Abstract
In this paper, we define the concept of level graphs on interval-valued fuzzy graphs (IVFGs) and introduce the notions of homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism of IVFGs by homomorphism. Also, an application of homomorphism of IVFGs has been presented by using coloring-FG.
1. Introduction
Zadeh [1] introduced the subject of the interval-valued fuzzy set (IVFS) in 1975 as an explanation of fuzzy sets (FSs) [2] so that the values of the membership degrees are shown through intervals of numbers. IVFS is useful in applications such as fuzzy control. One of the most extreme sections of fuzzy control in terms of calculation is defuzzification [3]. The fuzzy graph (FG) theory as a generalization of Euler’s graph theory was defined by Rosenfeld [4] in 1975. For the first time, the fuzzy relations between FSs were supposed by Rosenfeld, and he extended the structure of FGs and used it to achieve various graph theoretical subjects. After that, Bhattacharya [5] presented some remarks on FGs. Mordeson and Peng [6] explained some operations on FGs. Hongmei and Lianhua gave the definition of IVFG and so they introduced the notion of IVFGs [7]. For the first time, Akram and Dudek defined IVFGs [8]. In graph theory, we define a graph homomorphism that has a mapping between two graphs that relates to their structure. Different notions of graph coloring are generalized by homomorphisms. These concepts give essential information about an important class of constraint satisfaction problems [9].
Jiang et al. [10] introduced a new notion of vertex covering in cubic graphs. Turksen [11] defined IVFSs based on normal forms. Nagoorgani and Radha [12] presented isomorphism on fuzzy graphs. Jan et al. [13] studied some root-level modifications in IVFGs. Krishna et al. [14] investigated new results in cubic graphs. Talebi et al. [15, 16] defined Cayley fuzzy graphs and some operations on level graphs of bipolar fuzzy graphs. Kosari et al. [17] introduced the vague graph (VG) structure with an application in medical diagnosis. Kou et al. [18] defined some properties of VG. Qiang et al. [19] studied new kinds of domination in VGs. Rao et al. [20–22] presented some properties of vague incidence graphs and equitable domination in VGs. Shi et al. [23, 24] investigated the domination of product VGs with an application in transportation. Xu et al. [25] studied certain concepts of interval-valued intuitionistic fuzzy graphs. Rashmanlou and Pal [26] defined antipodal interval-valued fuzzy graphs. Bera et al. [27] proposed certain types of m-polar interval-valued fuzzy graphs. Zihni et al. [28] introduced interval-valued fuzzy soft graphs. Akram et al. [29–31] introduced the concept of interval-valued fuzzy line graphs, self-centered interval-valued fuzzy graphs, and certain types of interval-valued fuzzy graphs.
IVFG, belonging to the FG family, has good capabilities when faced with problems that cannot be explained by FGs and VGs. Since the principle of membership is not clear, neutrality is a good choice that can be well protected by an IVFG. Also, an IVFG is used to illustrate real-world phenomena using interval-valued fuzzy models in a variety of fields, including technology, social networking, and biological sciences. Therefore, in this paper, we introduce the concept of level graphs on IVFGs and define the notions of homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism of an IVFG by homomorphism. Finally, we have tried to find the most effective person in an electrical power department based on the performance of its staff.
2. Preliminaries
In this section, we present some basic concepts of IVFGs.
Definition 1. A fuzzy graph is a pair with a set X, that is a fuzzy set in X and is a fuzzy relation in , so thatfor all .
Suppose is a finite nonempty set. The set of all 2-element subsets of is shown by . A graph G is a pair on , so that , and are named vertex set and edge set, respectively.
Definition 2. A pair on graph is called interval-valued fuzzy graph (IVFG) so that is an IVFS on and is an IVFS on E to satisfy the following conditions:(1) so that shows the degree of membership of the node .(2), are the functions that satisfy for all .
Definition 3. An IVFG is called a strong interval-valued fuzzy graph (SIVFG) iffor all .
Definition 4. Let . For any , we have
Definition 5. Let . For any , we defineThen, is said to be -level set of .
The set is named the support .
Definition 6. Suppose and are two IVFGs. Then,(1)A homomorphism (HM) from to is a mapping with the following conditions:(i), (ii), , for all (2)An isomorphism from to is a bijective mapping (BM) with the following conditions:(i), (ii), , for all (3)A weak isomorphism (WI) from to is a BM with the following conditions:(i) is HM(ii),, (4)A co-weak isomorphism (CWI) from to is a BM with the following conditions:(i) is HM(ii), , for all
Definition 7. Suppose and are two IVFGs. Then, is an interval-valued subgraph (IVSG) of , if and .
Definition 8. A family of IVFSs on is called a -coloring of IVFG if(i).(ii), for .(iii)For each strong edge of , , for . We consider that a graph is -colorable if it can be colored by colors.
All the essential notations are denoted in Table 1.
3. Homomorphisms and Isomorphisms of Interval-Valued Fuzzy Graphs
In this part, we explain HM and isomorphisms on IVFGs by the HM of level graphs in IVFGs.
Theorem 1. Let be a finite nonempty set, , and . Then, if and only if is a graph, for all , .
Proof. Suppose is an IVFG. For any , , consider that . Then, and . Because is an IVFG,Also, we haveTherefore, and is a graph.
Conversely, suppose is a graph, for , . For any , let . Then, . Hence, . Thus, , , . This results thatandThus, is an IVFG.
Definition 9. Suppose and are two IVFGs on and , respectively, a mapping. For any , , if is a HM from to , then, is called HM mapping from to .
Theorem 2. Suppose and are two IVFGs. Then, is a HM from to if and only if is HM from to .
Proof. Let be a HM from to . Assume, , . If , then,Therefore implying is a mapping from to . For , let . Then,Hence,That implies Thus, is a HM from to .
Conversely, suppose is a HM from to . For the arbitrary element of , let , . Then, , so that, , because is a HM from to . We haveThat is,Now for arbitrary of , let , . Then,Hence, and . Because is a HM from to , we conclude that and . Therefore,
Theorem 3. Let and be IVFGs on and , respectively. Then, is a WI from to , if and only if, is a bijective HM from to and
Proof. Suppose is a WI from to . From the definition of HM, is a BH from to . By Theorem 2, is a bijective HM from to and also, by the definition of WI, we haveConversely, by hypothesis, is a BM andFor , let , . Then,That implies and . Because is a HM from to , we have and . Thus,
Theorem 4. Let and be IVFGs on and , respectively. Then, is a CWI from to , if and only if, is a bijective HM from to and
Proof. Let be a CWI from to . Then, is a BH from to . By Theorem 2, is a bijective HM from to . Also, by the description of CWI, we writeConversely, by hypothesis, we see that is a BM andFor arbitrary principle , consider , . Then, we have . Now, because is a HM from to , . Therefore,That implies is a CWI from to .
Corollary 1. Let and . If is a CWI from to , then, is an IH from to , for every , .
Example 1. Suppose and are two IVFGs, as in Figure 1. Consider the mapping of , defined by , . level graphs of and in Figure 1, if , therefore, is an IH from to , but is not a CWI.
Theorem 5. Let and , and is a mapping. For any , , if is an isomorphism from to a subgraph (SG) of , then, is a CWI from to an induced IVSG of .
Proof. The mapping from to a SG is an isomorphism, so is an IM. For arbitrary , we consider , . Then, and so .
Therefore,For , let , thatand . Thus, and .
Since is an isomorphism from to , we get and .
Therefore,Now suppose , . Then Because is injective and an isomorphism from to a SG of , we have and . Therefore,Now by and , we conclude the following statement:
Corollary 2. Let and be two IVFGs with , and a mapping. For , if, is an isomorphism from to a SG of , then, is a CWI from to .
Theorem 6. Suppose we have two graphs and that they are IVFGs, and is a BM. If for any , is an isomorphism from to , then, is an isomorphism from to .
Proof. By hypothesis, is a BM and an isomorphism from to .
By Theorem 3.7, is a CWI from to and is a CWI from to . Therefore, is an isomorphism from to .
Theorem 7. Suppose is an IVFG. Then, is a complete interval-valued fuzzy graph (CIVFG), if and only if, is a complete graph (CG) for .
Proof. If is a CIVFG and for , thenand soHence, . It shows that is a CG.
Conversely, consider is not a CIVFG. Then, there are so thatLet , and/or , for Then, . Hence, , for a , but . This implies that is not a CG. For the case , it follows similarly.
Theorem 8. Let . Then, does not have an isolated vertex (IV), for any if and only if for each so that , .
Proof. Suppose that for each , graph does not have an IV and there is a vertex so that for each , or . Let and , , for .
Then, and for each , , . Thus, is an IV in the graph , which is a contradiction.
Now let that for , , vertex is an IV in . If , then,and if , it is trivial that , hence,Thus, for any , , .
Theorem 9. A VG is -colorable there exists a HM from to the .
Proof. Let be -colorable with colors, so that . Let . We denote CIVFG with vertices set , so that the degree of membership vertex is and the degree of nonmembership vertex is . Now the mapping defined by is a graph homomorphism, becauseand by the definition of CIVFG, for and , we haveThen, , , for all .
Conversely, let is a HM. For a given , we define the set to beIf , let , otherwise . Thus, the IVFG is -colorable with coloring set .
4. Application
In graph theory, the subject of coloring is an important case of graph labeling, and there are more examples of coloring graph. In this part, we have expressed an application of the coloring of vertices in an IVFG, and we present this concept in the form of an example.
Example 2. Let be an IVFG (see Figure 2). In this example, we consider four sport groups such as so that they are vertices of the graph. We show each one of the sports groups with interval-valued. The interval-valued of the vertices are the good activity rate in the world cup games, so that areSuppose that , , , , and are the edges of graph .Here we want to see how many days they will need to exercise between these sport groups. Let be a set of sport groups, and \{AB, BC, AC, CD, BD \}. Now, form graph with vertices set , where are neighbors, if and only if, . For example, and . So , and so , are neighbors. By Theorem 9, there is a HM from to complete graph with . Therefore, these sport groups are needed to play 3 days, (see Figure 3).
In the next example, we want to identify the most effective employee of an electrical power department with the help of an interval-valued fuzzy influence digraph.
Example 3. Electricity is one of the discoveries that has changed the daily lives of everyone on the planet. Electricity is a staple of modern technology, and without it, most of the things we use every day will work. Other major areas where electricity is used are the production, supply, and distribution of food. Farmers use electricity for almost everything, including lights, pumps, electric motors, and so on. Electricity is very useful, especially in homes, offices, shopping malls, schools, supermarkets, and any other place we can think of. Today, it has become such a vital element of a society that depends on electricity that it is difficult for human beings to even imagine life without electricity. Access to electricity has become the most important thing in human society and has created great economic, political, and cultural changes in the world. Perhaps what was more important than electricity itself, and multiplied the importance of electricity, was the emergence of devices that received their driving forces from electricity and raised the level of well-being and quality of human life in society. It can be boldly said that the largest volume of living items in today’s society are electrical appliances. Most of these devices in human life play the role of converting electrical power into mechanical, thermal, and cooling power. Therefore, considering the importance of electricity in everyday human life, we have tried to consider an electrical power department and find the most efficient person in it with the help of an interval-valued fuzzy influence. Because worthy people should be considered to lead this organization, and with their leadership, electricity consumption should be saved and this very valuable blessing should not be wasted. Therefore, we consider the vertices of the interval-valued influence graph as the heads of each ward of the electrical power department and the edges of the graph as the degree of interaction and influence of each other. For this department, the set of staff is .(1)Eskandari has been working with Tahmasbi for 8 years and values his views on issues.(2)Farhadi has been responsible for maintaining the electrical department’s property for a long time, and everyone is satisfied with his performance.(3)In an electrical power department, it is very important to control the staff and check their treatment of clients. Maleki is the best person to do this.(4)Tahmasbi and Ashkani have a long history of conflict.(5)Due to Bozorgi’s long experience in finance, he is the best person for the job.
According to the above values, we consider an interval-valued influence graph. The nodes show each of the department’s staff that the lower bound indicates the ability of staff to perform related tasks and the upper bound shows the weakness and inability of each staff member. But the edges show the level of friendship and interest among employees. If this friendship is more, then the amount of influence on each other will naturally be more. For the weight of the edges, the lower bound means the amount of friendship and the upper bound means the amount of conflict. Names of employees and level of staff capability are shown in Tables 2 and 3. The adjacency matrix corresponding to Figure 4 is shown in Table 4.
Figure 4 shows that Ashkani has of the power needed to do the work of the board of directors of the electrical power department, but unfortunately he does not have of the necessary power to do so. The directional edge of Eskandari-Farhadi shows that there is friendship among these two employees, but unfortunately, they have conflict. Clearly, Eskandari has dominion over both Maleki and Bozorgi, and his dominance over both is . It is clear that Eskandari is the most influential employee of the electrical power department because he controls both the security guard and the financial affairs manager, who have of the power in the department.
5. Conclusion
IVFG, belonging to FG family, has good abilities when faced with problems that cannot be explained by FGs. It has a wide range of applications in the field of psychological sciences as well as in the reconnoiter of individuals founded on oncological behaviors. So, in this survey, we tried to show the new notion of level graph in IVFG. Also, we introduced the notion of HM, isomorphism, WI, and CWI of IVFG by HM. Finally, an application of HM of IVFGs has been presented by using coloring-FG. In our future work, we will introduce the new concepts of connectivity in IVFGs and investigate some of their properties. Also, we will study the new results of the complete dominating set, regular complete dominating set, and global dominating set on IVFGs.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.