1. Introduction
Since Zadeh’s seminal work [
1], the classical logic has been extended to fuzzy logic, which is characterized by a membership function in [0, 1] and provides a powerful alternative to probability theory to characterize imprecision, uncertainty, and obscureness in various fields. Gradually, it has been discovered that sometimes the membership function of the fuzzy set is not enough to reveal the characters of things because of the complexity of data and the ambiguity of the human mind. To overcome this shortcoming of the fuzzy set, Atanassov [
2] extended fuzzy set to an intuitionistic fuzzy set (IFS) by adding a non-membership function and a hesitancy function. An IFS is able to describe the things from three aspects of superiority, inferiority and hesitation, which are usually represented by the intuitionistic fuzzy numbers (IFNs) [
3]. To capture more useful information under imprecise and uncertain circumstances, Yager [
4,
5,
6] recently proposed the concept of Pythagorean fuzzy set (PFS) as a new evaluation format, which is characterized by the membership and the non-membership degree satisfying the condition that their square sum is not greater than 1. Zhang and Xu [
7] provided the detailed mathematical expression for PFS and put forward the concept of Pythagorean fuzzy number (PFN). The PFS is more general than the IFS because the space of PFS’s membership degree is greater than the space of IFS’s membership degree. For instance, when a decision-maker gives the evaluation information whose membership degree is 0.4 and non-membership degree is 0.9, it can be known that the IFN fails to address this issue because
. However,
that is to say, the PFN is capable of representing this evaluation information, as shown in
Figure 1. For this case, the PFS shows its wider applicability than the IFS. PFSs as a novel evaluation have been prosperously applied in various fields, such as the internet stocks investment [
8], the service quality of domestic airline [
7] and the governor selection of the Asian Infrastructure Investment Bank [
9].
A graph is a convenient way of interpreting information involving the relationship between objects. Fuzzy graphs are designed to represent structures of relationships between objects such that the existence of a concrete object (vertex) and relationship between two objects (edge) are matters of degree. The concept of fuzzy graphs was initiated by Kaufmann [
10], based on Zadeh’s fuzzy relations [
11]. Later, another elaborated definition of fuzzy graph with fuzzy vertex and fuzzy edges was introduced by Rosenfeld [
12] and obtaining analogs of several graph theoretical concepts such as paths, cycles and connectedness, etc., he developed the structure of fuzzy graphs. Mordeson and Peng [
13] defined some operations on fuzzy graphs and investigated their properties. Later, the degrees of the vertices of the resultant graphs, obtained from two given fuzzy graphs using these operations, were determined in [
14,
15]. Parvathi and Karunambigai considered intuitionistic fuzzy graphs (IFGs). Later, IFGs were discussed by Akram and Davvaz [
16]. After the inception of IFGs, many researchers [
17,
18] generalized the concept of fuzzy graphs to IFGs. Naz et al. [
19,
20] discussed some basic notions of single valued neutrosophic graphs along with its application in multi criteria decision-making. More recently, Akram et al. introduced many new concepts related to m-polar fuzzy graph, fuzzy soft graph, rough fuzzy graph, neutrosophic graph and their extensions [
21,
22,
23,
24,
25]. This paper proposes a new graph, called Pythagorean fuzzy graph (PFG). In particular, we solve decision-making problems, including evaluation of hospitals, partner selection in supply chain management, electronic learning main factors evaluation by using PFGs.
2. Pythagorean Fuzzy Graphs
Definition 1. A PFS in is said to be a Pythagorean fuzzy relation (PFR) in Z, denoted bywhere and represent the membership and non-membership function of , respectively, such that for all Definition 2. A PFG on a non-empty set Z is a pair , where is a PFS on Z and is a PFR on Z such thatand for all x, Remark 1. We call and the Pythagorean fuzzy vertex set and the Pythagorean fuzzy edge set of , respectively.
If is a symmetric on , then is called PFG.
If is not symmetric on , then is called Pythagorean fuzzy digraph.
The proposed concept of PFGs is a generalization of the notion of Akram and Davvaz’s IFGs [16].
Example 1. Consider a graph , where and . Let and be the Pythagorean fuzzy vertex set and the Pythagorean fuzzy edge set defined on V and E, respectively: By direct calculations, it is easy to see from Figure 2 that is a PFG. Definition 3. The degree and total degree of a vertex in a PFG is defined as and respectively, where The degree of a vertex
n is
and the total degree of a vertex
n in
Figure 2 is
.
Definition 4. Let and be two PFGs of the graphs and , respectively. The direct product of and is denoted by and defined as:
- (i)
- (ii)
Proposition 1. Let and be the PFGs of the graphs and , respectively. The direct product of and is a PFG of
Definition 5. Let and be two PFGs. Then, for any vertex, Theorem 1. Let and be two PFGs. If , then and if then for all
Proof. By definition of vertex degree of
, we have
Hence, Similarly, it is easy to show that, if , then ☐
Definition 6. Let and be two PFGs. For any vertex Theorem 2. Let and be two PFGs. If
- (i)
, then
- (ii)
, then
- (iii)
, then
- (vi)
, then
for all
Proof. The proof is straightforward using Definition 6 and Theorem 1. ☐
Example 2. Consider two PFGs and on and , respectively, as shown in Figure 3. Their direct product is shown in Figure 4. Since , so, by Theorem 1, we have Therefore,
In addition, by Theorem 2, we have Therefore,
Similarly, it is easy to find the degree and total degree of all the vertices in .
Definition 7. Let and be two PFGs of and , respectively. The Cartesian product of and is denoted by and defined as:
- (i)
- (ii)
- (iii)
Proposition 2. Let and be the PFGs of the graphs and , respectively. The Cartesian product of and is a PFG of
Definition 8. Let and be two PFGs. For any vertex Theorem 3. Let and be two PFGs. If and . Then for all
Proof. By definition of vertex degree of
, we have
Hence, ☐
Definition 9. Let and be two PFGs. For any vertex Theorem 4. Let and be two PFGs. If
- (i)
and , then ;
- (ii)
and , then
for all
Proof. By definition of vertex total degree of ,
- (i)
If
,
- (ii)
If
,
☐
Example 3. Consider two PFGs and as in Example 2, where and . Their Cartesian product is shown in Figure 5. Then, by Theorem 3, we have Therefore, .
In addition, by Theorem 4, we have Therefore, .
Similarly, we can find the degree and total degree of all the vertices in
Definition 10. Let and be two PFGs of the graphs and , respectively. The semi-strong product of and , denoted by , is defined as:
- (i)
- (ii)
- (iii)
Proposition 3. Let and be the PFGs of the graphs and , respectively. The semi-strong product of and is a PFG of
Definition 11. Let and be two PFGs. For any vertex Theorem 5. Let and be two PFGs. If , , , . Then, for all
Proof. By definition of vertex degree of
, we have
Analogously, it is easy to show that . Hence, ☐
Definition 12. Let and be two PFGs. For any vertex Theorem 6. Let and be two PFGs. If
- (i)
, , then
- (ii)
, , then
for all
Proof. By definition of vertex total degree of ,
- (i)
If
,
Analogously, we can prove (ii). ☐
Example 4. Consider two PFGs and as given in Example 2, where , , , , and their semi-strong product is shown in Figure 6. Thus, by Theorem 5, we have Therefore, .
In addition, by Theorem 6, we have Therefore, .
Similarly, we can find the degree and total degree of all the vertices in
Definition 13. Let and be two PFGs of and , respectively. The strong product of these two PFGs is denoted by and defined as:
- (i)
- (ii)
- (iii)
- (iv)
Proposition 4. Let and be the PFGs of the graphs and respectively. The strong product of and is a PFG of
Definition 14. Let and be two PFGs. For any vertex Theorem 7. Let and be two PFGs. If , , , . Then, for all where .
Proof. By definition of vertex degree of
, we have
Analogously, it is easy to show that Hence, ☐
Definition 15. Let and be two PFGs. For any vertex Theorem 8. Let and be two PFGs. If
- (i)
, , , then
- (ii)
, , , then
for all
Proof. For any vertex
- (i)
If
,
,
Analogously, we can prove (ii). ☐
Example 5. Consider two PFGs and as in Example 2, where , , , and their strong product is shown in Figure 7. Then, by Theorem 7, we must have Therefore,
In addition, by Theorem 8, we must have Therefore,
Similarly, we can find the degree and total degree of all the vertices in
Definition 16. Let and be two PFGs of and , respectively. The lexicographic product of these two PFGs is denoted by and defined as follows:
- (i)
- (ii)
- (iii)
- (iv)
Proposition 5. The lexicographic product of two PFGs of and is a PFG of
Definition 17. Let and be two PFGs. For any vertex Theorem 9. Let and be two PFGs. If and . Then, for all
Proof. For any vertex
Analogously, we can show that Hence, . ☐
Definition 18. Let and be two PFGs. For any vertex Theorem 10. Let and be two PFGs. If
- (i)
and , then
- (ii)
and , then
for all
Proof. For any vertex
- (i)
If
Analogously, we can prove (ii). ☐
Example 6. Consider two PFGs and as in Example 2, where and , and their lexicographic product is shown in Figure 8. Then, by Theorem 9, we must have Therefore,
In addition, by Theorem 10, we must have Therefore,
Similarly, we can find the degree and total degree of all the vertices in
Definition 19. The union of two PFGs and of the graphs and , respectively, is defined as follows:
- (i)
- (ii)
- (iii)
- (iv)
Theorem 11. The union of and is a PFG of if and only if and are PFGs of and , respectively, where and are the Pythagorean fuzzy subsets of and respectively and .
Definition 20. Let and be two PFGs. For any vertex there are three cases to consider.
Case 1: Either or . Then, no edge incident at x lies in . Thus, for Case 2: but no edge incident at x lies in . Then, any edge incident at x is either in or in . Similarly, Similarly,
Case 3: and some edges incident at x are in . Similarly, . In addition, Example 7. Consider two PFGs and on and , respectively, as shown in Figure 9. In addition, their union is shown in Figure 10. Since , thus, Therefore, . Therefore, .
Since but no edge incident at l lies in , Therefore, Therefore,
Since and , thus, Therefore, Therefore,
Definition 21. The ring-sum of two PFGs and of the graphs and , respectively, is defined as follows: Proposition 6. If and are the PFGs, then is the PFG.
Definition 22. Let and be two PFGs. For any vertex there are two cases to consider.
Case 1: If either or .
Case 2: If . Then, any edge incident at x is either in or in .
Definition 23. Let and be two PFGs of and , respectively. The join of and , denoted by , is defined as:
- (i)
- (ii)
- (iii)
where is the set of all edges joining the vertices of and ,
Theorem 12. The join of and is a PFG of if and only if and are PFGs of and , respectively, where and are the Pythagorean fuzzy subsets of and respectively, and .
Definition 24. Let and be two PFGs. For any vertex