. Let be crisp graphs with Let A1, A2, B1 and B2 be interval-valued fuzzy subset of V1, V2, E1 and E2 respectively. Then, G1 ∪ G2 = (A1 ∪ A2, B1 ∪ B2) is a product interval-valued fuzzy graph of if and only if G1 = (A1, B1) and G2 = (A2, B2) are product interval-valued fuzzy graph of respectively.

Proof. Let G1∪G2 = (A1∪A2, B1∪B2) be an product interval-valued fuzzy graph of Let xy ∈ E1. Then xy ∉ E2 and x,y ∈ V1. Hence

Therefore G1 = (A1, B1) is a product interval-valued fuzzy graphs. Similarly, we can prove that G2 = (A2, B2) is a product interval-valued fuzzy graph. By proposition (2.8) we get the converse. □

Proposition 2.10. Let G1 = (A1, B1) and G2 = (A2, B2) be product interval-valued fuzzy graph of respectively, and let Then G1∪G2 is complete if and only if G1 and G2 are complete.

Proof. It is obvious. □

Example 2.11. Let be graphs such that V1 = {a,b,c}, E1 = {ab,bc,ac}, V2 = {a,b,d} and E2 = {ab,bd}.

Consider two interval-valued fuzzy graphs G1 = (A1, B1) and G2 = (A2, B2) defined by

We have

FIGURE 1.Union of G1 and G2 (G1 ∪ G2)

It is clear that G1 and G2 are complete, but G1 ∪ G2 is not.

Definition 2.12. The joint G1+G2 = (A1+A2, B1+B2) of two interval-valued fuzzy graphs G1 = (A1,B1) and G2 = (A2,B2) is defined as follows:

where E′ denote the set of all arcs joining the vertices V1 and V2.

Proposition 2.13. If G1 and G2 are product interval-valued fuzzy graph, then G1 + G2 is a product interval-valued fuzzy graph.

Proof. In view of Proposition (2.8) it is sufficient to verify when xy ∈ E′. In this case we have:

Example 2.14. In previous example we have

FIGURE 2.Join of G1 and G2 (G1 + G2)

It is clear that G1 and G2 are complete, but G1 + G2 is not complete.

Proposition 2.15. Let G1 and G2 be product interval-valued fuzzy graph such that Then, G1 + G2 is complete if and only if G1 and G2 are both complete.

Proof. Let G1 and G2 be complete and u, v ∈ V1. Then

If u, v ∈ V2 then we have the same argument as above. Now Suppose that u ∈ V1 and v ∈ V2. We get whereas Thus, Also whereas Therefore, Hence, G1 + G2 is complete.

Conversely, assume that G1 + G2 is complete. First we show that G1 is complete. Let u, v ∈ V1. Then

Since G1 + G2 is complete,

Now using (5),(6),(7) and (8) we get Therefore G1 is complete. Similarly, we may prove that G2 is complete. □

Proposition 2.16. Let G1 and G2 be product interval-valued fuzzy graph of respectively, such that Then,

Proof. Let u ∈ V1. Then, and

Hence Similarly we can prove that for all u ∈ V2. Now suppose that uv ∈ X1. Then u, v ∈ V1, and we have

Also we have Hence Similarly we can prove that for all uv ∈ X2.

Now assume that (u, v) ∈ X′. Then u ∈ V1 and v ∈ V2. Thus

Also since u ∈ V1 and v ∈ V2. Therefore □

Proposition 2.17. Let G1 and G2 be product interval-valued fuzzy graph. Then

Proof. Let u ∈ V1. Then, and Hence, for all u ∈ V1. Similarly we can prove when u ∈ V2. Now suppose that uv ∈ X1, then

If uv ∈ X2, then u, v ∈ V2, and hence

If uv ∈ X′, then u ∈ V1, and v ∈ V2. Hence,

Therefore Similarly, we get Let be two graphs whose vertex sets are V1 and V2, respectively. Consider a new graph whose vertex set is V1 × V2 and edge set is a subset of (V1 × V2) × (V1 × V2).

Let G1 and G2 be product interval-valued fuzzy graph of , respectively. If v1 ∈ V1 and v2 ∈ V2, we define:

Also, if u1, v1 ∈ V1 and u2, v2 ∈ V2, then we define:

So, is an interval-valued fuzzy sub set on V = V1 × V2 and is an interval-valued fuzzy subset of (V1 × V2) × (V1 × V2). In fact G = (A,B) is an interval-valued fuzzy graph of that is denoted by G1 × G2. □

Proposition 2.18. Let G1 and G2 be product interval-valued fuzzy graph. Then G1 × G2 is a product interval-valued fuzzy graph.

Proof. Let u1, v1 ∈ V1 and u2, v2 ∈ V2. Then we have

Similarly, we can prove that for all u1, v1 ∈ V1 and u2, v2 ∈ V2. This complete the proof. □

Definition 2.19. The product interval-valued fuzzy graph G1 × G2 is referred to as the multiplication of the product interval-valued fuzzy graphs G1 and G2.

Proposition 2.20. Let G1 and G2 be product interval-valued fuzzy graph. Then, G1 × G2 is complete if and only if both G1 and G2 are complete.

Proof. Let G1 and G2 be complete, u1, v1 ∈ V1 and u2, v2 ∈ V2. Then

Similarly, we can prove that if u1, v1 ∈ V1 and u2, v2 ∈ V2, then

Therefore G1 × G2 is complete. Conversely, let G1 × G2 be complete. We will prove that G1 and G2 both are complete. Suppose that G1 is not complete. Then, there exist u1, v1 ∈ V1 for which one of the following inequalities hold.

Assume that

Now by considering ((u1, u2), (v1, v2)) ∈ (V1 × V2) × (V1 × V2), we have

This is a contradiction, since G1 × G2 is complete. Similarly, if a contradiction can be obtained. Hence G1 is complete. By the same argument as above we can prove that G2 is complete. □

Proposition 2.21. Let V1 = {v11, v12, · · · , v1n} and V2 = {v21, v22, · · · , v2n} be the vertex sets of graphs G1 and G2, respectively . Further, let G = (A,B) be the multiplication of G1 and G2. Then the following equations have solutions in [0, 1].

(i) (i = 1, 2, · · ·, n, j = 1, 2, · · · , m),

(ii) (i, k = 1, 2, · · ·, n, j,l = 1, 2, · · ·, m),

(iii) (i = 1, 2, · · ·, n, j = 1, 2, · · · , m)

(iv) (i, k = 1, 2, · · ·, n, j,l = 1, 2, · · ·, m).

Proof. Let G be the multiplication of product interval-valued fuzzy graphs G1 and G2. Then Now we have

where If v1i, v1k ∈ V1 and v2j, v2k ∈ V2, then

where Therefore, the equations (i) and (ii) have solutions in [0, 1]. Similarly, by the same argument as above we can prove that the equations (iii) and (iv) have solutions. □

Theorem 2.22. Let G∗ be a product of two graphs Let G = (A,B) be a product interval-valued fuzzy graph of G∗ where are normal. Moreover, suppose that the following equations have solutions in [0, 1],

(i) (i = 1, 2, · · ·, n, j = 1, 2, · · · , m),

(ii) (i, k = 1, 2, · · ·, n, j,l = 1, 2, · · ·, m,)

(iii) (i = 1, 2, · · ·, n, j = 1, 2, · · · , m),

(iv) (i, k = 1, 2, · · ·, n, j,l = 1, 2, · · ·, m).

Then G is the multiplication of a product interval-valued fuzzy graph of and a product interval-valued fuzzy graph of

Proof. Define

We will prove the following.

(i) G1 = (A1,B1) is a product interval-valued fuzzy graph of

(ii) G2 = (A2,B2) is a product interval-valued fuzzy graph of

(iii)

(iv)

If v1i, v1k ∈ V1 then for all v2j, v2l ∈ V2 we have

Hence, for all j, l. Since is normal, for some p, q, a and b. Thus zpq × wab = 1, and so zpq = wab = 1 (Since zpq.wab ∈ [0, 1]). Replacing j by a and l by b, we get

Similarly, we can get This prove that G1 = (A1,B1) is a product interval-valued fuzzy graph of Similarly, we can prove that G2 = (A2,B2) is a product fuzzy graph of Now if v1i ∈ V1 and v2i ∈ V2, then

This prove that If v1i,v1k ∈ V1 and v2i,v2k ∈ V2, then

Thus, Similarly, we can prove that □

 

3. Application of related theorems

An interval-valued fuzzy set is an extension of Zadeh’s fuzzy set theory whose range of membership degree is [0, 1]. The interval-valued fuzzy graph is a generalized structure of a fuzzy graph which gives more precision, flexibility, and compatibility with a system when compared with the fuzzy graphs. The natural extension of the research work on interval-valued fuzzy graph is product interval-valued fuzzy graphs. Note that one of the most widely studied classes of interval-valued fuzzy graphs is product interval-valued fuzzy graph. They show up in many contexts. These results can be applied in database theory, geographical information system roughness in graphs, roughness in hyper- graphs, soft graphs, and soft hypergraphs. Fuzzy cognitive maps (FCMs) are used in science, engineering, and the social sciences to represent the causal structure of a body of knowledge (be it empirical knowledge, traditional knowledge, or a personal view); for some examples. An FCM of the type that we shall consider in this paper is described by a set of factors and causal relationships between pairs of factors. A factor can have a direct positive or direct negative impact (or both) on another factor or on itself. In addition, a numerical weight is assigned to each direct impact; these weights are usually taken to be in the interval [0, 1]. Graph-theoretic tools are used to analyze FCMs. In particular, algorithms for computing a transitive closure of the FCM, from which all, not just direct, impacts together with their weights can be read. Two models can be constructed in the probabilistic model, the absolute value of the weight of an impact is interpreted as the probability that the impact occurs, while in the fuzzy model, it is interpreted as the degree of truth. In both cases, the FCM is represented as an interval-valued fuzzy directed graph; the definition of the transitive closure, however, depends on the model. Here, product interval-valued fuzzy graphs are introduced to improve the solution of the problems. The problem of the probabilistic transitive closure of an interval-valued fuzzy directed graph is an interval-valued version of the network reliability problem called s, t-connectedness (for all pairs of vertices s and t). Some of these results mentioned in the paper will help the reduction-recovery algorithm, complete state enumeration, the basic inclusion-exclusion algorithm, and the boolean algebra approach. This adaptation is far from trivial, as care must be taken to generate not only directed paths, but rather all minimal directed walks, and to distinguish between positive and negative minimal directed walks.

 

4. Conclusions

Graph theory has several interesting applications in system analysis, operations research, computer applications, and economics. Since most of the time the aspects of graph problems are uncertain, it is nice to deal with these aspects via the methods of fuzzy systems. It is known that fuzzy graph theory has numerous applications in modern science and engineering, neural networks, expert systems, medical diagnosis, town planning and control theory. In this paper, we have introduced product interval-valued fuzzy graphs and proved several interesting results which are analogous to interval-valued fuzzy graphs. In our future work, we will focus on applications of product interval-valued fuzzy graphs in other sciences.