1. Introduction
Fuzzy sets (FSs), coined by Zadeh [
1], have become one of the emerging areas in contemporary technologies of information processing. Recent studies spread across various areas from control, pattern recognition, and knowledge-based systems to computer vision and artificial life. Fuzzy sets implicate capturing, portraying and dealing with linguistic notions-items with indefinite boundaries and thus it came forth as a new approach to incorporate uncertainty. Picture fuzzy sets (PFSs), created by Cuong [
2,
3], a direct extension of Atanassov’s [
4] intuitionistic fuzzy sets (IFSs), which itself extends the Zadeh’s notion of fuzzy sets, are the sets characterized by not only membership
and nonmembership
functions but also neutral function
where the difference
is called refusal function. Thus PFSs may be satisfactory in cases whenever expert’s evaluations are of types: yes, abstain, no, refusal. Cuong suggested ‘voting’ as a paradigm of his proposed concept. Since picture fuzzy sets are suitable for capturing imprecise, uncertain and inconsistent information, therefore they can be applied to many decision-making processes such as: solution choice, financial forecasting, estimating risks in business etc. Son [
5] introduced some clustering algorithms based on picture fuzzy sets with applications to time series and weather forecasting. Thong [
6] developed a hybrid model relating picture fuzzy clustering for medical diagnosis. There have been significant attempts which explored this concept, one can see [
7,
8,
9,
10,
11,
12,
13]. It is noteworthy that the constraint
in IFS limits the selection of orthopairs from a triangular region. In order to increase the adeptness of IFS, Yager and Abbasov [
14] proposed the notion of Pythagorean fuzzy sets (PyFSs) in
which replace the constraint of IFS with
This notion also limits the selection of orthopairs from unit circular region in the first quadrant. In
Yager [
15] introduced
q-rung orthopair fuzzy sets (
q-ROFSs) as a new generalization of orthopair fuzzy sets (i.e., IFS and PyFS), which further relax the constraint of orthopair membership grades with
Analogously, since picture fuzzy sets confine the selection of triplets only from a tetrahedron, as shown in
Figure 1, the spherical fuzzy sets (SFSs), proposed by Gündoğdu and Kahraman [
16], have given strength to the idea of picture fuzzy sets. Although both PFS and SFS can easily reflect the ambiguous character of subjective assessments, they still have apparent variations. The membership functions
and
of PFSs are required to meet the condition
However, these functions in SFSs satisfy the constraint
Which indicates that SFSs somehow expanded the space of admissible triplets. In
Li et al. [
17] proposed the
q-RPFS model which inherits the virtues of both
q-ROFS and PFS. Being a suitable model for capturing imprecise and inconsistent data, it not only assigns three membership degrees to an element, but also alleviate the constraint of picture and spherical fuzzy sets to a great extent with
Figure 1 shows that the space of admissible triplets expands with increasing
For example, if an expert provides the positive, neutral and negative membership values to an object as
and
respectively. It is immediately seen that
and
Such a case can neither be explained by PFS nor by SFS. However, it is appropriate to use
q-RPFS because
for sufficiently large value of
Thus immense number of triplets are qualified for
q-RPF model due to its elastic bounding constraint. It can be observed that the surface
bounds a portion of first octant, whose volume approaches to the unit cube’s volume as the parameter
q approaches to infinity. Of course the constraint condition of
q-RPF model provides a sense of interdependence of membership functions
and
This fact makes this notion considerably more close to natural world than that of prior notions. Some remarkable attempts can be seen in [
18,
19,
20,
21].
Graphs theory, a dynamic field in both theory and applications, allows graphs as the most important abstract data structures in many fields. Graphs can not translate all the phenomenons of real world scenarios adequately due to the uncertainty and vagueness of parameters. These positions direct to define fuzziness in graph theoretic concepts. Zadeh [
22] originated the idea of fuzzy relation. Kaufmann [
23] introduced fuzzy graph to state uncertainty in networks. Rosenfeld [
24] presented more concepts relating fuzzy graphs. Parvathi and Karunambigai introduced intuitionistic fuzzy graphs(IFGs) [
25]. Afterward, IFGs were examined by Akram and Davvaz [
26]. Naz et al. [
27] discussed the notion of Pythagorean fuzzy graphs. Habib et al. [
28] investigated
q-ROFGs. The flexible nature of these notions make them vast research area, so far in [
29,
30,
31,
32,
33,
34]. To manage the cases requiring opinions of types: yes, abstain, no and refusal in graph-theocratic concepts in a broad manner, recently, Akram and Habib [
35] introduced the concept of
q-RPFGs and defined their regularity. The concept of regularity of fuzzy graphs has led to many developments in their structural theory as they play important role in combinatorics and theoretical computer science. First, Gani and Radha [
36] defined degree, total degree and regularity of fuzzy graphs. Degree and total degree of an edge is introduced by Radha and Kumaravel [
37]. Cary [
38] initiated the idea of perfectly regular and perfectly edge regular fuzzy graphs. Several related works on regularity and edge regularity can be viewed in [
35,
39,
40,
41,
42,
43,
44,
45,
46]. In this paper, we discuss some properties of
q-RPFGs, namely edge regularity, total edge regularity and perfect edge regularity. We introduce and investigate these properties for square
q-RPFGs and
q-RPF line graphs. Furthermore, we provide a brief characterization on structural relationships between regularity and edge regularity of several
q-RPFGs. In addition, we present the idea of ego-networks to extract knowledge from large social networks under
q-rung picture fuzzy environment with algorithm, as an application of the proposed concept.
We now review the concept of
q-RPFS [
17] which is necessary to proceed further.
Definition 1. Let X be a universe of discourse. A q-rung picture fuzzy set(q-RPFS) on X given byis characterized by a positive membership function a neutral/abstinence membership function and a negative membership function such that Moreover, is called a q-rung picture fuzzy index or degree of refusal membership of u to the set where The
q-rung picture fuzzy set based models may be satisfactory in conjunctures when human opinions concerning responses: yes, abstain, no, and refusal are encountered. Voting can be considered as a paradigm of such environments as human voters can be split into four classes of those who: vote for, abstain, vote against, refusal of voting. The graphical structure of
q-RPFS in
Figure 1 provides a gradual increase in spaces bounded by surfaces
obtained by varying
q in the first octant. For
the space bounded by surface
in the first octant is equivalent to the volume occupied by a tetrahedron ABCD. For
the
q-RPFS reduces to spherical fuzzy set which covers more space of acceptable triplets than PFS as it is bounded by surface
in the first octant, which is equivalent to the volume occupied by unit sphere in the first octant. It can be seen that for
the
q-RPFS can accept more triplets than picture and spherical fuzzy sets as it covers up more space and for
the space of
q-RPFS is vast than all preceding spaces. Since the volume occupied by a surface covers the volume occupied by all prior surfaces; therefore, any element belongs to a particular
q-RPFS, must qualify for all picture fuzzy sets of higher rungs (i.e., greater than
q).
2. Edge Regular q-Rung Picture Fuzzy Graphs
In this section, we first provide some basic definitions relating regularity of
q-RPFGs, defined in [
35], which will be used for further developments.
Definition 2. A q-rung picture fuzzy graph on a non-empty set X is a pair with a q-rung picture fuzzy set on and a q-rung picture fuzzy relation on X such thatand for all , where and represents the positive membership, neutral membership, and negative membership functions of , respectively. Example 1. Consider a graph such that and Let be a 4
-RPFS on P and be a 4
-RPFR on defined by | a | b | c | d | e | | | | | | | | |
| 0.8 | 0.7 | 0.55 | 0.7 | 0.9 | | | 0.5 | 0.6 | 0.55 | 0.7 | 0.5 | 0.5 |
| 0.7 | 0.85 | 0.8 | 0.9 | 0.4 | | | 0.7 | 0.7 | 0.8 | 0.3 | 0.8 | 0.25 |
| 0.6 | 0.67 | 0.77 | 0.5 | 0.55 | | | 0.65 | 0.6 | 0.56 | 0.65 | 0.66 | 0.75 |
Routine computations show that displayed in Figure 2, is a 4
-RPFG. Definition 3. Let be a q-RPFG on underlying crisp graph . If and for all , then is called a strong q-rung picture fuzzy graph.
Definition 4. Let be a q-RPFG on underlying crisp graph . If and for all , then is called a complete q-rung picture fuzzy graph.
Definition 5. Let be a q-RPFG defined on The order of is denoted by and defined as Definition 6. Let be a q-RPFG defined on The size of is denoted by and defined as Example 2. The order and size of q-rung picture fuzzy graph displayed in Figure 2 are and respectively. Definition 7. Let be a q-RPFG defined on The degree of a vertex u of is denoted by and defined as Definition 8. Let be a q-RPFG defined on The total degree of a vertex u of is denoted by and defined as Example 3. Consider a 4
-RPFG displayed in Figure 2. The degree, and total degree of vertex a in is given as and respectively. Definition 9. Let be a q-RPFG defined on If each vertex of has same degree, that is, for all then is called -regular q-RPFG.
Example 4. Consider a 3-RPFG as displayed in Figure 3. We see that the degree of each vertex in is Hence, is -regular.
Definition 10. Let be a q-RPFG defined on If each vertex of has same total degree, that is, for all then is called -totally regular q-RPFG.
Example 5. Consider a 2
-RPFG as displayed in Figure 4. is -totally regular 2-RPFG since
Definition 11. A perfectly regular q-RPFG is a q-RPFG that is both regular and totally regular.
Example 6. Consider a 3
-rung picture fuzzy graph as shown in Figure 5. Clearly, 3-RPFG is regular since and is totally regular since Hence, is perfectly regular 3-RPFG.
Definition 12. A q-RPFG defined on is said to be partially regular if the underlying graph is regular.
Definition 13. A q-RPFG defined on is said to be full regular if is both regular, and partially regular.
The concept of edge regularity has been explored by many researchers on fuzzy graphs, and several of its generalizations. We now give a description on edge regular q-RPFGs. First, we state some definitions in this context.
Definition 14. Let be a q-RPFG defined on The edge degree of in is denoted by and defined asThis is equivalent to Example 7. Consider a 4
-RPFG displayed in Figure 6. The degree of edge in can be computed as Theorem 1. [35] For any q-RPFG defined on the following relation for degrees of vertices of must hold:for all The following theorem is developed to define a comprehensive relationship between the degrees of edges, and the degrees of vertices of q-RPFGs.
Theorem 2. For any q-RPFG defined on if for all then the following relation for degrees of edges of must hold:for all That is, the sum of degrees of all edges is equal to times the sum of degrees of all vertices of Proof Let
and
be a
q-RPFG defined on
The degree of an edge
of
q-RPFG
can be defined as
For all
we have
This completes the proof. □
Next, we state some well known results regarding degrees of edges in q-RPFGs.
Theorem 3. Let be a q-RPFG on a cycle Then, Theorem 4. Let be a q-RPFG on Then,where for all Theorem 5. Let be a q-RPFG on a k-regular graph Then, For proofs of above theorems, readers are referred to [
37,
41,
44].
Definition 15. The minimum edge degree of q-RPFG is defined as where is minimum μ-edge degree of is minimum η-edge degree of and is minimum ν-edge degree of
Definition 16. The maximum edge degree of q-RPFG is defined as where is maximum μ-edge degree of is maximum η-edge degree of and is maximum ν-edge degree of
Example 8. Consider a q-rung picture fuzzy graph as shown in Figure 6. By routine computations, it is easy to see that the minimum, and maximum edge degree of are Definition 17. Let be a q-RPFG defined on The total edge degree of in is denoted by and defined asThis is equivalent to Example 9. Consider a 4
-RPFG displayed in Figure 6. The total degree of edge in can be computed as Theorem 6. For any q-RPFG defined on if for all then the following relation for total degrees of edges of must hold:for all Proof. The proof directly follows from Theorem 2 and Definition 17. □
Theorem 7. Let be a q-RPFG on Then,where for all Proof. The total degree of an edge
in a
q-RPFG is
Therefore,
where
for all
This completes the proof. □
The concept of edge degree leads to defining edge regularity of q-RPFGs. Formally, we have the following definition:
Definition 18. Let be a q-RPFG defined on If each edge of has same degree, which isthen is called -edge regular q-RPFG. Example 10. Consider a 3-RPFG defined on where be a 3-rung picture fuzzy set on and be a 3-rung picture fuzzy relation on defined by | a | b | c | | | | | |
| 0.4 | 0.2 | 0.9 | | | 0.2 | 0.2 | 0.2 |
| 0.5 | 0.3 | 0.1 | | | 0.1 | 0.1 | 0.1 |
| 0.6 | 0.4 | 0.6 | | | 0.6 | 0.6 | 0.6 |
We see that Hence, the 3-RPFG displayed in Figure 7, is -edge regular. Definition 19. Let be a q-RPFG defined on If each edge of has same total edge degree, that isthen is called -total edge regular q-RPFG. Example 11. Consider a 3-RPFG as displayed in Figure 8. We see that is -total edge regular 3-RPFG since
Remark 1. - 1.
Any connected q-RPFG with two vertices is edge regular.
- 2.
A q-RPFG is edge regular if and only if
Remark 2. A complete q-rung picture fuzzy graph need not be edge regular.
For example, consider a 4-RPFG as displayed in Figure 9. It is clear that is a complete 4-RPFG. However, is not edge regular since
Next, we present a necessary condition for a complete q-RPFG to be edge regular.
Theorem 8. Let be a complete q-rung picture fuzzy graph on and and are constant functions; then, is an edge regular q-RPFG.
Proof. Let
be a complete
q-rung picture fuzzy graph defined on
where
Then, for all
and
Let
for all
The completeness of
implies that each vertex
u of
is connected with
vertices by edges, with membership values
Thus, degree of each vertex
can be written as
By definition of edge degree, we have
for all
Hence,
is an edge regular
q-RPFG. This completes the proof. □
Remark 3. An edge regular q-rung picture fuzzy graph may not be total edge regular.
For example, consider a 6-RPFG as displayed in Figure 10. We see that is edge regular as However, is not total edge regular 6-RPFG since
Remark 4. A total edge regular q-rung picture fuzzy graph may not be edge regular.
For example, consider a 3-rung picture fuzzy graph as displayed in Figure 8. We see that is -total edge regular. However, is not an edge regular 3-RPFG since Moreover, it is easy to observe that there may exist such q-RPFGs which are both edge regular, and total edge regular—neither edge regular nor total edge regular. Thus, there does not exist any relationship between edge regular q-RPFGs, and total edge regular q-RPFGs. Next, we illustrate a theorem providing characterization for edge regularity of q-rung picture fuzzy graphs.
Theorem 9. Let be a q-rung picture fuzzy graph on and and are constant functions; then, the following are equivalent:
- (a)
is an edge regular q-RPFG.
- (b)
is a total edge regular q-RPFG.
Proof. Let
be a
q-RPFG on
Suppose that, for all
Assume that
is a
-edge regular
q-RPFG. Then, for all
By definition of total edge degree,
for all
Hence,
is total edge regular
q-RPFG.
Conversely, suppose that
is a
-total edge regular
q-RPFG. Then, by definition of total edge regular
for all
Hence,
is edge regular
q-RPFG. This proves that
and
are equivalent.
On the other hand, assume that
and
are equivalent. Suppose, on the contrary, that
and
are not constant functions. Then, there exist at least one edge
in Q such that
Let
be a
-edge regular
q-RPFG. Then,
Therefore,
Since therefore, Hence, is not total edge regular, which gives a contradiction to our assumption.
Now, let
be a total edge regular
q-RPFG. Then, by definition of total edge regular,
The fact that
is not edge regular leads a contradiction to our assumption. Hence,
and
are constant functions. This completes the proof. □
Definition 20. A q-RPFG defined on is said to be partially edge regular if the underlying graph is edge regular.
Example 12. Consider a 5-RPFG displayed in Figure 11a, and its underlying crisp graph displayed in Figure 11b. We see that G is 2-edge regular graph. Thus, is partially edge regular 5-RPFG.
Definition 21. A q-RPFG defined on is said to be full edge regular if is both edge regular, and partially edge regular.
Example 13. Figure 7 can be considered as an example of full edge regular 3-RPFG. Since as well as its underlying crisp graph G are edge regular. Remark 5. An edge regular q-rung picture fuzzy graph may not be partially edge regular (or full edge regular).
For example, consider the 6-RPFG displayed in Figure 12 and its underlying crisp graph displayed in Figure 12 We see that is edge regular as However, is not partially edge regular 6-RPFG since its underlying graph is not edge regular.
Remark 6. A partially edge regular q-rung picture fuzzy graph may not be edge regular (or full edge regular).
For example, consider a 5-RPFG displayed in Figure 11. We see that is partially edge regular as its underlying graph G is edge regular. However, is not edge regular 5-RPFG since The above remarks show that there does not exist any relationship between edge regular, and partially edge regular (or full edge regular) q-RPFGs. In the following theorem, we develop a relationship between edge regular, and partially edge regular q-RPFGs.
Theorem 10. Let be a q-rung picture fuzzy graph on If and are constant functions, then is edge regular if and only if is partially edge regular.
Proof. Let
be a
q-RPFG such that for all
Then, by definition of edge degree
for all
Hence,
is edge regular. Assume that
is a
-edge regular
q-RPFG. Then, for all
,
This shows that G is an edge regular q-RPFG. Hence, is partially edge regular q-RPFG.
Conversely, assume that is partially edge regular q-RPFG. Let G be a p-edge regular graph. Then, for all Consequently, is edge regular q-RPFG. This completes the proof. □
Corollary 1. Let be a q-RPFG such that and are constant functions. If is an edge regular q-RPFG or a partially edge regular q-RPFG, then is a full edge regular q-RPFG.
Theorem 11. Let be a strong q-RPFG such that and are constant functions. Then, is an edge regular q-RPFG if and only if is partially edge regular q-RPFG.
Proof. Let
be a strong
q-RPFG defined on
. Then, for all
and
Let
and
be constant functions. Then, for all
and
Combining both facts, we have
Thus,
and
are constant functions. Consequently, the result follows from Theorem 10. □
Corollary 2. Let be a strong q-RPFG such that and are constant functions. If is an edge regular q-RPFG or a partially edge regular q-RPFG, then is a full edge regular q-RPFG.
Next, we state some results regarding order, and size of edge regular q-RPFG.
Theorem 12. The size of -edge regular q-RPFG on a p-edge regular graph is where
Theorem 13. If is -total edge regular, and -partially edge regular q-RPFG, then where
Theorem 14. If is -edge regular, and -total edge regular q-RPFG, then where
For proofs of above results, readers are referred to [
37,
41].
2.1. Perfectly Edge Regular q-RPFGs
In this section, we introduce perfect edge regular
q-RPFGs taking as a point of departure the respective definition of perfect edge regular fuzzy graphs by Cary [
38].
Definition 22. A perfect edge regular q-RPFG is a q-RPFG that is both edge regular, and total edge regular.
Example 14. Consider a 5-rung picture fuzzy graph as shown in Figure 13. Clearly, is -edge regular, and -total edge regular. Hence, is perfect edge regular 5-RPFG.
Theorem 15. If is perfect edge regular q-RPFG, then and are constant functions.
Proof. Let
be a perfect edge regular
q-RPFG defined on
Then,
must be edge regular and total edge regular, i.e., for all
and
By definition of total edge degree,
Since
and
for all
therefore,
and
are constant functions. This completes the proof. □
Perfectly edge regular q-RPFGs can be classified by the following theorem.
Theorem 16. A q-RPFG is perfect edge regular if and only if it satisfies the following conditions:
- 1.
- 2.
for all
Proof. Suppose that is perfect edge regular q-RPFG. Then, it must be regular as well as total edge regular. The fact that is edge regular implies Condition 1 so that each edge has the same degree. Condition 2 is followed directly by Theorem 15.
Conversely, suppose that Condition 1 and 2 hold. It is straightforward to see that Condition 1 guarantees the edge regularity of
Let for any two edges
and
By definition of total edge degree,
Thus,
is total edge regular, as
and
are arbitrary edges. Consequently,
is perfect edge regular. This completes the proof. □
Remark 7. The above theorem does not constitute a sufficient condition for a q-RPFG to be perfect edge regular.
For example, consider a 4-RPFG as displayed in Figure 14. We see that for all However, since and therefore, is neither edge regular nor totally egde regular. Hence, is not a perfect edge regular 4-RPFG.
The following theorem provides a necessary condition for the converse of the above theorem to hold.
Theorem 17. If is edge regular q-RPFG, and and are constant functions, then is perfect edge regular q-RPFG.
Proof. The proof is directly followed from Theorem 9, which proves that is total edge regular q-RPFG. Moreover, by our assumption, is edge regular. Hence, is perfect edge regular. This completes the proof. □
We next illustrate some results relating order and size of perfect edge regular q-RPFG.
Theorem 18. If is perfect edge regular q-RPFG, and and are constant functions, then the size of is
Unlike the order of perfectly regular q-RPFG, we can no longer give an explicit formula for the order of perfect edge regular q-RPFG. However, we can bound its order.
Theorem 19. If is perfect edge regular q-RPFG, then the order of is bounded as: Proof. Upper bound of
is obvious. We now prove its lower bound. By definition of
q-RPFG,
and
Thus, we have
and
For each
we obtain
This completes the proof. □
2.2. Edge Regular Square q-RPFGs
Following [
47], we define square
q-RPFGs. In this section, we concentrate on edge regularity of square
q-RPFGs.
Definition 23. Let be a q-RPFG defined on The square q-RPFG of is denoted by and defined as:
- 1.
If then and
- 2.
If and are joined by a path of length less than or equal to 2 in then
Example 15. Consider a 6-RPFG defined on where is a 6-rung picture fuzzy set on and is a 6-rung picture fuzzy relation on defined by | a | b | c | d | e | | | | | | | |
| 0.8 | 0.82 | 0.9 | 0.8 | 0.6 | | | 0.8 | 0.6 | 0.5 | 0.8 | 0.6 |
| 0.75 | 0.93 | 0.85 | 0.7 | 0.85 | | | 0.7 | 0.6 | 0.7 | 0.7 | 0.8 |
| 0.85 | 0.55 | 0.59 | 0.9 | 0.9 | | | 0.7 | 0.6 | 0.9 | 0.5 | 0.6 |
By routine computations, it is easy to see that in Figure 15b is a square 6-rung picture fuzzy graph of where the 6-rung picture fuzzy relation can be defined as follows: | | | | | | | | |
| 0.8 | 0.6 | 0.5 | 0.8 | 0.6 | 0.6 | 0.8 | 0.82 |
| 0.7 | 0.6 | 0.7 | 0.7 | 0.8 | 0.5 | 0.7 | 0.85 |
| 0.7 | 0.6 | 0.9 | 0.5 | 0.6 | 0.9 | 0.9 | 0.59 |
The following theorem defines the degree of an edge in square q-RPFG.
Theorem 20. For any q-RPFG the degree of an edge in its square q-RPFG is given by
- 1.
- 2.
For such that ,
Proof. Assume that is a square q-RPFG of
- 1.
If
then by definition of square
q-RPFG
The degree of edge
in
is
- 2.
If
such that
then degree of edge
in
is
This completes the proof. □
Remark 8. If is edge regular q-RPFG, then may not be edge regular.
For example, consider a 3-RPFG and its square graph as shown in Figure 16. We see that is -edge regular 3-RPFG. However, since Hence, is not edge regular.
Remark 9. If is edge regular q-RPFG, then may not be edge regular.
For example, consider a 5-RPFG and its square graph as shown in Figure 17. We see that is -edge regular 5-RPFG. However, since Hence, is not edge regular.
Lemma 1. The square q-RPFG of a complete q-RPFG is the q-RPFG itself.
Proof. The proof is obvious. □
Theorem 21. If be complete q-RPFG. Then, is an edge regular q-RPFG if and only if is an edge regular q-RPFG.
Proof. Let be complete q-RPFG. By Lemma 1, and are same. It follows that is an edge regular q-RPFG if and only if is an edge regular q-RPFG. □
Lemma 2. The square q-RPFG of a complete bipartite q-RPFG is complete.
Proof. Let be a q-RPFG defined on a complete bipartite graph Then, and for all and where Let be the square q-RPFG of Then, we have the following:
If then and
If and are joined by a path of length less then or equal to 2 in then and
Since each pair of vertices in a complete bipartite graph must connected by a path of length less then or equal to 2 (see
Figure 18), therefore, above axioms imply that for all
Hence, the square
q-RPFG
is complete. This completes the proof. □
Theorem 22. If is a complete bipartite q-RPFG such that and are constant functions, then its square graph is edge regular.
Proof. Let be a complete bipartite q-RPFG such that and are constant functions. Assume that is the square q-RPFG of Then, by Lemma 2, is a complete q-RPFG.
Let
and
for all
By definition of square
q-RPFG, we have
Hence,
and
are constant functions. Thus, by Theorem 8,
is edge regular. This completes the proof. □
Theorem 23. If is a q-RPFG on a cycle of length n such that and are constant functions, then its square q-RPFG is edge regular.
Proof. Let
be a
q-RPFG defined on a cycle
(see
Figure 19). Assume that
and
are constant functions, and
and
for all
where
The degree of edge
can be computed as:
Hence,
is edge regular. Let
be the square
q-RPFG of
To prove that
is edge regular, there arise three cases:
Case-I When we have a complete q-RPFG By Lemma 1, is itself.
Case-II When each pair of non-adjacent vertices is connected by two distinct paths of length 2. Thus, its square graph will be a complete q-RPFG with and are constant functions. In this case, is edge regular by Theorem 8.
Case-III Let
Consider a vertex u in
There are exactly two vertices which are at distance 2 from
Therefore, u is adjacent to exactly two more vertices in
Hence,
is 4-regular. Then, for each
in
we have
Hence,
is
-edge regular. This completes the proof. □
2.3. Edge Regular q-RPF Line Graphs
Following [
45], this section gives a brief discussion on edge regular
q-rung picture fuzzy line graphs (
q-RPFLGs). Akram and Habib [
35] defined the
q-RPFLGs as follows:
Definition 24. Let be a q-RPFG defined on The q-rung picture fuzzy line graph with underlying graph where and can be defined as
- (i)
For each vertex in - (ii)
For each edge in
Theorem 24. Let be a q-RPFG such that and are constant functions. Then, for all
Proof. Let
be a
q-rung picture fuzzy line graph of a
q-RPFG
For any
This completes the proof. □
Remark 10. Consider a 6-RPFG as displayed in Figure 20a. Its corresponding line graph is displayed in Figure 20b. We see that is -edge regular 6-RPFG. However, since therefore, is not an edge regular 6-RPFG.
We now develop a necessary condition for q-RPFLG to be perfect edge regular.
Theorem 25. Every q-rung picture fuzzy line graph of an edge regular q-RPFG is perfect edge regular only if and are constant functions in
Proof. Let
be an edge regular
q-RPFG defined on
Then, for each edge
in
Let
and
be constant functions. Then, for each edge
in
Consider
as a
q-RPFG as a
q-rung picture fuzzy line graph of
Then, by definition of
q-rung picture fuzzy line graph, each vertex
of
corresponding to an edge
of
has membership value
and each edge of
has membership value
Let
and
be the end points of an edge
in
Since
is edge regular with constant functions
and
therefore, each vertex u of
is common vertex of α (say) edges in
(see
Figure 21).
Let
be the vertex of
corresponding to an edge x in
The edge x has a vertex
in common with
edges
Similarly,
is common with
edges
in
Thus, each vertex
in
is common vertex of
edges. Hence, for all
in
By Theorem 24, we have
Then, by Equation (
1),
which implies that
is an edge regular q-RPFG. Moreover,
which shows that
is total edge regular
q-RPFG. Consequently,
is perfect edge regular
q-RPFG. This completes the proof. □
Remark 11. Consider a 4-RPFG as displayed in Figure 22a, and its line graph Figure 22b. We see that is -edge regular 4-RPFLG. However, since therefore, is not an edge regular 4-RPFG.
Theorem 26. If is an edge regular q-rung picture fuzzy line graph of and and are constant functions, then is edge regular q-RPFG.
Proof. The proof is similar to the proof of Theorem 25. □
Theorem 27. Let be a strong q-rung picture fuzzy graph such that and are constant functions. If is an edge regular q-RPFG, then its line graph is an edge regular q-rung picture fuzzy graph.
Proof. Let be a strong edge regular q-rung picture fuzzy graph such that and are constant functions. Then, the functions and must be constants. Consequently, the result follows from Theorem 25. □
3. Relations between Regular and Edge-Regular q-RPFGs
The fact that edge regularity property is a strong analog of regularity in fuzzy graphs is shown by many researchers. This section provides some additional properties relating regularity and edge regularity of q-RPFGs.
Remark 12. Every regular q-RPFG need not be edge regular.
For example, consider a 5-RPFG displayed in Figure 23. It is clear that is -regular 5-RPFG, but is not edge regular as
Remark 13. Every edge regular q-RPFG need not be regular.
For example, consider a 6-RPFG as shown in Figure 10. It is clear that is -edge regular 6-RPFG but is not regular as The above remarks show that one form of regularity can not imply another form. We now develop some results between regularity and edge regularity.
Theorem 28. Let be a regular q-RPFG on Then, is edge regular if and only if and are constant functions.
Proof. Let be a -regular q-RPFG. Then, for all Assume that and are constant functions, that is, and By definition of edge degree for all Hence, is edge regular.
Conversely, assume that is -edge regular. Then, for all By definition of edge degree Thus, for all Hence, and are constant functions. This completes the proof. □
Theorem 29. Let be a q-RPFG on such that and are constant functions. If is full regular q-RPFG, then is full edge regular q-RPFG.
Proof. Let be a q-RPFG on and and for all Assume that is full regular q-RPFG, that is, and for all Then, Hence, G is edge regular graph. Now, Hence, is edge regular q-RPFG. Consequently, is full edge regular q-RPFG. □
Remark 14. The converse of the above theorem need not be true.
For example, consider a 4-RPFG as displayed in Figure 24. We see that is -edge regular q-RPFG, and G is 1-edge regular graph. Thus, is full edge regular q-RPFG. However, since therefore, is not regular. Moreover, G is not regular. Thus, is not full regular.
Theorem 30. Let and are constant functions in a q-RPFG If is regular, then is perfect edge regular.
Proof. Let be a q-RPFG on with and for all Assume that is -regular. Then, for all By definition of edge degree, for all Moreover, by definition of total edge degree, for all Hence, is perfect edge regular. □
Remark 15. The converse of above theorem need not be true.
For example, consider 11-RPFG as shown in Figure 25. We see that is -edge regular, and -total edge regular 11-RPFG, but is not regular 11-RPFG as
Theorem 31. [35] Let be a q-RPFG such that and are constant functions. Then, is regular q-RPFG if and only if is partially regular q-RPFG. Theorem 32. If a q-RPFG is perfect edge regular, then is regular if and only if is partially regular.
Proof. Let be perfect edge regular. Then, by Theorem 15, and are constant functions. Thus, Theorem 31 implies that is regular if and only if is partially regular. □
Remark 16. The converse of above theorem need not be true.
For example, consider a 5-RPFG as shown in Figure 26. We see that is -regular 5-RPFG. Moreover, is partially regular. However, it is easy to observe that is neither edge regular nor total edge regular.
Theorem 33. Let be the k-partially regular q-RPFG. Then, and are constant functions if and only if both regular, and perfect edge regular.
Proof. Consider a q-RPFG defined on such that G is k-regular. Assume that and for all Then, for all Hence, is regular q-RPFG. Thus, by Theorem 30, is perfect edge regular.
Conversely, let
be both regular, and perfect edge regular. Then, suppose that
for all
and
for all
By definition of edge degree,
Hence,
and
are constant functions. This completes the proof. □
Lemma 3. If is perfectly regular (or regular) q-RPFG, and and are constant functions, then is perfect edge-regular.
Proof. Let be a perfectly regular q-RPFG defined on and and be constant functions. Then, for all Since is regular q-RPFG, therefore, for all Then, for any edge in Hence, is edge regular q-RPFG. Now, the total degree of an edge in q-RPFG is for all Thus, is total edge-regular q-RPFG. Consequently, is perfect edge-regular q-RPFG. This completes the proof. □
Remark 17. If is totally regular q-RPFG, and and are constant functions, then may not perfect edge-regular.
For example, consider a 3-RPFG as shown in Figure 27. We see that is totally regular since and and are constant functions as and for all However, leads to be not edge-regular 3-RPFG. Thus, is not perfect edge-regular.
Theorem 34. If is complete perfectly regular q-RPFG, then is perfect edge-regular.
Proof. Let be a complete perfectly regular q-RPFG defined on Then, for each vertex u of In addition, completeness of implies that, for every edge of Combining the two facts above, we obtain for all It shows that and are constant functions. Hence, by Lemma 3, is perfect edge-regular q-RPFG. This completes the proof. □
Next, we relate the concepts of regularity and edge regularity in q-RPF line graphs.
Observation 1. The total degree of an edge x in is equal to the sum of the membership values of corresponding vertex and its adjacent vertices in
Theorem 35. Let be a q-RPFG such that and are constant functions. If is a regular q-RPFG, then its line graph is an edge regular q-RPFG.
Proof. Let be a regular q-RPFG such that and are constant functions. Then, by Theorem 28, is an edge regular q-RPFG. Consequently, by Theorem 25, the q-RPF line graph is edge regular. □
Remark 18. The converse of the above theorem need not be true.
Consider a 4-rung picture fuzzy graph and its line graph as shown in Figure 28. We see that and are constant functions, that is, for each in Since each edge of has degree therefore, is -edge regular 4-rung picture fuzzy line graph of However, leads to the fact that the 4-RPFG is not regular.
4. Applications
Graph theory, graph-partitioning, and graph-based computing are characterized by detecting different network structures. The unstoppable growth of social networks, and the huge number of connected users, has made these networks some of the most popular, and successful domains for a large number of research areas. The different possibilities, volume, and variety that these social networks offer has made them essential tools for everyday working, and social relationships. The number of users registered in different social networks, and the volume of information generated by them in a more explanatory manner are increasing day by day. The analysis over the whole network becomes extremely difficult due to this fact. For instance, if every people has 5 close friends, then in a town of 10,000 people, there will be 50,000 close friendship ties to study. In order to extract the knowledge from such a large network, some relevant works have focused the attention on ‘ego-networks’. On the other hand, if someone does not want to understand a whole community, but just what individual people do, that is, one only finds some people in a community interesting (leaders, teenagers, artists, etc.), also lead to study ego-networks, which tells us about social structure of entire population, and its sub-populations.
An Ego-network is a social network composed of one user centering the graph (called ego), all the users connected to this ego (called alters), and all the relations between these alters (called ties). Ego can be considered as an individual ‘focal’ node. It can be a person, a group, an organization, or a whole society. A network has as many egos as it has nodes.
q-Rung Picture Fuzzy Social Network
An ordinary social network cannot represent the power of employees, and the degree of relations among employees within an organization. As the powers, and relationships have no defined boundaries, it is desired to represent them in the form of fuzzy set. The fuzzy social networks are used to express interactions between different nodes. It is obvious that the fuzzy graph model is not enough to fully illustrate any phenomenon represented by networks/graphs. Several extensions of fuzzy set have been introduced in this context. Thus, to deal with the situations where opinions are not only yes or no, but there are some abstinence, and refusal too; recently, the q-rung picture fuzzy graph model was introduced, providing a vast depiction space of triplets. We now discuss the concept of ego-networks under q-rung picture fuzzy environment.
Algorithm 1 illustrates the extraction of
q-rung picture fuzzy ego-networks from a
q-rung picture fuzzy social network
The complexity of algorithm is
where
Algorithm 1 Extraction of q-rung picture fuzzy ego-networks |
INPUT: A q-RPF social network OUTPUT: A q-RPF ego-network |
function fordo for do if is q-RPF ego then choose all such that and Define a q-RPF vertex set and a q-RPF edge set end if end for end for end function
|
We can use q-RPFG to examine the social relationships among different groups of people. By the concept of q-RPF ego-networks, we can focus on a single entity to investigate its potential with other members of a social group. We can also examine the percentage of relationships under a q-rung picture fuzzy environment.
Consider a large company in Japan. The typical structure of executive titles is illustrated as a set of employees
P = {
Chairman (C), CEO, President (C), Deputy president (DP)/Senior executive (SE), Executive vice president (EVP), Senior vice president (SVP), Vice president (VP)/general manager (GM)/Department head (DH), Deputy general manager (DGM), Manager (M)/Section head (SH), Assistant manager (AM)/Team leader (TL), Staff (S)} for that company. Let
be the 4-RPFS on
P given by
Table 1.
Figure 29 displays a 4-RPFG representing the social network composed of 11 different employees.
There are 11 different
q-RPF ego networks/graphs associated with the given
q-RPF network/graph, where the vertices/nodes represent the powers of employees in the company, and edges/links represent their relationships. The membership functions
and
in triplet
, assigned to each node, indicate their positive impact, their neutral behavior, and their negative impact within the company, respectively. For example,
tells the status level of the CEO as: the CEO possesses
positive impact, and
negative impact within the company. A total of
of his behavior is neutral. While the refusal degree
of CEO indicates that only
, he has the ability to refuse to participate in the company’s matters. However, the positive, neutral, and negative membership degrees
and
of edges depict the positive associations, abstinence to associate, and negative associations among employees. For example, the edge between president and assistant manager is assigned by triplet
The relationships between them can be translated by a means of
q-rung picture fuzzy analysis as follows: they have only
positive attitude. The neutral and negative behavior have same the weight, i.e.,
and
of interactions translate the extent of the clash between them. Similar considerations can be extracted from other edges. The corresponding adjacency matrix is shown in
Table 2, where the degrees for each edge
in
are evaluated using the following relations:
The CEO is a company’s top decision-maker, and all other executives answer to him/her. Next, we want to study the powers of the CEO in this company, and his associations with other executives. For this, we consider the CEO as ego, and extract a
q-rung picture fuzzy ego network from a
q-rung picture fuzzy social network (see
Figure 29) according to Algorithm 1.
Figure 30 displays a
q-rung picture fuzzy ego network
for selected ego, i.e., CEO (colored node).
The CEO influences the executives who are board members, and jointly supervise the activities of company. The titles designate an individual as an officer of the company with specific responsibilities that make them legally accountable in their position. The following investigations provide a q-RPF social network analysis for the CEO:
The impact of CEO is equally distributed with the vice president, and deputy president as the positive, neutral, and negative associations of both are and respectively.
The CEO greatly refuses to value the inputs of executive vice present for strategic initiatives as there is refusal part while no negative associations between them, where the positive associations are and is the neutrality in their behavior.
By means of the positive and negative associations of the CEO with the Chairman, the founder of the company, we can this interpret as: the Chairman works on the CEO’s opinion, for , he works opposite of his opinion, for , his behavior is neutral with the CEO, and, for , the chairman refuses to trust his part.
The reporting relationships between general manager, and CEO is positive, neutral, and negative, while, for of his own part, the general manager refuses to report to him.
Between the vice president and deputy president, there is less positive and high negative associations, which indicate their history of conflict.
The manager being a section head has to resolve all cases in his section to answer to the CEO. The absence of edge with the executive vice president show that he has no concern with the executive vice president in company’s matters.
The similar investigations can be seen between executives (alters) in the
q-RPF ego-network (
Figure 30).