Abstract
Due to the absence of a neutral function, there are drawbacks to the existing definition of an intuitionistic fuzzy graph (Int-FG). In that definition of Int-FG, membership function and nonmembership function are involved. In this study, lower truncation (Low-T) and upper truncation (Up-T) are applied to picture fuzzy graphs (Pic-FGs). Pic-FG has an additional neutral membership function. Furthermore, Low-T and Up-T of subdivision of the picture fuzzy line graph (Pic-FLnG) are also discussed. The degree of an edge in both Up-T and Low-T of Pic-FG is discussed. Some related theorems related to the degree of Up-T and Low-T of Pic-FG are discussed.
1. Introduction
In 1965, Zadeh [1] considerably developed the fuzzy set (FS) idea as a generalization of the crisp set which deals with imprecise or vague data. In real-world situations, the theory of the crisp set may fail to deal with vague data. The true membership degree of FS lies in the closed interval [0, 1]. FS is widely used in the medical and biological sciences. To improve fuzzy theory, Atanassov [2] established the intuitionistic fuzzy set (Int-FS) as a generalization of FS. Int-FS is useful in various domains, having membership and nonmembership degrees whose sum does not exceed 1. It is observed that Int-FS cannot deal with neutrality. Cuong [3] proposed the Pic-FS, which gives more satisfactory results. A picture fuzzy set (Pic-FS) is an extension of Int-FS due to its neutrality degree as an extra term.
Graphs have a wide range of applications in real-world problems and are used in computer science, sociology, circuit analysis, network traffic routing, biological networks, operations research, and social networks. Graphs are excellent techniques for information transfer with features such as entity relationships. Nodes can be represented by objects, and edges can be represented by relationships. The graph cannot deal with vague situations. To fulfill this limitation, Kauffman [4] initiated the concept of FG. Mahapatra et al. [5] worked on the colouring of the COVID-19-affected region based on fuzzy directed graphs. Gani and Malarvizhi [6] discussed the concept of truncation in FG. Mahapatra et al. [7] investigated Radio FG. Nowadays, a few authors are working in fuzzy analysis [8–12]. Mahapatra et al. [13] studied the edge colouring of FGs.Shao et al. [14] discussed the application of the water supplier system in Int-FG. Ghani et al. [15] presented the subdivision and middle Int-FG. The edge degree and edge regular properties of FG truncations are discussed by Radha and Kumaravel [16]. Mahapatra et al. [17] studied extended neutrosophic planar graphs. Mahapatra et al. [18] worked on predicting links in social networks by using a neutrosophic graph to look at the way people connect with each other. Furthermore, Mahapatra et al. [19] worked on a new way of link prediction in social networks. The concept of Pic-FG is foundation of Zuo et al. [20] which is extension of Int-FG. Shoaib et al. [21] studied some properties of Pic-FG. The following are the major points of this article:(i)In this paper, we study the Low-T and Up-T of Pic-FG.(ii)Furthermore, Low-T and Up-T of subdivision Pic-FLnG are discussed.(iii)Pic-FG is the extension of Int-FG. In this study, we initiate the degree of an edge in truncations of Pic-FG.
The following is the structure of this paper:
We presented some basic definitions which will help to understand the paper in Section 2. In Section 3, we study the Low-T and Up-T of Pic-FG. In Section 4, we present the Up-T and Low-T of Pic-FG. We study the degree of an edge in truncations of Pic-FLnG in Section 5. At the end, we write conclusion and some future plans in Section 6.
The basic notations of this paper are shown in Table 1.
2. Preliminaries
Definition 1. [20] Let X be a nonempty universal set, then, FS is defined as
{ , , }.
Definition 2. [20] Let X be a nonempty universal set, then, Int-FS is defined as follows:
is called Int-FS, where and , and and satisfy the axiom
(for all b X).
Definition 3. [20] Let X be a nonempty universal set, then, Pic-FS is defined as:
is called Pic-FS, where , and and are known as degree of truth, neutral, and falsity membership of b, respectively.
, , and satisfy the axiomThe refusal membership degree = 1 − ( + + ).
Definition 4. [20] A Pic-FG , which is of crisp graph with (, , ) be a picture fuzzy subset on V and (, , ) be a picture fuzzy relation on V is defined as(i), , and , where 0 (b) + (b) + (b) 1b .(ii)The functions , , and :E [0, 1] are defined by
Example 1. Suppose a Pic-FG shown as in Figure 1 having four vertices z, , x, and y and four edges zy, , xz, and such that
be the picture fuzzy V-set and
be the picture fuzzy E-set.
3. Truncation and Subdivision Pic-FG
Definition 5. Let be a picture fuzzy subset of a set . Low-Ts and Up-Ts of at the level , , are the picture fuzzy subsets and defined by:whereConsider .
Let be Pic-FG with underlying crisp graph (Und-CG) . It is known as Low-T of the Pic-FG at level , where and can be a proper subset of and , respectively. Take and , then, is an Pic-FG with Und-CG . It is called as Up-T of the Pic-FG at level .
Theorem 1. For any level , , the Low-T of is equal as the subdivision of .
Proof. Suppose is a Pic-FG with its underlying V-set and crisp graph as .
We claim that V-set = V-set . Let b is a node in Pic-FG. Then, by using the definition of Low-T of , we haveBy using the definition of Sud(),
Case 1. Consider the node in , such that , , and .
Subcase 1. If b V, by equations (6)–(9), , , and . Hence, , , and . By the definition of , , if , , and , i.e., is vertex in such that
Subcase 2. If , let ; equations (6) and (7) , by equation (8), . Hence, . In as , that is, is an edge in which taken together as the nodes of , by Subcases 1 and 2, is node of if , and .
Case 2. Suppose the node in in such a way , , and ; equation (5) , i.e., as .
Subcase 3. Let ; equations (6) and (7) . Hence, . , (or) , by definition of is a node of , ; i.e., is node in such that .
Subcase 4. Consider , ; equations (6) and (7) . Hence, , i.e., . So, by definition of ; i.e., is edge in such that . Therefore, from Subcases 3 and 4 and by definition of is vertex of such that , so, by Cases 1 and 2, .
Claim 1. Now, our claim is E-set is equal E-set .
Take to be edge in . By using the definition of ,By using the definition of ,
Case 3. Let be such that .As , equation (12) , equation (12) , and equation (14) By definition of ,By equations (12), (14), and (16), we have,So, is an edge in and
Case 4. Suppose is an edge in be such that
Subcase 5. If lies on , equations (12) and (14)So, . Considering
Subcase 6. If does not lie on , . It yields, by using Subcases 5 and 6, if is an arc in such that , then, .If Hence, by Cases 3 and 4,. So, .
Theorem 2. For any level , .
4. Truncation and Pic-FLnG
Theorem 3. For any level set , .
Proof. Let is a Pic-FG with its underlying set and crisp graph as .
Case 5. Take by applying the definition of equations (6) and (7),Hence, .So, a node in gives an edge in which in turn in form of a node in .Similarly, it can be proved thatHence, .So, a node in gives an edge in which in turn in form of a node in .Similarly, it can be proven that
Case 6. Let be node in , where and they consists common node in .
Subcase 7. If .We know that by using equations (6) as union of and is nonempty, and have a common node, where . So, the respective and will be node in and union of and is nonempty. Therefore, will be arc in such that the membership value of is , indeterminate value of is , and the nonmembership value of is .
Subcase 8. LetSo, the respective and will not become node in , and will not become an arc in . So, by using Subcases 7 and 8,Similarly, it can be proven thatBy using equations (18), (21), and (27),
Theorem 4. For any level , .
5. Degree of an Edge in Truncations of Pic-FG
5.1. Degree of an Edge in Lower Truncation of Pic-FG
5.2. Degree of an Edge in Upper Truncation of Pic-FG
Theorem 5. Let be Pic-FG such that , , ba E, where . Then, for any ba, = .
Proof. Since
Theorem 6. Let be a Pic-FG such that , , and , baE, where , and are constants. Then, for any ba.
Proof. Let .Let .Let .Hence, = .
Let .Let .Let .Hence, .
6. Conclusion
Graph theory has many applications in other fields of mathematics. Int-FGs are the backbone of many real systems, such as scheduling, networks, and image segmentation. However, due to the missing neutrality function, Int-FG is limited to representing some systems. To improve real systems, Pic-FG works better. Pic-FG is a generalization of Int-FG due to the extra neutrality membership function. To improve the fuzzy theory, we have introduced the Up-T and Low-T of Pic-FG and Pic-FLnG. In this paper, the degree of an edge in truncations of Pic-FG is also discussed. We have noticed some limitations to this work. The focus of this research was only on Pic-FGs and their related network systems. It is not always possible to collect the real data. In the future, we will apply some operations to bipolar Pic-FG.
Data Availability
The data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The 4th author is thankful to University Savar, Dhaka, Bangladesh, for the unlimited support during this research.