Abstract
In this modern era, graph theory has become an integral part of science and technology. It has enormous applications in handling various design-based problems. In this study, we present a new approach that increases the ability of graph theory to deal with uncertain challenges. The spherical fuzzy graph addresses the uncertainty domain more broadly, going beyond classical fuzzy graphs other than generalized fuzzy graph types. The concept of the spherical fuzzy planarity value provides a new method to evaluate the edge intersection and order in the graph. The ability of the spherical fuzzy planar graph to model complex relationships can enable more precise and reliable network designs and analyses. In this work, the concepts of spherical fuzzy multigraph , spherical strong and weak edges, and the value of planarity for spherical fuzzy graphs are introduced. Moreover, the concept of the degree of planarity within the context of spherical fuzzy planar graphs and the notion of strong and weak faces are introduced. Additionally, we delve into the construction of spherical fuzzy dual graphs, which can be realized in cases where the fuzzy graph is planar or possesses a degree of planarity . This notion also serves as the foundation for certain basic theorems. We talk about some significant findings related to this subject. We discuss some major results linked to this topic. To show the worth and importance of our work, we also provide a real-world application.
1 Introduction
One of the most advanced fields of science is “graph theory,” which plays a vital role in the applications of other branches of science like chemistry, biology, physics, electrical engineering, computer science, discrete mathematics, astronomy, and operations research. Graph theory research has experienced significant advancements recently due to its diverse range of applications. It is also helpful in image segmentation, networking, data mining, structuring, organizing, communication, etc. For instance, a data set can be represented graphically in the form of a model, like a tree containing vertices and edges. Similarly, the concept of a graph can be used to organize network design. In many different types of graph structure applications, including design problems for electrical transmission lines, utility lines, subways, circuits, and trains, crossing through edges can be problematic. It is actually essential to run paths of communication at different levels for it to cross them. Although the positioning of the electrical cables is not extremely difficult, the cost of some types of lines can increase if subway tunnels are constructed beneath them. Specifically, circuits with a few layers in their structure are simpler to produce. Planar graphs become a framework for these applications. All the abovementioned applications utilize the concept of planar graphs. Crossing has certain benefits, such as saving space and being affordable, but it also has some disadvantages. In city road planning, due to crossing, there are increased chances of accidents because of the increasing rush of vehicles day by day. Moreover, the expenditure on crossing routes underground is high, but traffic jams are reduced on underground routes. In urban planning, it would be protective for human lives not to cross the routes. However, due to a lack of space, such crossing of routes is permitted. Generally, we use such linguistic terms as “congested, “very congested,” and “high congested”. The word congested has no definite meaning. All the abovementioned terms have some membership degrees. The choice of navigating between a congested road and a non-congested one is more favorable than navigating between two congested routes. In a fuzzy planar graph, the “congested” edges would represent strong connections between vertices, possibly indicating a high level of interaction or influence, while the “low congested” edges would represent weaker connections, perhaps suggesting a lower level of interaction or influence.
These days, science and technology deal with complicated models for which appropriate data are limited. To deal with such a phenomenon, we use mathematical models to tackle different kinds of systems that contain elements of uncertainty and vagueness. The generalization of the ordinary set theory, namely, fuzzy sets, is the foundation for handling such types of models. The concept of the fuzzy set, introduced by Lotif A. Zadeh [1] in 1965, revolutionized how we handle ambiguity and partial data in various fields. Numerous implications for fuzzy sets extend the range of investigation areas across various educational fields. In 1983, Atanassov presented the idea of an intuitionistic fuzzy set to overcome the lack of fuzzy sets as it provides a degree of truthfulness as well as degree of falseness with the constraint that . To deal with such types of models, which require more space, did not support. In order to address this requirement, Yager [2] introduced the concept of Pythagorean fuzzy sets , which extends the space by introducing additional limitations . People often have a variety of opinions such as Yes, Abstain, No, and Refusal; a picture fuzzy set has been supported, as initiated by Cuong [3, 4], which is the generalization of , which extends the space with the new limitations , where , , and denote the degree of truthfulness, degree of abstinence, and degree of falseness, respectively. Furthermore, represents the degree of refusal.
However, aspects such as being young, smart, tall, short, healthy, and successful in a certain field cannot be easily quantified. It is possible to express these qualitative and vague predicates by defining appropriate boundaries. To enlarge the space for uncertain and vague information, Gundogdu and Kahraman [5] expanded the concept of by introducing the notion of a spherical fuzzy set with the new limitations. , and . The notion of with various practical applications in decision-making problems was investigated by Ashraf et al. [6]. Ashraf et al. [7] likely extended the Dombi aggregation operators to the context of , providing a framework for aggregating information or making decisions in scenarios where uncertainty is represented using .
Based on Zadeh’s fuzzy relation [8], Kaufmann [9] introduced the concept of fuzzy graphs in 1973. Then, in fuzzy graphs, various graph-based theoretical concepts were initiated by Rosenfeld [10]. In his study of fuzzy graphs, Bhattacharya [11] made lots of remarkable perspectives about how they differ from classical graph theory. He proved that not all ideas in the field of fuzzy graphs have a direct equivalent or parallel in classical graph theory. In Mordeson and Nair [12], the idea of the complement of a fuzzy graph was introduced. A few operations on fuzzy graphs were also presented. Subsequently, the original fuzzy graph was redefined as the complement of the complement, following this change in the definition of complement. Nagoorgani and Malarvizhi [13] introduced the notion of isomorphism on fuzzy graphs. One of the most important tools of graphing is its dual graph. Abdul-Jabbar et al. [14] introduced the concept of fuzzy dual graphs. Shannon and Atanassov [15] put forward the concepts of intuitionistic fuzzy relations and intuitionistic fuzzy graphs. Several operations on intuitionistic fuzzy graphs were introduced by Parvathi et al. [16]. Akram et al. [17–20] proposed some additional advanced concepts: intuitionistic fuzzy hypergraphs, intuitionistic fuzzy cycles, strong intuitionistic fuzzy graphs, and intuitionistic fuzzy trees. Alshehri and Akram [21] gave an idea of intuitionistic fuzzy planar graphs. The concept of Pythagorean fuzzy graphs was introduced by Naz et al. [22] along with some applications. Naz and Akram [23] presented the idea of the Pythagorean fuzzy energy of . Moreover, some operations of were defined by Akram et al. [24]. Akram et al. [25] suggested many graphs in a Pythagorean fuzzy environment.
Akram et al. [26] developed the concept of a spherical fuzzy graph . In addition, some operators on , namely, symmetric difference and rejection, were defined. The term energy of defined by Yahya and Mohamed [27] and some bounds of energy of were extracted too. Yager [28] proposed the idea of the fuzzy multiset. Akram et al. [26] developed the concept of a . Recently, the idea of a fuzzy planar graph was introduced by Pal et al. [29] and Samanta et al. [30]. Furthermore, some properties were discussed too. Pramanik et al. [31] looked into a few unique planar fuzzy graphs. Moreover, planar fuzzy graphs have some extensions discussed in [32, 33]. Their ability to capture and represent uncertainty makes them a valuable tool in scenarios where traditional crisp graphs may fall short; refer [34–49].
The item that follows is the subscription to our suggested research project:
• This research project aims to introduce the concept of a based on a spherical fuzzy multiset.
•The concept of the degree of vertex in is defined and elaborated with an example.
•An idea of a strong edge in as well as of complete is initiated.
•Under a spherical fuzzy environment, planar graphs are discussed, and the planarity of is defined.
• We discuss the concept of duality in .
• Some results regarding as well as the planarity of are presented.
• Finally, to utilize an idea of in an MCDM problem.
2 Motivation
The motivation behind spherical fuzzy sets lies in their ability to provide a richer, more flexible, and accurate framework for handling uncertainty. By addressing the limitations of classical fuzzy sets and , open new avenues for research and application in various fields, ultimately leading to reliable and effective decision-making processes. Despite significant progress in the study of , there has been limited effort directed toward the exploration and development of . Inspired by the potential of fuzzy planar graphs and intuitionistic planar graphs, we developed the concept of . This paper is organized as follows.
In Section 3, some basic definitions are presented. In Section 4, we define the concept of , degree of the vertex in , strong edge in , complete , strength of the spherical fuzzy edge, planar spherical fuzzy graph, strong , and strong and weak spherical fuzzy faces of illustrated with examples. In addition, some results regarding planarity are also present. In Section 5, we introduce the notion of of and illustrate with the example. In Section 6, the application of is presented. In Section 8, we put our recommended task to its conclusion.
Some symbols and notations are used, which are presented in Table 2 along with their meanings.
| Symbol | Meaning |
|---|---|
| An underlying set of vertices | |
| Degree of truthfulness | |
| Degree of abstinence | |
| Degree of falseness | |
| Degree of refusal | |
| Spherical fuzzy multiset. | |
| Spherical fuzzy multiedge set. | |
| Intersecting value at the point | |
| Spherical fuzzy planarity value |
| Notation | Meaning |
|---|---|
| Degree of the vertex | |
| Spherical fuzzy graphs | |
| Spherical fuzzy multigraph | |
| Spherical fuzzy planar graph | |
| Spherical fuzzy face | |
| Spherical fuzzy dual graph |
3 Preliminaries
Definition 3.1[1] Let be an underlying set of vertices. A fuzzy set is characterized by a membership function and is defined as . The fuzzy binary relation is a fuzzy subset on given as . A fuzzy graph is a pair of mappings and such that for all , , where and denote the membership degrees of the vertex and of the edge in , respectively.
Example 3.2Given the set , the fuzzy set and fuzzy relation are defined as follows: the fuzzy set isThe fuzzy relation isThe graph is shown in Figure 1.
FIGURE 1
Definition 3.3[28] Let be a non-empty set of vertices. A fuzzy set is said to be a fuzzy multiset if the fuzzy set is characterized by a membership function named as count membership, such that , where is defined as the collection of every crisp multiset taken from the interval [0, 1]. The value of in is a crisp multiset taken from [0, 1] for all . Furthermore, the entries , , , …, in construct a decreasingly ordered sequence, i.e., , for all .
Definition 3.4Let be a mapping on an underlying set of vertices and be a fuzzy multiset of such that for all , where . Then, is called a fuzzy multigraph, where and denote the membership degrees of the vertex and of the edge in , respectively.
Example 3.5Let be a non-empty set. Then, a fuzzy multiset is given in Equation 1The fuzzy multigraph is shown in Figure 2. There are two edges between and which is called the fuzzy multigraph.
FIGURE 2
Definition 3.6[26] Let the underlying set of vertices be . A spherical fuzzy set on a universe is an object having a form aswhere [0,1] denote the degree of truthfulness of in , [0,1] denote the degree of abstinence of in , and [0, 1] denote the degree of falseness of in with the following conditionMoreover, is the degree of refusal of in for all . A spherical fuzzy relation on .
Example 3.7Let and for all be a spherical fuzzy set and spherical fuzzy edge set in Spherical fuzzy graph shown in Figure 3, and defined by Tables 1, 2, respectively.
FIGURE 3
TABLE 1
| 0.5 | 0.8 | 0.3 | 0.4 | |
| 0.4 | 0.2 | 0.6 | 0.7 | |
| 0.1 | 0.1 | 0.4 | 0.8 |
Spherical fuzzy set .
TABLE 2
| 0.5 | 0.3 | 0.3 | 0.4 | |
| 0.2 | 0.5 | 0.2 | 0.3 | |
| 0.1 | 0.6 | 0.3 | 0.7 |
Spherical fuzzy edge set .
Definition 3.8[26] Let the underlying set of vertices be . A spherical fuzzy multiset is characterized by functions named as “count truthness membership” of , “count abstinence membership” of , and “count falseness membership” of given by , , and , respectively, where is the set of all crisp multisets drawn from the unit interval [0, 1], such that, for every the truthfulness membership sequence and abstinence membership sequence denoted as and , respectively, construct decreasing sequences. Similarly, the terms of form a sequence represented as such that ; . An is denoted byThe orders of membership for truthfulness and abstinence are listed in descending order, but the sequence for falsehood may not follow the same order.
4 Spherical fuzzy planar graphs
Definition 4.1[26] Let the underlying set of vertices be and be a spherical fuzzy multiset on further, let be a spherical fuzzy multiset of such thatfor all . Then, is called . By noting that, between two vertices (say) and more than one edge may be present. Furthermore, denotes the degree of truthfulness, denotes the degree of abstinence, and denotes the degree of falseness of an edge and represents the number of edges between two vertices.
Example 4.2In Figure 4, a such that and . Let and for all be and defined by Table 3 and Table 4, respectively.
FIGURE 4
TABLE 3
| 0.3 | 0.2 | 0.3 | 0.4 | |
| 0.4 | 0.1 | 0.5 | 0.3 | |
| 0.5 | 0.3 | 0.2 | 0.6 |
.
TABLE 4
| 0.2 | 0.2 | 0.3 | 0.2 | 0.2 | 0.2 | 0.3 | 0.2 | |
| 0.1 | 0.3 | 0.2 | 0.3 | 0.1 | 0.1 | 0.2 | 0.3 | |
| 0.4 | 0.5 | 0.5 | 0.6 | 0.3 | 0.5 | 0.5 | 0.6 |
.
Definition 4.3Let be and be a spherical fuzzy multiedge set in . The degree of a vertex is represented by and is defined asfor all .
Example 4.4
In Example 4.2, the degrees of the vertices of
are computed as follows:
• deg=(0.9,0.9,2.0).
• deg=(0.6,0.3,1.2).
• deg =(0.9,0.9,1.9).
• deg=(1.2,1.1,2.7).
Definition 4.5Let a spherical fuzzy multiedge set be defined asin . A multiedge of is strong ifandfor all and for all , .
Example 4.6Consider a as shown in Figure 5, where and are and spherical fuzzy multiedge set are defined in Table 5 and in Table 6, respectively.By direct computations, we can easily summarize that in this these are , strong edges.
FIGURE 5
TABLE 5
| 0.2 | 0.3 | 0.1 | |
| 0.3 | 0.5 | 0.3 | |
| 0.4 | 0.6 | 0.5 |
.
TABLE 6
| 0.1 | 0.2 | 0.1 | 0.1 | 0.1 | |
| 0.2 | 0.2 | 0.2 | 0.2 | 0.3 | |
| 0.5 | 0.3 | 0.3 | 0.5 | 0.6 |
Spherical fuzzy multiedge set .
Definition 4 7Letbe a spherical fuzzy multiedge set in . A is complete iffor all i = 1, 2, …, k and for all
Example 4.8Let be a as shown in Figure 6, where and are and spherical fuzzy multiedge set defined in Table 7 and in Table 8, respectively.Using basic calculations, from Figure 6, it is clear that it is complete .
FIGURE 6
TABLE 7
| 0.3 | 0.2 | 0.4 | 0.2 | 0.2 | |
| 0.1 | 0.1 | 0.3 | 0.1 | 0.2 | |
| 0.5 | 0.4 | 0.6 | 0.5 | 0.5 |
.
TABLE 8
| 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | |
| 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |
| 0.5 | 0.5 | 0.5 | 0.6 | 0.5 | 0.5 | 0.6 | 0.6 |
Spherical fuzzy multiedge set .
Definition 4.9Strength of the spherical fuzzy edge is denoted byand can be calculated as given in Equation 2.where i = 1, 2, …, k and for all .
Example 4.10Let be a as shown in Figure 7, where and are and spherical fuzzy multiedge set defined in Table 9 and in Table 10, respectively.
From
Equation 2, the strength of the edges is computed as follows.
• For an edge , =(1, 0.5, 0.6).
• For an edge , =(0, 0.5, 0.3).
• For an edge , =(1, 1, 1).
• For an edge , =(1, 1, 0.8).
• For an edge , =(1, 1, 0.6).
FIGURE 7
TABLE 9
| 0.2 | 0.1 | 0.3 | 0.2 | |
| 0.4 | 0.2 | 0.4 | 0.4 | |
| 0.3 | 0.3 | 0.5 | 0.1 |
.
TABLE 10
| 0.1 | 0 | 0.1 | 0.1 | 0.1 | |
| 0.1 | 0.1 | 0.2 | 0.2 | 0.2 | |
| 0.2 | 0.1 | 0.3 | 0.4 | 0.2 |
Spherical fuzzy multiedge set .
Definition 4.11Let be a . A spherical fuzzy strong edge is an edge if 0.5, or , otherwise weak.
Definition 4.12Let be a and the spherical fuzzy multiedge set contains two edges and which are intersected at a point , where and are fixed integers. The intersecting value at the point is defined as given in Equation 3.If the number of intersecting points increases, then the value of the planarity decreases. Thus, for is inversely proportional to the value of spherical planarity.
Example 4.13Let be a as shown in Figure 8, where and are and spherical fuzzy multiedge set, which are defined in Table 11 and in Table 12, respectively, as below.The intersecting value at the point is given by Equation 4.For an edge , we can write from Equation 2 as follows:By substituting the values from Tables 9, 10, we get Equation 5.Similarly, for an edge ,By putting the values from Tables 9, 10, we get Equation 6.Substituting the values from Equations 8, 9 in Equation 7, we get the intersecting value as
FIGURE 8
TABLE 11
| 0.5 | 0.2 | 0.4 | 0.5 | |
| 0.2 | 0.4 | 0.3 | 0.2 | |
| 0.4 | 0.3 | 0.1 | 0.3 |
.
TABLE 12
| 0.1 | 0.2 | 0.3 | 0.1 | 0.3 | |
| 0.2 | 0.1 | 0.1 | 0.2 | 0.1 | |
| 0.2 | 0.3 | 0.3 | 0.2 | 0.2 |
Spherical fuzzy multiedge set .
Definition 4.14For , , , …, are intersecting points between the edges of ; is said to be having the value of spherical planarity defined asVividly, is bounded, and ; ; and . For certain geometrical representations of a if there is no intersecting points, then its value of spherical planarity is (1, 1, 1). Thus, the crisp graph of this becomes a crisp planar graph. If decreases and increases, the number of intersecting points between edges increases and the nature of spherical planarity decreases. Conversely, if increases and decreases, the number of intersecting points between edges decreases and the nature of spherical planarity increases. We summarize that every is a with a certain value of spherical planarity.
Example 4.15An is said to be a crisp graph where and . Then, , as shown in Figure 9.The intersecting value at the point is given byFor an edge , we can write from Equation 2 as follows:By substituting the values from Tables 13, 14, we getSimilarly, for an edge ,By putting the values from Tables 13, 14, we getSubstituting the values from Equations 8, 9 in Equation 7, we get intersecting value as
FIGURE 9
TABLE 13
| 0.7 | 0.6 | 0.8 | 0.5 | |
| 0.4 | 0.5 | 0.4 | 0.4 | |
| 0.3 | 0.4 | 0.6 | 0.7 |
.
TABLE 14
| 0.5 | 0.5 | 0.6 | 0.4 | 0.5 | 0.5 | |
| 0.3 | 0.3 | 0.4 | 0.3 | 0.3 | 0.3 | |
| 0.4 | 0.4 | 0.5 | 0.6 | 0.7 | 0.3 |
Spherical fuzzy multiedge set .
Let be complete . Then, the spherical planarity value of is defined by where , having a value less than 1, and where is the number of intersecting points for edges in .
Proof. For complete , we havefor all . Let ,…, be the intersecting points along the edges in . For any edge in complete and
In this way, for point the intersecting value at point along edges and , and hence , and , where . Then,
Here, represents the number of intersecting for edge in . So, p has a value less than 1.
Definition 4.17A is called strong if the spherical fuzzy planarity value of the graph is 0.5, 0.5, or 0.5.
For , with then spherical-valued strong edges in containing the number of intersecting value are at most 1.
Proof. Let be with where , and . Let us assume that contains two intersecting values, and , which correspond to two spherical, strongly valued edges. For a strong edge , , and . Accordingly, for two intersecting spherical-valued strong edges and .andthat is, , and , , and . Then, , and therefore, , and which is contradiction as . So, the intersecting value among spherical strong edges can never be 2. The level of planarity diminishes as the number of cutting spherical strong edges increases. Moreover, if the number of intersecting points of strong-valued strong edges is 1, then in this case, the level of planarity is assumed as . Accordingly, we found that spherical-valued strong edges in containing the number of intersecting point are at most 1.
Example 4.19Two are shown in Figures 10, 11. In Figure 10, a with one crossing among two strong edges that there is one point in where two strong edges and intersect or cross each other. The value of the spherical planarity of is (0.52,0.5,0.5). Hence, this is strong and the number of intersecting values at point is 1. In Figure 3.8, a is considered with two intersecting points among strong edges and . The value of the spherical planarity of this graph is (0.5, 0.5, 0.5). Hence, the is not strong. So, the graph has no intersecting point, and then the must be strong.
FIGURE 10
FIGURE 11
Let be a having a value of spherical fuzzy planarity . If then two spherical strong multi-valued edges in do not have any crossing between them.
Proof. A is called with spherical fuzzy planarity . Take the value of spherical fuzzy planarity where two spherical strong-valued edges intersect. For any spherical-valued strong edge , , and . For a minimum value of , , , , and . , and . Then, the value of , and . Hence, contains no intersecting point between spherical-valued strong edges.
The above theorem motivated to define the term strong planar graph.
Definition 4.21A is called strong if .
A containing complete or spherical graph is not strong .
Proof. The =(, ) is said to be complete corresponding to the crisp graph with vertices such that and the set B and values of , and are given in Equations 10-13.
From Theorem 4.16, the value of planarity for complete is the number of intersecting values for edges in . So, it has only one intersecting point that it can be desisted from . Then, . Hence, is not strong.
Definition 4.23Let be and be a rational number. An edge (say) is said to be the considerable edge if the following condition must be held.andOtherwise, it is not a considerable edge. For , a multiedge is said to be considerable edge if , , , for each edge in .
A is said to be a strong , where h be a considerable number. Then, the considerable edges in have at most intersecting points (here is the greatest integer not exceeding x).
Proof. Let be the value of spherical fuzzy planarity and . Let be a considerable edge; it is seen that and . So, and . In this case, , and . Let , ,…, be the intersecting points among the considerable edges. Let be the intersecting point between the considerable edges and . Then, and ., and
Since is a strong , we have,This implies thatand .This implies that
The crucial parameter of a is its face. The face of a is a region bounded by spherical fuzzy edges. Every is characterized by spherical fuzzy edges at its boundary. If all the edges in the boundary of a have membership values of truthfulness, abstinence, and falseness (1,1,1), respectively, it becomes a crisp face. If one of these edges is eliminated, will not exist. So, the existence of a depends on the minimum value of the strength of the spherical fuzzy edges in its boundary.
Definition 4.25Let be a andfor all . A of is a region bounded by the set of spherical fuzzy edges , of a geometric representation of . The truthfulness degree, abstinence degree, and falseness degree of the are defined as follows: respectively.
Definition 4.26A is called strong if its value of truthfulness and value of abstinence are greater than 0.5 and degree of falseness is less than 0.5; otherwise, the face is weak. Every has an infinite region, which is called outer . Other faces are called inner .
Example 4.27
Suppose
as shown in
Figure 12. The
has the following faces:
• is bounded by the edges , and .
• is bounded by the edges , and .
• is surrounded by the edges , and .
•Outer is surrounded by the edges , and .
FIGURE 12
5 Spherical fuzzy dual graph
The concept of duality is very useful in elaborating many models, like integrated circuits, drainage systems for basins, and others. A graph is planar if and only it has a dual graph. This means that for any planar graph, there exists a dual graph, and for any graph with dual graph, it must be planar. This concept works well to deal with a wide range of complicated and significant circumstances. By inspiring this concept, we are going to present the idea of a of . In a , vertices correspond to the strong of the , and between two vertices, every spherical fuzzy edge corresponds to every edge in the boundary between two faces of the . The formal definition is given below.
Definition 5.1Let be and spherical fuzzy multiedge setLet , , …, be the strong of . The of is a , where , and the vertex of is considered for face of . The values of truthfulness, abstinence, and falseness of vertex are given by the mapping : such that:andrespectively. Between two faces and of , there may exist more than one same edge. Hence, between two vertices and in there may exist more than one edge. Let , , and represent the degree of truthfulness, degree of abstinence, and degree of falseness of the path edge between and , respectively.
Example 5.2
Let the underlying set of vertices
, the
and the spherical fuzzy multiedge set
,
. Then, the
has the following faces.
• is bounded by the edges and .
• is bounded by the edges and .
• is bounded by the edges and .
• is bounded by the edges , , and .
• is surrounded by the edges , , and
• is bounded by the edges , , and
Algorithm 1
INPUT: 1. Let L represent a collection of electric connections, where L = , ,…,
2. The set of edges that connect the units L = is denoted by , and the set of intersecting points is denoted by .
3. Compute the strength of the edge = by the equation
4. Calculate the value of intersecting points.
5. Determine the value of planarity for .
FIGURE 13
6 Application
In real-world power distribution systems, maintaining efficient and reliable connections between transformers and households is crucial. Using the spherical fuzzy multigraph model allows us to better visualize and optimize these complex networks. By representing transformers as vertices and the wires connecting them to households as edges, we can account for uncertainties or varying degrees of connectivity in the system. This approach provides insights into how power is distributed, identifies potential vulnerabilities, and suggests improvements in efficiency and reliability. In the example of a transformer system represented as a spherical fuzzy planar graph, the transformer is considered a node and the electrical connections (wires) between components are treated as edges. The number of intersecting points (where wires cross) directly impacts the damage rate. The more intersections there are, the higher the likelihood of damage, as each crossing increases the risk of overheating or failure. Using the spherical fuzzy graph, we can model the uncertainty and degree of risk associated with each connection, helping minimize the damage rate by optimizing the placement of wires and reducing intersections. A vertex denotes each transformer , while the edge denotes each electric connection made between the transformer via a small wire as given in Figure 14.
FIGURE 14
The rate of decomposition grows with the number of crossings. , , , , , and are crossings among the pairs of wire , , , , , , respectively. The strength of the wire , , , , , , and . For crossings, the points of intersection are
Thus, the spherical fuzzy planarity value . When the planarity value is at minimum, it indicates that graph has a high number of edge crossings, which can make it harder to interpret and manipulate, as shown in Figure 15. The spherical fuzzy planarity value indicates a high number of edge crossings at the intersection, which increases complexity and the risk of damage. Reducing the number of these crossings can improve the system’s planarity and lower the destruction rate. To mitigate the risks of electric hazards, high-quality wires are recommended for critical connections, specifically between and , and and . These high-quality wires help minimize the potential for overheating and failures at the intersection. This model aids in monitoring and detecting destruction rates in real-time, enabling proactive management of the system and identification of high-risk areas, such as the intersection, to reduce potential hazards. By applying these strategies, the intersection’s risk can be minimized, improving both the safety and efficiency of the system. Through meticulous examination and the implementation of enhanced security measures, it is feasible to diminish the percentage of destruction, thereby safeguarding numerous lives.
FIGURE 15
6.1 Comparison with already existing methods
There are already existing methods, such as fuzzy planar graph and interval-valued planar graph, which can be used to determine if a discrete process is planar or if an interval representing a continuous process is planar, respectively. However, spherical planar graphs can be understood by combining principles from spherical fuzzy sets and planar graphs.
In planar graphs, interval-valued planar graphs are considered simultaneously [40]. However, we use a more versatile technique that involves three types of degree of membership such as truthfulness, falseness, and abstinence memberships, which gives more clear information about uncertain data.
7 Limitation and advantages
The proposed technique of spherical planar graph is restricted to undirected graphs only.
The spherical fuzzy planar graph extends traditional fuzzy graph theory by incorporating membership, non-membership, and hesitancy degrees, offering a more comprehensive representation of uncertainty. Compared to standard fuzzy graphs, it provides higher precision in modeling ambiguous or imprecise relationships. Its planar structure simplifies visualization and analysis of complex networks, reducing computational complexity. The approach supports dynamic adaptability, making it suitable for systems with evolving uncertainties. It also enhances decision-making capabilities by capturing partial truths more effectively than traditional fuzzy models. Furthermore, its multi-dimensional representation is particularly useful for handling conflicting and incomplete information in engineering and optimization problems. The proposed study, applying spherical fuzzy planar graphs to transformer systems, effectively models uncertain and dynamic connections between components like wires and nodes. It can address network reliability analysis, fault detection, and load balancing in power systems by capturing imprecise data and ambiguous relationships. This approach also aids in optimizing energy distribution and minimizing losses through flexible modeling of uncertainty. Additionally, it supports scalability and adaptability, making it suitable for evolving smart grid systems and renewable energy integration challenges.
8 Conclusion
Graph theory has enormous applications to problems in transportation, operations research, computer science, image capture, data mining, etc. Sometimes, to deal with uncertainty and vagueness in various network problems, various graph theoretical concepts are used based on Zadeh’s fuzzy relations. as the generalization of fuzzy graphs, intuitionistic fuzzy graphs, Pythagorean fuzzy graphs, and picture fuzzy graphs can be used to tackle various models based on real-world problems more effectively due to the enlargement of the space of uncertainty. In this article, the idea of spherical fuzzy graphs is utilized, as are the notions of , , and . The concept of spherical fuzzy planarity value provides a new method to evaluate the edge intersection. Furthermore, the spherical fuzzy planarity value has been defined based on strong, weak, and considerable edges. In addition, we present the idea of a . Planar graphs are very useful in designing circuits as well as various network models. In this article, we delve into potential synergies between and two distinct yet interconnected fields: neutral networks and geographical information system. By harnessing this strength of , we aim to enhance the capabilities of neutral networks in modeling complex relationships, while also extending the functionality of for spatial analysis and decision-making. We can deal with such problems using a planar graph. Planar graphs can be used to build circuits and road networks. We gave an example of transformer connections to check crossing between edges, so the planarity value can also be calculated. Our future plans regarding our research work are as follows:
Soft fuzzy planar graphs, vague planar graphs, hesitant , spherical fuzzy hypergraphs, and single-valued .
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
HG: conceptualization and writing–original draft. SH: project administration, supervision, and writing–original draft. Sadaf: investigation, methodology, and writing–original draft. AK: project administration and writing–review and editing. JS: data curation, supervision, and writing–review and editing.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was supported by the National Natural Science Foundation of China (No. 62407011).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
spherical fuzzy multigraph, spherical fuzzy planar graph, spherical fuzzy planarity value, spherical fuzzy dual graph, fuzzy graph theory
Citation
Guan H, Hameed S, Sadaf, Khan A and Shafi J (2025) A spherical fuzzy planar graph approach to optimize wire configuration in transformers. Front. Phys. 13:1413813. doi: 10.3389/fphy.2025.1413813
Received
07 April 2024
Accepted
27 January 2025
Published
19 March 2025
Volume
13 - 2025
Edited by
Igor Kondrashuk, University of Bío-Bío, Chile
Reviewed by
Sarfraz Ahmad, COMSATS University Islamabad, Lahore Campus, Pakistan
Ugur Kadak, Gazi University, Türkiye
Updates
Copyright
© 2025 Guan, Hameed, Sadaf, Khan and Shafi.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Saira Hameed, saira.math@pu.edu.pk
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.