1. Introduction
We divide the Introduction Section into four main paragraphs. In the first paragraph, we provide some details about the fuzzy sets. In the second paragraph, detail is given about the complex version of fuzzy sets, namely complex fuzzy sets, which is an extension of fuzzy sets. In the third paragraph, detail is given about graph theory in terms of different types of fuzzy sets. In the fourth paragraph, we give our presented approach by combining the two different approaches given in the second and third paragraphs.
Fuzzy set theory was conferred by Zadeh [
1] to solve difficulties in dealing with uncertainties. Since then, the theory of fuzzy sets and fuzzy logic have been examined by many researchers to solve many real life problems involving ambiguous and uncertain environment. Atanassov [
2] proposed the extended form of fuzzy set by adding a new component, called “intuitionistic fuzzy sets” (
-sets). The idea of
-sets is more meaningful as well as intensive due to the presence of degree of truth and falsity membership. Applications of these sets have been broadly studied in other aspects such as image processing [
3], multi-criteria decision making [
4], pattern recognition [
5], etc.
Buckley [
6] and Nguyen et al. [
7] combined complex numbers with fuzzy sets. On the other hand, Ramot et al. [
8,
9] extended the range of membership to “unit circle in the complex plane”, unlike others who limited the range to
. Zhang et al. [
10] studied some operation properties and
-equalities of complex fuzzy sets. Some applications of complex fuzzy sets have been considered in reasoning schemes [
11], image restoration [
12] and decision making [
13]. Further, this concept has been studied in intuitionistic fuzzy sets [
14]. Alkouri and Salleh studied some operations on complex Atanassov’s intuitionistic fuzzy sets in [
15] and also studied complex Atanassov’s intuitionistic fuzzy relation in [
16]. Ali et al. [
17] introduced complex intuitionistic fuzzy classes.
Fuzzy graphs were narrated by Rosenfeld [
18] and Mordeson [
19]. After that, some opinion on “fuzzy graphs” were given by Bhattacharya [
20]. He showed that none of the concepts of crisp graph theory have similarities in fuzzy graphs. Thirunavukarasu et al. [
21] extended this concept for complex fuzzy graphs. Shannon and Atanassov [
22] and Akram and Davvaz [
23] defined intuitionistic fuzzy graphs. Later, several authors worked on intuitionistic fuzzy graphs and added many useful results to this area, for instance, Akram and Akmal [
24], Alshehri and Akram [
25], Karunambigai et al. [
26], Myithili et al. [
27], Nagoorgani et al. [
28] and Parvathi et al. [
29,
30]. See also [
31,
32,
33,
34,
35].
Inspired by the fact that complex intuitionistic fuzzy sets generalize intuitionistic fuzzy sets, in this paper, we provide the new idea of complex intuitionistic fuzzy graphs with some fundamental operations. We also describe homomorphisms of complex intuitionistic fuzzy graphs. Finally, we provide an application.
3. Complex Intuitionistic Fuzzy Graphs
In this section, we provide definition and operations of complex intuitionistic fuzzy graphs.
Definition 6. A complex intuitionistic fuzzy graph (cif-graph) with an underlaying set V is defined to be a pair , where is a cif-set on V and is a cif-set on such thatfor all Definition 7. Let be a cif-graph. The order of a cif-graph is defined by The degree of a vertex x in is defined by Example 1. Consider a graph such that . Let be a cif-subset of V and let be a cif-subset of as given: - (i)
By routine calculations, it can be observed that the graph shown in Figure 1 is a cif-graph. - (ii)
Order of cif-graph
- (iii)
Degree of each vertex in is
Definition 8. The Cartesian product of two cif-graphs is defined as a pair , such that:
- 1.
for all
- 2.
for all , and ,
- 3.
for all , and .
Definition 9. Let and be two cif-graphs. The degree of a vertex in can be defined as follows: for any , Example 2. Then, their corresponding Cartesian product is shown in Figure 4. Proposition 1. The Cartesian product of two cif-graphs is a cif-graph.
Proof. The conditions for are obvious, therefore, we verify only conditions for .
Let
, and
. Then,
Similarly, we can prove it for and □
Definition 10. The composition of two -graphs is defined as a pair , , such that:
- 1.
for all , ,
- 2.
for all , and ,
- 3.
for all , and .
- 4.
for all , and .
Definition 11. Let and be two cif-graphs. The degree of a vertex in can be defined as follows: for any , Example 3. Consider the two cif-graphs, as shown in Figure 5. Then, their composition is shown in Figure 6. Proposition 2. The composition of two cif-graphs is a cif-graph.
Definition 12. The union of two -graphs is defined as follows:
- 1.
for and .
- 2.
for and .
- 3.
for .
- 4.
for and .
- 5.
for and .
- 6.
for .
Example 4. Consider the two cif-graphs, as shown in Figure 7. Then, their corresponding union is shown in Figure 8. Proposition 3. The union of two cif-graphs is a cif-graph.
Definition 13. The join of two cif-graphs, where , is defined as follows:
- 1.
if
- 2.
if
- 3.
if where is the set of all edges joining the vertices of and .
Example 5. Consider the two cif-graphs, as shown in Figure 9. Then, their corresponding join is shown in Figure 10. Proposition 4. The join of two cif-graphs is a cif-graph.
Proposition 5. Let and be cif-graphs of the graphs and and let Then, union is a cif-graph of if and only if and are cif-graphs of the graphs and , respectively.
Proof. Suppose that
is a cif-graph. Let
Then,
and
Thus,
This shows that is a cif-graph. Similarly, we can show that is a cif-graph. The converse part is obvious. □
Proposition 6. Let and be cif-graphs of the graphs and and let Then, join is a cif-graph of if and only if and are cif-graphs of the graphs and , respectively.
Proof. The proof is similar to the proof of Proposition 5. □
5. Complement of cif-Graphs
In this section, we discuss complements of cif-graphs.
Definition 15. The complement of a weak cif-graph of is a weak cif-graph on , is defined by
- (i)
- (ii)
for all ,
- (iii)
Example 7. Consider a cif-graph , as shown in Figure 12. Then, the complement of is shown in Figure 13. Definition 16. A cif-graph G is called self complementary if .
The following propositions are obvious.
Proposition 9. Let be a self complementary cif-graph. Then, Proposition 10. Let be a cif-graph. If then G is self complementary.
Proposition 11. Let and be cif-graphs. If there is a strong isomorphism between and , then there is a strong isomorphism between and .
Proof. Let
f be a strong isomorphism between
and
. Then,
is a bijective map that satisfies
Since
is a bijective map,
f is also bijective map such that
f for all
Thus
By definition of complement, we have
Thus, f is a bijective map which is a strong isomorphism between and . This ends the proof. □
The following Proposition is obvious.
Proposition 12. Let and be cif-graphs. Then, if and only if
Proposition 13. Let and be cif-graphs. If there is a co-strong isomorphism between and , then there is a homomorphism between and .
6. Application
Intuitionistic fuzzy sets are the valuable generalization of fuzzy sets. We combine complex intuitionistic fuzzy sets with the graph theory. Complex intuitionistic fuzzy graphs have many applications in database theory, expert systems, neural networks, decision making problems, GIS-based road networks, facility location problems and so on. In the following, we propose an assumption based application that can be utilized in a physical way.
Consider a cellular company that has a plan to fix the minimum number of towers in a city, such that the maximum numbers of the users can be attracted. For this purpose, the following are some of the parameters that can be taken in account:
Suitable place to fix a tower
Transportation means
Users
Connectivity with the main server
Urban area or hilly area
Any other existing cellular network
Available recourses
Expenditures and outcomes
Suppose a team selected five places where they are interested in placing a tower, so that they can facilitate maximum numbers of the users. They observe the following two situations:
For Situation we proceed as follows:
Let be the set of places where the team is interested in fixing a tower as a vertex set. Suppose that of the experts on the team believe that should have a tower and of the experts believe that there is no need to fix tower at the place after carefully observing the different parameters. Thus, in this way, we can find the amplitude term for both membership and non-membership functions. Now, the phase term that represents the period needs to be found. Let of the experts believe that in a particular time can attract the maximum number of users (Profit) and of the experts have the opposite opinion. We model this information as
Thus, the team finalizes its opinion about the place
Now, they visit the place
. After careful observation, they model the information as
. It means that
of the experts are in the favor of
, even though it will produce only
of profit, while
are opposed to
, even though it will produce
profit. Similarly, they model all the other places as
and
We denote this model as
The complex membership of the vertices denotes the positive characteristics and complex non-membership of the vertices denotes the negative characteristics of a certain parameter for a certain place. Now, finding the absolute values, we have
To find the optimal choice, we find the score function of the absolute values of
Thus, we have
Since the scores for
and
are equal, we find the accuracies of
and
and
thus
, which is the most suitable choice to fix a tower. This is the application of complex intuitionistic fuzzy graph, where it has no edge, as shown in
Figure 14.
Now, for Situation we proceed as follows:
If a tower is fixed between places
and
, it will represent the edge
of the vertex
To find the model of
, we use Definition 6 and find that
Similarly, we find the other edges and we denote this model as
If we consider the edge
. In this case, the amplitude term shows that
of the experts believe that there should be a tower between these two places and
of the experts believe the opposite. The phase terms show that
of the experts believe that in a certain time if a tower is fixed between these two places it will produce maximum profit, while
of the experts believe the opposite. Absolute values of the edges are:
To find the optimal choice, we find the score function of the absolute values of the edges. Thus, we have
is the greatest, and hence most suitable choice to fix the tower. This is the case where complex intuitionistic fuzzy graph has edges, as shown in
Figure 15.