#### Abstract

Fuzzy graph (FG) models embrace the ubiquity of existing in natural and man-made structures, specifically dynamic processes in physical, biological, and social systems. It is exceedingly difficult for an expert to model those problems based on a FG because of the inconsistent and indeterminate information inherent in real-life problems being often uncertain. Vague graph (VG) can deal with the uncertainty associated with the inconsistent and determinate information of any real-world problem, where FGs many fail to reveal satisfactory results. Regularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology, and economy; so, adjacency sequence (AS) and fundamental sequences (FS) of regular vague graphs (RVGs) are defined with examples. One essential and adequate prerequisite has been ascribed to a VG with maximum four vertices is that it should be regular based on the adjacency sequences concept. Likewise, it is described that if and its principal crisp graph (CG) are regular, then all the nodes do not have to have the similar AS. In the following, we obtain a characterization of vague detour (VD) g-eccentric node, and the concepts of vague detour g-boundary nodes and vague detour g-interior nodes in a VG are examined. Finally, an application of vague detour g-distance in transportation systems is given.

#### 1. Introduction

The graph concept stands as one of the most dominant and widely employed tools for the multiple real-world problem representation, modeling, and analyses. To represent the objects and the relations between them, the graph vertices and edges are applied, respectively. FG-models are beneficial mathematical tools for addressing the combinatorial problems in various fields involving research, optimization, algebra, computing, environmental science, and topology. Thanks to the natural existence of vagueness and ambiguity, fuzzy graphical models are strikingly better than graphical models. Rosenfeld [1] proposed the idea of FG in 1975. Akin to the set theory, the historical past of the FG theory is the fuzzy set theory developed by Zadeh [2] in 1965. The notion of vague set theory, the generalization of Zadeh’s fuzzy set theory, was introduced by Gau and Buehrer [3] in 1993. The VSs describe more possibilities than fuzzy sets. VS is more effective for the existence of the false membership degree. Kauffman [4] represented FGs based on Zadeh’s fuzzy relation [5, 6]. Akram et al. [7, 8] described several concepts and results of FGs. Samanta et al. [9, 10] represented fuzzy competition graphs and some remarks on bipolar fuzzy graphs. Gani et al. [11, 12] investigated on irregular and regular fuzzy graphs. Sunitha et al. [13, 14] studied complement of a fuzzy graph. VG notion was introduced by Ramakrishna in [15]. Borzooei et al. [16, 17] investigated new concepts of VGs. Gary Chartrand [18] discussed the concepts of detour center of a graph. The notion of detour number, detour set, detour nodes, and detour basis in a graph were established by Gary Chatrand, G. L. Johns, and P. Zhang [19]. Interior nodes and boundary nodes are discussed by G. Chatrand, D. Erwin, G. L. Johns, and P. Zhang [20]. Fuzzy detour g-distance was given by J. P. Linda and M. S. Sunitha [21]. Ghorai et al. [22, 23] defined detour g-interior nodes in bipolar fuzzy graphs and characterization of regular bipolar fuzzy graphs. The idea of strong arcs in FG was given by Bhutani and Rosenfeld [24], and types of arc in FG were given by Mathew and Sunitha [25]. The notion of bridge, trees, cycles, and end nodes were described by Rosenfeld [1]. Rosenfeld and Bhutani [24] represented the notion of g-distance in FG. Also, the notions of g-boundary node, g-interior node, and g-eccentric node were defined by Linda and Sunitha [26].

A VG is referred to as a generalized structure of an FG that conveys more exactness, adaptability, and compatibility to a system when coordinated with systems running on FGs. Also, a VG is able to concentrate on determining the uncertainly coupled with the inconsistent and indeterminate information of any real-world problem, where FGs may not lead to adequate results.

The concepts of regularity play an important part in both graph theory and application in the vague environment. The highly regular graphs characterization has also been applied to the question of heterogeneity, yet all of these fail to shed enough light on real-world situations; hence, in this paper, AS and FS of RVGs are defined with examples. One essential and adequate prerequisite has been ascribed to a VG with maximum four vertices is that it should be regular based on the AS concept. Detection of nodes on the network boundary is necessary for correct operation in many wireless applications. Also, nodes close to network boundary are often assumed to provide the best candidate for beacon nodes in virtual co-ordinate construction. Focusing on these applications, a research is carried out for boundary nodes and interior ones in vague graph. A characterization of vague detour g-eccentric node and the concepts of vague detour g-boundary nodes and vague detour g-interior nodes in a VG are examined. Finally, an application of vague detour g-distance in transportation systems is given. Recently, some research has been conducted by the authors in the continuation of previous works related to VGs, bipolar fuzzy graphs, and intuitionistic fuzzy graphs which are mentioned in [27–38]. Shoaib et al. [39] introduced new concepts of pythagorean fuzzy graphs.

#### 2. Preliminaries

A graph denotes a pair satisfying . The elements of and are the nodes and edges of the graph , correspondingly. An FG has the form of , where and as is defined by and and is a symmetric fuzzy relation on and denotes the minimum.

The FS of an FG ([40]) is described as follows:in which the elements are taken from the interval .

*Definition 1 (see [3]). *A VS is a pair on set that and are real-valued functions which can be defined on , so that , .

*Definition 2 (see [15]). *A pair is called to be a VG on a CG , where is a VS on and is a VS on so that and , for every edge .

*Definition 3 (see [16]). *The complement of a VG is a VG , where and are defined by the following:

*Definition 4 (see [17]). *Let be a VG.(i)Then, OND of is defined by , that and . If all the nodes of have the same OND , then, is called to be -regular.(ii)The CND of in is denoted as , where and . If every node of has equal closed neighborhood degree , then is called to be -totally regular.

*Definition 5 (see [17]). *Let be a VG. If and , , then is called a complete-VG and if and , then, is called a strong-VG of the CG .

*Definition 6 (see [7]). *Let be a line graph of a CG . Suppose be a VG of . Then, a vague line graph of is described as follows: and are vague subsets of and , respectively , and

*Definition 7 (see [16]). *Let be a VG and .(i)A path in is a sequence of distinct nodes so that , , and the length of the path is .(ii)If be a path of length between and , then, and are defined as and . is said to be the strength of connectedness between two nodes and in , where and . If and , then, the arc is said to be a strong arc. A path is strong path if all arcs on the path are strong.

*Definition 8 (see [16]). *A connected-VG is said to be a vague tree if has a vague spanning subgraph which is also a vague tree and for all arcs not in , and . The vague spanning subgraph of is a maximum spanning subgraph of if has no vague spanning subgraph different from contains .

Notations are shown in Table 1.

#### 3. New Concepts in Regular Vague Graphs

*Definition 9. *The AS of a node in a VG is described by , where and signify the sequence of TMVs and FMVs of the edges neighbor to a arranged in ascending order, respectively.

*Example 1. *Consider the VG as Figure 1. The AS of the nodes are

*Remark 1. *(i)The elements number in or is the degree of in .(ii)The sum of all elements in and the sum of all elements in is in VG , i.e.,

*Remark 2. *If is a RVG, then does not have to be regular and all the nodes do not need to have the same AS.

Now, we present an example that proves the correctness of the above remark.

*Example 2. *Consider the VG in Figure 2.

In this example, is a -RVG, but is not regular and the nodes in do not share the same AS.Hence, in a RVG, all the nodes do not need to have the similar AS.

*Remark 3. *If and are RGs, then all the nodes do not need to have the similar AS.

*Example 3. *Consider the VG as Figure 3.

Here, . So, is a RVG. Correspondingly, we observe that is regular. But .

*Definition 10. *The FS of a VG is defined as , whereThe elements of and are ordered in descending and ascending order, respectively. The node part of the FS is characterized as the FFS and the edge part of the FS is labeled as the SFS and described by and , respectively.

Theorem 1. *Let be a VG. If , then is a RVG iff , i.e., RG and all the nodes have the similar AS, being the number of nodes in .*

*Proof. *Suppose is a RG and all the nodes have the similar AS. Then, is a RVG.

On the contrary, let be a -RVG. If there is no edge among the nodes or if the number of nodes is one, then there is nothing to prove. Thus, we suppose that . Then, and . Consider the three cases for .

*Case 1. *If , then, have two nodes and . So, is 1-regular, and and have the same AS .

*Case 2. *If , then there are three nodes with edges in (see Figure 4). Let the edge-MV , , and be , , and , respectively.

Since is -RVG, , for . Hence, we haveSimplifying the above equations, we haveSo, each node has the similar AS, i.e., , for and is 2-RG. Here, is a cycle and each edge has the MV .

*Case 3. *If , then, there are four nodes of with edges (see Figure 5). Suppose the MV of the edges is , , , , , and , respectively. Since is -RVG, , for . Hence, we haveOn simplification, we obtainLet , , , and , , and .

Since that , at least one of the edges-MV should be nonzero. If all the edges-MVs are non-zero, then is 3-RG and every node of AS has the elements in ascending order.

If any two edges-MVs are nonzero, then is 2-RG and each node AS has those two nonzero MVs in the ascending order. If one edge-MVs is nonzero, then is 1-RG and each node AS has that nonzero MV.

Theorem 2. *Let be a -RVG of the -regular CG and every node has the same AS , and , . Then, the VLG of is regular iff either is constant or has exactly three values so that , , for and , , for .*

*Proof. *Let the VLG of be -regular. Let for some , where be any edge of . Then, every and is neighbor with edges with MVs . So, the node in is neighbor with edges with MVs , each appearing twice. So, , whereandThis holds for every edge of . Since is -regular, for each node set in . Thus, and , i.e.,By equation (13), we haveSimplifying equations (13)–(18), we have the following:If and , then is constant; otherwise, has three values so that and .

On the contrary, suppose is constant or have just three values so that , , for , and , , for . If is constant, let , . Then, and , . So, and , . Otherwise, suppose , and , . Next, and .

So, and , . Therefore, in all the cases, and , . Thus, for any node , we haveHence, is a -regular.

*Remark 4. *The RVG complement does not have to be regular. This can be observed in the following example.

*Example 4. *Consider the VG as Figure 6.

Here, . So, is -regular. But in its complement , , , .

So, is not regular.

#### 4. Vague Detour -Distance and Vague Detour -Periphery

*Definition 11. *The length of a strong path between and in a CVG is said to be a vague detour (VD) -distance if there is no other strong path longer than between and , and we show this by . Any strong path whose length is is named a vague -detour.

*Example 5. *Let be a CVG of the graph , where and . For the VG in Figure 7, it is seen that all arcs except and are strong arcs and the VD -distance of two nodes are given as follows:

*Definition 12. *The length of any smallest strong path from to is named the vague geodesic distance, described by .

The VD -eccentricity for a node is , . The set of all VD -eccentricity nodes of , described by . The VD -radius of , denoted by , is defined as min , . If , then the node is the VD -central node of . The VD -diameter of , denoted by , is defined as , . If , then the node is named the VD -peripheral node of .

*Example 6. *For the connected-VG in Figure 7, , , , , , and , .

*Definition 13. *The vague subgraph of the VG , whose nodes are only the VD -peripheral nodes is named a VD -periphery of , and it is denoted by .

*Definition 14. *If every node of a CVG is a VD -eccentric node, then is said to be a VD -eccentric vague graph. The vague subgraph of formed by the set of all vague -eccentric nodes of is named a VD -eccentric vague subgraph of , it is described by

*Example 7. *For the VG of Figure 8, all nodes , and are VD -periphery nodes since , . Also, we have

Theorem 3. *A VG is a VD -self-centered if and only if each node of is a VD -eccentric.*

*Proof. *Assume is a VD -self centered graph and let be a node in . Let . So, . Since is a VD -self centered vague graph, , and this implies that . So, is a VD -eccentric vertex of . Conversely, let each vertex of is a VD -eccentric node. If possible, suppose be not VD -self-centered vague graph. Then, , and there exists a node so that . Also, let Let be a VD in . So, there must have a node on for which the node is not a VD -eccentric node of . Also, cannot be a VD -eccentric node of each other node. Again, if be a VD -eccentric node of a node (say), it means . Then, there exists an extension of a vague -detour up to or up to . But this contradicts the facts that . Therefore, and is a VD -self centered vague graph.

Theorem 4. *If is a VD -self-centered graph, then , where is the number of nodes of .*

*Proof. *Let be a VD -self-centered graph. If possible, let . Let and be two distinct VD -peripheral path. Let , . So, there exists a strong path between and because of connectedness of . Then, there exists nodes on and , whose , but this is impossible because . Hence, and are not distinct. Since and are arbitrary, so there exists a node in such that is a common in all VD peripheral paths. So, , which is impossible, because is a VD -self-centered. Therefore, .

Corollary 1. *For a CVG , if and only if the VD -eccentricity of every node of is , .*

*Proof. *Let . Then, , . So, each node of is a VD -periphery node of . So, is a self-centered vague graph and . Therefore, the VD -eccentricity of every node of is .

Conversely, let the VD -eccentricity of every node of is . So, . All nodes of are VD -peripheral nodes, and hence .

Corollary 2. *For a CVG , if and only if the VD -eccentricity of every node of is , .*

*Proof. *Let . So, all nodes of are VD -eccentric node. Hence, is self-centered vague graph and . So, the VD -eccentricity of every node of is .

Conversely, let the VD -eccentricity of every node of is . So, . Therefore, all nodes of are VD -peripheral nodes as well as VD -eccentric node. So, .

Theorem 5. *In a CVG , a node is a VD -eccentric node if and only if is a VD -peripheral node.*

*Proof. *Let be a VD -eccentric node of and let . Let and be two VD -peripheral nodes, then . Let and be any and vague -detour in , respectively. There arise two cases: Case 1**:** if is not internal node in , i.e., there is only one node, say which is adjacent to . So, . Since is connected, is connected to a node of , say . Therefore, either or . Thus, in any case, the path from to or to through and is longer than . But it is impossible, since is a VD -eccentric node of . So, , i.e., is a VD -peripheral node of . Case 2**:** if is an internal node in , then a connection between to and to because of connectedness of . Then, vague -detour can be extended to or . This is impossible because is a VD -eccentric node of . Hence, , i.e., is a VD -peripheral node of . Conversely, we suppose that be a VD -peripheral node of . So, a VD -peripheral node, say (distinct from ). Therefore, is a VD -eccentric node of .

*Definition 15. *In a CVG , a node is said to be a VD -boundary node of a node , if , for each in , where is a neighbor of .

The set of all VD -boundary nodes of described by .

*Example 8. *Consider the CVG in Figure 9.

In this example, we haveHere, , , and are the VD -boundary nodes of .

Theorem 6. *A CVG is a vague tree if and only if is a vague -detour graph.*

*Proof. *Let be a vague tree. Then, between any two nodes in , there is exactly one vague strong path. So, , for any two nodes in . Hence, is a vague -detour graph.

On the contrary, let be a vague -detour graph, which has nodes. Then, , for any two nodes in . If , then is a vague tree. Let . If possible, let be not a vague tree. So, there exists two nodes in for which there is at least two strong paths between and . Let and be two vague strong paths. So, has a cycle in . If nodes and are adjacent nodes in , then we have and . This contradicts the fact that . Hence, is a vague tree.

*Definition 16. *A node in a VG is called a VEN of if is the only strong neighbor of , where .

*Example 9. *For the VG in Figure 9, the nodes , , and are VENs of .

Theorem 7. *A node in a vague tree is a VD -boundary node if and only if is a VEN.*

*Proof. *Let a node be a VD -boundary node for a node in a vague tree . Let be a maximum vague spanning tree in , which is unique in .

#### 5. Application of Vague Detour -Distance in the Transportation System

Today, the issue of transportation plays a very important role in human life. If this is done faster and easier, then it can affect the quality of life and human health. Unfortunately, in the past, due to the lack of sufficient vehicles and bad road conditions, many problems were created for human beings, one of the most important of which is the lack of timely transfer of patients to private hospitals and clinics for treatment. Therefore, in this paper, we intend to express the importance and application of vague detour -distance in the transportation of a patient to the most appropriate hospital in the shortest possible time. For this purpose, we consider four hospitals in Iran (Babol City) named Yahya Nejad (B), Beheshti (C), Mehregan (D), and Rouhani (E), which are shown in the graph with the symbols of , , , and . Suppose that a patient lives in location and must be transported by ambulance to one of these four hospitals in the shortest possible time. In this vague graph, the nodes represent the hospitals and the edges also shows the amount of traffic generated at a certain hour of the day. The weight of the nodes and edges is shown in Tables 2 and 3. The location of hospitals is shown in Figure 10.

The node shows that this hospital has of the medical facilities and equipment needed to treat a patient, but does not have of the necessary tools. The edge indicates that of this route has traffic and congestion caused by vehicles, but of it is free of cars and vehicles. The vague detour -distance for Figure 11 is as follows:

It is clear that has the lowest value, so we conclude that it can be the best choice because, firstly, Mehregan hospital has the most medical equipment and facilities compared to other hospitals, and secondly, A-Mehregan route has the most empty space for patient transfer to medical centers by ambulance. Therefore, governments should provide conditions for patients to be transported to hospitals without stress due to the congestion of roads and intercity traffic, so that they can be treated as soon as possible.

#### 6. Conclusion

A VG is an indiscriminately comprehensive structure of an FG that offers higher precision, adaptability, and compatibility to a system when coordinated with systems running on FGs. VGs are so useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, medical science, and traffic plan. Eccentric nodes and -boundary nodes are a practical interest in several areas. In wireless networking, boundary nodes are used to find efficient routes within ad hoc mobile networks. They have also been used in document summarization and in designing secure systems for electrical grids. Hence, in this paper, the adjacency sequence of RVGs is defined with examples. Also, a characterization of vague detour -eccentric node and the concepts of vague detour -boundary nodes, vague detour -interior nodes in a VG are examined. In our future work, we will introduce vague incidence graphs and study the concepts of perfect dominating set, regular perfect dominating set, and independent perfect dominating set on vague incidence graph. Likewise, we will try to define the average connectivity index, parameter and the new concepts of vague connectivity-enhancing node, vague detour g-interior node, and vague detour g-boundary node of a vague incidence tree using maximum vague spanning tree.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare no conflicts of interest.