DEGREE OF VERTICES IN VAGUE GRAPHS

Abstract

A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we define two new operation on vague graphs namely normal product and tensor product and study about the degree of a vertex in vague graphs which are obtained from two given vague graphs G₁ and G₂ using the operations cartesian product, composition, tensor product and normal product. These operations are highly utilized by computer science, geometry, algebra, number theory and operation research. In addition to the existing operations these properties will also be helpful to study large vague graph as a combination of small, vague graphs and to derive its properties from those of the smaller ones.

1. Introduction

Graphs and hypergraphs have been applied in a large number of problemsincluding cancer detection, robotics, human cardiac functions, networking and designing. It was Zadeh [25] who introduced fuzzy sets and fuzzy logic into mathematics to deal with problems of uncertainty. As most of the phenomena around us involve much of ambiguity and vagueness, fuzzy logic and fuzzy math-ematics have to play a crucial role in modeling real time systems with some level of uncertainty. The most important feature of a fuzzy set is that a fuzzy set A is a class of objects that satisfy a certain (or several) property. Gau and Buehrer [5] proposed the concept of vague set in 1993, by replacing the value of an ele-ment in a set with a subinterval of [0,1]. Namely, a true-membership function tv(x) and a false membership function fv(x) are used to describe the boundaries of the membership degree. The initial definition given by Kaufmann [6] of a fuzzy graph was based on the fuzzy relation proposed by Zadeh [26]. Later Rosenfeld [15] introduced the fuzzy analogue of several basic graph-theoretic concepts. Mordeson and Nair [7] defined the concept of complement of fuzzy graph and studied some operations on fuzzy graphs. Akram et al. [2,3,4] introduced vague hypergraphs, certain types of vague graphs and regularity in vague intersection graphs and vague line graphs . Ramakrishna [9] introduced the concept of vague graphs and studied some of their properties. Pal and Rashmanlou [8] studied irregular interval-valued fuzzy graphs. Also, they de-fined antipodal interval-valued fuzzy graphs [10], balanced interval-valued fuzzy graphs [11], some properties of highly irregular interval-valued fuzzy graphs [12] and a study on bipolar fuzzy graphs [14]. Rashmanlou and Yang Bae Jun investigated complete interval-valued fuzzy graphs [13]. Samanta and Pal defined fuzzy tolerance graphs [16], fuzzy threshold graphs [17], fuzzy planar graphs [18], fuzzy k-competition graphs and p-competition fuzzy graphs [19], irregular bipolar fuzzy graphs [20], fuzzy coloring of fuzzy graphs [21]. In this paper, we defined two new operation on vague graphs namely normal product and tensor product and studied about the degree of a vertex in vague graphs which are obtained from two given vague graphs G1 and G2 using the operations cartesian product, composition, tensor product and normal product. For further details, reader may look into [1,22,23,24].

 

2. Preliminaries

By a graph G∗ = (V,E), we mean a non-trivial, finite, connected and undirected graph without loops or multiple edges. Formally, given a graph G∗ = (V,E), two vertices x, y ∈ V are said to be neighbors, or adjacent nodes, if xy ∈ E. A fuzzy subset μ on a set X is a map μ : X → [0,1]. A fuzzy binary relation on X is a fuzzy subset μ on X × X. A fuzzy graph G is a pair of functions G = (σ, μ) where σ is a fuzzy subset of a non-empty set V and μ : V × V → [0,1] is a symmetric fuzzy relation on σ, i.e. μ(uv) ≤ σ(u) ∧ σ(v). The degree of a vertex u in fuzzy graph G is defined by dG(u) = Σu≠v μ(uv) = Σuv∈E μ(uv). The order of a fuzzy graph G is defined by O(G) = Σu∈V σ(u).

The main objective of this paper is to study of vague graph and this graph is based on the vague set defined below.

Definition 2.1 ([5]). A vague set on an ordinary finite non-empty set X is a pair (tA, fA), where tA : X → [0, 1], fA : X → [0, 1] are true and false membership functions, respectively such that 0 ≤ tA(x)+fA(x) ≤ 1, for all x ∈ X. Note that tA(x) is considered as the lower bound for degree of membership of x in A and fA(x) is the lower bound for negative of membership of x in A. So, the degree of membership of x in the vague set A is characterized by interval [tA(x), 1−fA(x)]. Let X and Y be ordinary finite non-empty sets. We call a vague relation to be a vague subset of X × Y , that is an expression R defined by

where tR : X × Y → [0, 1], fR : X × Y → [0, 1], which satisfies the condition 0 ≤ tR(x, y) + fR(x, y) ≤ 1, for all (x, y) ∈ X × Y .

Definition 2.2 ([9]). Let G∗ = (V,E) be a crisp graph. A pair G = (A,B) is called a vague graph on a crisp graph G∗ = (V,E), where A = (tA, fA) is a vague set on V and B = (tB, fB) is a vague set on E ⊆ V × V such that

for each edge xy ∈ E.

If G is a vague graph, then the order of G is defined and denoted as

and the size of G is

The open degree of a vertex u in a vague graph G = (A,B) is defined as d(u) = dt(u), df (u)) where If all the vertices have the same open neighborhood degree n, then G is called an n-regular vague graph.

Definition 2.3. Let G1 = (A1,B1) and G2 = (A2,B2) be two vague graphs of = (V1,E1) and = (V2,E2) respectively.

(1) The cartesian product G1×G2 of G1 and G2 is defined as pair (A1×A2,B1×B2) such that

(2) The composition G1 ◦ G2 of G1 and G2 is defined as pair (A1 ◦ A2,B1 ◦ B2) such that

where E◦ = E ∪ {(u1, u2)(v1, v2) | u1v1 ∈ E1, u2 ≠ v2}.

 

3. Degree of vertices in vague graphs

Operation in fuzzy graph is a great tool to consider large fuzzy graph as a combination of small fuzzy graphs and to derive its properties from those of the smaller ones. Also, they are conveniently used in many combinatorial applications. In various situations they present a suitable construction means. For example in partition theory we deal with complex objects. A typical such object is a fuzzy graph and a fuzzy hypergraph with large chromatic number that we do not know how to compute exactly the chromatic number of these graphs. In such cases, these operations have the main role in solving problems. Hence, in this section, at first we define two new operations on vague graphs namely normal product and tensor product. Then we study about the degree of a vertex in vague graphs which are obtained from two given vague graphs G1 and G2 using the operations cartesian product, composition, tensor product and normal product.

Definition 3.1. The normal product of two vague graphs Gi = (Ai,Bi) on Gi = (Vi,Ei), i = 1, 2 is defined as a vague graph (A1 ● A2,B1 ● B2) on G = (V,E) where V = V1 × V2 and E = {((u, u2)(u, v2)) | u ∈ V1, u2v2 ∈ E2} ∪ {((u1, z)(v1, z)) | u1v1 ∈ E1, z ∈ V2}∪{((u1, u2)(v1, v2)) | u1v1 ∈ E1, u2v2 ∈ E2} such that.

Definition 3.2. The tensor product of two vague graphs Gi = (Ai,Bi) on Gi = (Vi,Ei), i = 1, 2, is defined as a vague graph (A1 ⊗ A2,B1 ⊗ B2) on G = (V,E) where V = V1 × V2 and E = {(u1, u2), (v1, v2) | u1v1 ∈ E1, u2v2 ∈ E2} such that

Now, we derive degree of a vertex in the cartesian product. By the definition of cartesian product for any vertex (u1, u2) ∈ V1 × V2,

Theorem 3.3. Let G1 = (A1,B1) and G2 = (A2,B2) be two vague graphs. If tA1 ≥ tB2 , fA1 ≤ fB2 and tA2 ≥ tB1 , fA2 ≤ fB1 then

Proof. From the definition of a vertex in the cartesian product we have

Also we have

Hence, dG1×G2 (u1, u2) = dG1 (u1) + dG2 (u2). □

Example 3.4. Consider the vague graphs G1, G2 and G1 × G2 as follows.

Since tA1 ≥ tB2 , fA1 ≤ fB2 , tA2 ≥ tB1 and fA2 ≤ fB1 . By Theorem 3.3, we have

So, dG1×G2 (u1, u2) = (0.5, 1.2).

Hence, dG1×G2 (u1, v2) = (0.5, 1.2).

Similarly, we can find the degrees of all the vertices in G1 × G2. This can be verified in the Figure 1.

Figure 1.Cartesian product of G1 and G2

Now we calculate the degree of a vertex in composition. By the definition of composition for any vertex (u1, u2) ∈ V1 × V2 we have

Theorem 3.5. Let G1 = (A1,B1) and G2 = (A2,B2) be two vague graphs. If tA1 ≥ tB2 , fA1 ≤ fB2 , tA2 ≥ tB1 and fA2 ≤ fB1 , then dG1◦G2 (u1, u2) = |V2|dG1 (u1) + dG2 (u2) for all (u1, u2) ∈ V1 × V2.

Proof.

Similarly we can show that

Hence, dG1◦G2 (u1, u2) = dG2 (u2) + |V2|dG1 (u1). □

Example 3.6. Consider the vague graphs G1, G2 and G1 ◦ G2 as follows.

Here, tA1 ≥ tB2 , fA1 ≤ fB2 , tA2 ≥ tB1 and fA2 ≤ fB1 . By Theorem 3.5, we have

Therefore, dG1◦G2 (u1, u2) = (0.6, 2.1).

So, dG1◦G2 (u1, v2) = (0.6, 2.1).

In the same way, we can find the degree of all the vertices in G1 ◦ G2. This can be verified in the Figure 2.

Figure 2.Composition of G1 and G2

Degree of a vertex in the tensor product is as follows.

By definition of tensor product for any (u1, u2) ∈ V1 × V2 we have

Theorem 3.7. Let G1 = (A1,B1) and G2 = (A2,B2) be two vague graphs. If tB2 ≥tB1 and fB2 ≤fB1 then dG1⊗G2 (u1, u2) = dG1 (u1). Also, if tB1 ≥tB2 and fB1 ≤fB2 then dG1⊗⊗G2 (u1, u2) = dG2 (u2).

Proof. Let tB2 ≥ tB1 , fB2 ≤ fB1 then we have

Hence, dG1⊗G2 (u1, u2) = dG1 (u1). Similarly if tB1 ≥ tB2 and fB1 ≤ fB2 , then dG1⊗G2 (u1, u2) = dG2 (u2). □

Example 3.8. In this example we obtain the degree of vertices of G1 ⊗ G2 by Theorem 3.7.

Consider the vague graphs G1 and G2 in Figure 3. Here tB2 ≥ tB1 , fB2 ≤ fB1 . By Theorem 3.7 we have

So, dG1⊗G2 (u1, u2) = (0.2, 0.5) and dG1⊗G2 (v1, v2) = (0.2, 0.5). Similarly, we can find the degree of all the vertices in G1 ⊗ G2. This can be verified in the Figure 3.

Figure 3.Tensor product of G1 and G2

Finally, we derive the degree of a vertex in normal product. By the definition of normal product for any (u1, u2) ∈ V1 × V2 we have

Theorem 3.9. Let G1 = (A1,B1) and G2 = (A2,B2) be two vague graphs. If tA1 ≥ tB2 , fA1 ≤ fB2 , tA2 ≥ tB1 , fA2 ≤ fB1 , tB1 ≤ tB2 and fB1 ≥ fB2 then dG1●G2 (u1, u2) = |V2|dG1 (u1) + dG2 (u2).

Proof.

In the same way we can show that

Hence, dG1●G2 (u1, u2) = |V2|dG1 (u1) + dG2 (u2). □

Example 3.10. In this example we obtain the degree of vertices of G1 ● G2 by Theorem 3.9.

Consider the vague graphs G1 and G2 in Figure 4. Here tA1 ≥ tB2 , fA1 ≤ fB2 , tA2 ≥ tB1 , fA2 ≤ fB1 , tB1 ≤ tB2 and fB1 ≥ fB2 . So, by Theorem 3.9 we have

Therefore, dG1●G2 (u1, u2) = (0.6, 2).

So, dG1●G2 (u1, v2) = (0.6, 2).

Figure 4.Normal product of G1 and G2

Similarly, we can find the degree of all the vertices in G1 ● G2. This can be verified in the Figure 4.

 

4. Conclusion

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 found the degree of vertices in G1 × G2, G1 ◦ G2, G1 ⊗ G2 and G1 ● G2 in terms of the degree of vertices in G1 and G2 under some conditions and illustrated them through examples. This will be helpful when the graphs are very large and it can help us in studying various properties of cartesian product, composition, tensor product and normal product of two vague graphs.