Regular product vague graphs and product vague line graphs

Vague graph is a generalized structure of fuzzy graph which gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we introduced the notion of regular, totally regular product vague graphs, and product vague line graph. We proved that under some conditions regular and totally regular product vague graph becomes equivalent. Some properties of product vague line graph are investigated. We showed that a product vague graph is isomorphic to its corresponding product vague line graph under some conditions. Subjects: Advanced Mathematics; Combinatorics; Discrete Mathematics; Mathematical Logic; Mathematics & Statistics; Science

ABOUT THE AUTHORS Ganesh Ghorai is an assistant professor in the Department of Applied Mathematics, Vidyasagar University, India. His research interests include fuzzy sets, fuzzy graphs, and graph theory.
Madhumangal Pal is a professor of Applied Mathematics, Vidyasagar University, India. He received jointly with G. P. Bhattacherjee the "Computer Division Medal" from the Institute of Engineers (India) in 1996 for best research work. He received the Bharat Jyoti Award from International Friendship Society, New Delhi, in 2012. He has published more than 250 articles in international and national journals and 31 articles in edited books and in conference proceedings. His specializations include computational graph theory, genetic algorithms and parallel algorithms, fuzzy correlation and regression, fuzzy game theory, fuzzy matrices and fuzzy algebra. He is the editorin-chief of Journal of Physical Sciences and Annals of Pure and Applied Mathematics, and a member of the editorial boards of several other journals.

PUBLIC INTEREST STATEMENT
The theoretical concepts of graphs are highly utilized by computer science applications, especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing, and networking. The vague graphs are more flexible and compatible than fuzzy graphs due to the fact that they have many applications in networks. In this paper, the concept of vague sets is applied to define and study many important properties of regular, totally regular product vague graphs and product vague line graphs. fuzzy sets. Application of higher order fuzzy sets makes the solution-procedure more complex, but if the complexity on computation-time, computation-volume, or memory-space are not matters of concern, then we can achieve better results. In a fuzzy set, each element is associated with a pointvalue selected from the unit interval [0, 1], which is termed as the grade of membership in the set. Instead of using point-based membership as in fuzzy sets, interval-based membership is used in a vague set. The interval-based membership in vague sets is more expressive in capturing vagueness of data. There are some interesting features for handling vague data that are unique to vague sets. For example, vague sets allow for a more intuitive graphical representation of vague data, which facilitates significantly better analysis in data relationships, incompleteness, and similarity measures.

Preliminaries
In this section, we point out some basic definitions of graphs. The readers are encouraged to see these references (Balakrishnan, 1997;Harary, 1972;Mordeson & Nair, 2000) for further study.
Definition 2.1 Harary (1972) A graph is an ordered pair G * = (V, E), where V is the set of vertices of G * and E is the set of all edges of G * . Two vertices x and y in an undirected graph G * are said to be adjacent in G * if xy is an edge of G * . A simple graph is an undirected graph that has no loops and not more than one edge between any two different vertices.
We write xy ∈ E to mean (x, y) ∈ E, and if e = xy ∈ E, we say x and y are adjacent. Formally, given a graph G * = (V, E), two vertices x, y ∈ V are said to be neighbors or adjacent nodes, if xy ∈ E. The neighborhood of a vertex v in a graph G * is the induced subgraph of G * consisting of all vertices adjacent to v and all edges connecting two such vertices. The neighborhood of v is often denoted by An isomorphism of the graphs G * 1 and G * 2 is a bijection between the vertex sets of G * 1 and G * 2 such that any two vertices v 1 and v 2 of G * 1 are adjacent in G * 1 if and only if (v 1 ) and (v 2 ) are adjacent in G * 2 . If G * 1 and G * 2 are isomorphic, then we denote it by G * 1 ≅ G * 2 .
The line graph L(G * ) of a simple graph G * is a graph which represents the adjacentness between edges of G * . For a graph G * , its line graph L(G * ) is a graph such that: Definition 2.2 Gau and Buehrer (1993) A vague set on a non-empty set X is a pair (t A , f A ), where t A :X → [0, 1], f A :X → [0, 1] are true and false membership functions, respectively, such that In the above definition, t A (x) is considered as the lower bound for degree of membership of x in A (based on evidence for), and f A (x) is the lower bound for negation of membership of x in A (based on evidence against). Therefore, the degree of membership of x in the vague set A is characterized by So, a vague set is a special case of interval-valued sets studied by many mathematicians and applied in many branches of mathematics. Vague sets also have many appli- is called the vague value of x in A and is denoted by V A (x). We denote zero vague and unit vague value by 0 = [0, 0] and 1 = [1, 1], respectively.
It is worth to mention here that interval-valued fuzzy sets are not vague sets. In interval-valued fuzzy sets, an interval-valued membership value is assigned to each element of the universe considering the "evidence for x" only, without considering "evidence against x". In vague sets both are independently proposed by the decision-maker. This makes a major difference in the judgment about the grade of membership.
A vague relation is a further generalization of a fuzzy relation. (2009) Let X and Y be ordinary finite non-empty sets. We call a vague relation

Definition 2.3 Ramakrishna
Definition 2.5 Ramakrishna (2009) We call A the vague vertex set of G and B as the vague edge set of G, respectively.

Regular and totally regular product vague graphs
Throughout the paper, G * represents a crisp graph and G is a product vague graph of G * .  defined the product vague graphs as follows.
Here after we use xy ∈ E to denote (x, y) ∈ E throughout the paper.
Note that, every product vague graph is also a vague graph. Figure 1.
The product vague graph G of Figure 1 is not strong. uv). If all the vertices of G have the same open neighborhood degree (r 1 , r 2 ), then G is called (r 1 , r 2 )-regular product vague graph.
If each vertex of G has the same closed neighborhood degree (g 1 , g 2 ), then G is called (g 1 , g 2 )-totally regular product vague graph. Now, we give some examples which show that product vague graphs may be both regular and totally regular, neither totally regular nor regular and totally regular but not regular. In other words, there is no relation between regular and totally regular product vague graphs. First we give an example of a product vague graph which is both regular and totally regular (see Figure 2). Figure  2). Here, Hence, G is both (0.3,0.4)-regular and (0.8,0.8)-totally regular product vague graph. Now, we give an example of a product vague graph which is neither regular nor totally regular (see Figure 3). 0.8, 0.65). Hence, G is neither regular nor totally regular product vague graph.

Example 3.8 Let us consider a product vague graph
The following example shows that a product vague graph may be totally regular but not regular (see Figure 4). Similarly, we can give example of a product vague graph which is regular but not totally regular (see Figure 5).
We now state the following propositions without proof.   (i) G is (r 1 , r 2 )-regular vague graph, (ii) G is (g 1 , g 2 )-totally regular vague graph.
Proof Let us assume that A = (t A , f A ) is constant function. Therefore, let t A (v) = a 1 and f A (v) = a 2 for all v ∈ V, where a 1 , a 2 ∈ [0, 1].
We will now show that the statements (i) and (ii) are equivalent.
Conversely, let (i) and (ii) are equivalent. Suppose A is not constant function. This means there exist at least two vertices u, v ∈ V such that t A (u) ≠ t A (v) and f A (u) ≠ f A (v).
Let G be a (r 1 , r 2 )-regular product vague graph. Then, Hence, G is not totally regular which is a contradiction to the assumption that (i) and (ii) are equivalent. Therefore, A must be constant.
In a similar way, we can show that if A is not constant function, then G totally regular does not imply G is regular. ✷ Proposition 3.14 Let G = (V, A, B) be a product vague graph which is both regular and totally regular.
Proof Let G be a (r 1 , r 2 )-regular and (g 1 , g 2 )-totally regular product vague graph. Now, The converse of the Proposition 3.14 is not true always. For example, consider the product vague graph G = (V, A, B Figure 6). Here, A is constant but G is neither regular nor totally regular. G is (r 1 , r 2 )-regular implies that deg t (v 1 ) = r 1 and deg This means, deg t (v 1 ) = t B (e 1 ) + t B (e 2 ) = r 1 and deg f (v 1 ) = f B (e 1 ) + f B (e 2 ) = r 2 .

Product vague line graphs
In this section, we first define a product vague intersection graph of a product vague graph. Finally we define the product vague line graphs.

Definition 4.1 Let P(S) = (S, T) be an intersection graph of a simple graph G
be a product vague graph of G * . We define a product vague intersection graph P(G) = (A 1 , B 1 ) of P(S) as follows: (i) A 1 and B 1 are vague subsets of S and T, respectively, In other words, any product vague graph of P(S) is called a product vague intersection graph.
The following proposition is immediate.
Proposition 4.2 Let G = (V, A, B) be a product vague graph of G * = (V, E) and P(G) = (A 1 , B 1 ) be a product vague intersection graph of P(S). Then the following holds: (a) P(G) is a product vague graph of P(S), A, B) is a product vague graph we have by Definition 4.1, Hence, P(G) is a product vague graph.
(b) Let us define a mapping :V → S by (v i ) = S i for i = 1, 2, … , n. Then clearly is one to one map- Hence, is an isomorphism of G onto P(G), i.e. G ≅ P(G). ✷ This proposition shows that any product vague graph is isomorphic to a product vague intersection graph. The product vague line graph of a product vague graph is defined as below.  A 1 , B 1 ) of G is defined as follows: (i) A 1 and B 1 are vague subsets of Z and W, respectively, A, B) be a product vague graph of G * (see Figure 7).
Let A 1 and B 1 be vague subsets of Z and W, respectively. Then we have  Proof Since G = (V, A, B) is a product vague graph and L(G) = (A 1 , B 1 ) is a product vague line graph, Proof Follows from the Propositions 4.6 and 4.7. ✷ Definition 4.9 Let G 1 = (V 1 , A 1 , B 1 ) and G 2 = (V 2 , A 2 , B 2 ) … be two product vague graphs of the graphs G * 1 = (V 1 , E 1 ) and G * 2 = (V 2 , E 2 ), respectively. A homomorphism between G 1 and G 2 is a mapping A bijective homomorphism with the property that t B 1 (xy) = t B 2 ( (x) (y)) and f B 1 (xy) = f B 2 ( (x) (y)) for all xy ∈ V 2 1 , is called a (weak) line-isomorphism.
If is both (weak) vertex isomorphism and (weak) line-isomorphism, then is called a (weak) isomorphism of G 1 onto G 2 . If G 1 is isomorphic to G 2 , then we write G 1 ≅ G 2 .
Proof Obvious. ✷ (ii) If is a weak isomorphism of G onto L(G), then is an isomorphism.
Proof Suppose that is a weak isomorphism of G onto L(G). By Proposition 4.10, is an isomorphism of G * onto L(G * ). By Proposition 4.6, G * is a cycle, (by Harary, 1972), Theorem 8.2).