# Decision-Making Analysis Based on Fuzzy Graph Structures

Authors: Ali N. A. Koam; Muhammad Akram; Peide Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A graph structure is a useful framework to solve the combinatorial problems in various fields of computational intelligence systems and computer science. In this research article, the concept of fuzzy sets is applied to the graph structure to define certain notions of fuzzy graph structures. Fuzzy graph structures can be very useful in the study of various structures, including fuzzy graphs, signed graphs, and the graphs having labeled or colored edges. The notions of the fuzzy graph structure, lexicographic-max product, and degree and total degree of a vertex in the lexicographic-max product are introduced. Further, the proposed concepts are explained through several numerical examples. In particular, applications of the fuzzy graph structures in decision-making process, regarding detection of marine crimes and detection of the road crimes, are presented. Finally, the general procedure of these applications is described by an algorithm.


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## Body

## 1. Introduction

Decision-making is a process of solving problems for choosing the best alternative. Multicriteria decision-making (MCDM) can be explicated as a discipline of operation research in which the satisfactory solution is opted regarding some significant factors which can be considered as conflicting criteria for the decision-making problems. The concept of MCDM is widely used in medical, economics, engineering, and social sciences. The process of MCDM can be classified according to the number of decision makers (DMs), i.e., individual decision-making and group decision-making (GDM). Multicriteria group decision-making (MCGDM) is a procedure to deal with the problems to find the most suitable alternative relative to some criteria by a cooperative group of decision makers. In MCGDM, the individual decision matrices of all DMs are merged to get a group satisfactory solution.Most of the classical frameworks of formal modeling and reasoning are crisp and deterministic in the character. Crisp is yes or no type and does not deal more or less type. In the traditional dual logic, a statement is either true or false and nothing is in between. In the set theory, an element either belongs to the set or not. A fuzzy set, proposed by Zadeh [1], is the class of objects with continuity of grades of membership. It is characterized by the membership function that assigns each element grade of membership between zero and one. Nowadays, in modeling uncertain systems of the industry, society, and nature, the theory of fuzzy sets plays a vital role. In the decision making-process, it is used as a strong mathematical tool facilitating the approximate reasoning.A more convenient way to illustrate data and information regarding interactions between elements or objects is a graph [2, 3]. Due to uncertainty in representation of elements or in their relations, it is quite natural to develop a fuzzy graph. A fuzzy graph is defined as a symmetric and binary fuzzy relation on the fuzzy subset [4]. Rosenfeld [5] considered fuzzy graphs based on fuzzy relations [6]. Later on, Bhattacharya [7] gave remarks on fuzzy graphs. Sunitha and Vijayakumar [8] worked on the complement of fuzzy graphs. Mordeson and Nair [9] discussed various properties of fuzzy graphs and fuzzy hypergraphs. Nagoor Gani and Radha [10, 11] introduced the notions of regular fuzzy graphs, totally regular fuzzy graphs, and degree and total degree of a vertex in some fuzzy graphs. Bhutani and Battou [12] studied novel concepts of M-strong fuzzy graphs. Matrix representation of graphs under fuzzy information was proposed by Chen [13]. For further studies on fuzzy graphs, the readers are referred to [14–34].Sampathkumar [35] introduced the concept of graph structures. Graph structures prove to be very useful in the study of various areas of computational intelligence and computer science. The lexicographic product was first considered by Hausdorff [36]. The lexicographic product of two fuzzy graphs was introduced by Radha and Arumugam [37]. Dinesh [38] worked on fuzzy graph structures and discussed few related concepts. Ramakrishnan and Dinesh [39–41] discussed generalized fuzzy graph structures. Akram and Sitara [42, 43] introduced the semistrong min-product and maximal product of fuzzy graph structures and investigated their corresponding properties. Residue product of fuzzy graph structures was studied by Akram et al. [44]. In this research article, the concept of fuzzy sets is applied to a graph structure to define certain notions of fuzzy graph structures. The notions of the fuzzy graph structure, lexicographic-max product, and degree and total degree of a vertex in lexicographic-max product are introduced. Further, the proposed concepts are explained through several numerical examples. In particular, applications of the fuzzy graph structures in decision-making process, regarding detection of marine crimes and detection of the road crimes, are presented. Finally, the general procedure of these applications is described by an algorithm.The contents of this article are as follows: In Section2, some fundamental concepts of graph structures and fuzzy graphs are reviewed. The certain notions of fuzzy graph structures are defined, and their interesting properties are investigated in the same section. Section 3 illustrates the applicability of fuzzy graph structures in real-life phenomena. The whole article is concluded in Section 4.


## 2. Certain Notions of Fuzzy Graph Structures

Definition 1 (see [35]).
A graph structure (GS)G∗=V,R1,…,Rn consists of a nonempty set V together with relations R1,R2,…,Rn on V which are mutually disjoint such that each Ri, 1≤i≤n, is symmetric and irreflexive. One can represent a graph structure G∗=V,R1,…,Rn in the plane just like a graph where each edge is labeled as Ri, 1≤i≤n.Definition 2 (see [38]).
Letα be the fuzzy set on set V and ν1,ν2,…,νn be fuzzy sets on R1,R2,…,Rn, respectively. If 0≤νivu≤αv∧αu∀u,v∈V, i=1,2,…,n, then G^=α,ν1,ν2,…,νn is called the fuzzy graph structure (FGS) of the graph structure G∗. If vu∈suppνi, then vu is named as the νi-edge of FGS G^.Definition 3.
LetG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ be two fuzzy graph structures (FGSs) having underlying crisp graph structures (GSs) G^1∗=V1,R1′,R2′,…,Rn′ and G^2∗=V2,R1″,R2″,…,Rn″, respectively. A fuzzy graph structure G^=α,ν1,ν2,…νn with an underlying crisp graph structure G^∗=V,R1,R2,…,Rn, where V=V1×V2 and Ri=v1,w1v2,w2/v1=v2,w1w2∈Ri″orv1v2∈Ri′, is defined.
The fuzzy vertex setα is defined as αv,w=α1v∨α2w, for all v,w∈V=V1×V2.
Fuzzy relationsνi are defined as(1)νiv1,w1v2,w2=νi′v1v2,v1v2∈Ri′,α1v1∨νi″w1w2,v1=v2,w1w2∈Ri″,i=1,2,…,n.
G^=α,ν1,ν2,…νn is called the lexicographic-max product of G^1 and G^2 and is denoted by G^1G^2max.Example 1.
Consider two FGSsG^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″ with underlying crisp GSs G^1∗=V1,R1′,R2′ and G^2∗=V2,R1″,R2″, respectively, which are shown in Figure 1, where R1′=v1v2, R2′=v3v4, R1″=w1w2, and R2″=w2w3.
The lexicographic-max product of the above FGSsG^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″ is shown in Figure 2.
In the lexicographic-max product,νi′ and νi″ edges belong to the νi fuzzy set.Figure 1
Two fuzzy graph structures. (a)G1=α1,ν1′,ν2′. (b) G2=α2,ν1′′,ν2′′.
(a)(b)Figure 2
Lexicographic max-product of two FGSs.Theorem 1.
IfG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ are two effective FGSs such that α2≥α1 and νi′ and νi″ are constant functions having similar values, then the lexicographic-max product of G^1 and G^2 is an effective FGS.Proof.
LetG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ be two effective FGSs such that α2≥α1 and νi′ and νi″ be constant functions having similar values, then the definition of the lexicographic-max product providesCase 1:v1v2∈Ri′:(2)νiv1,w1v2,w2=νi′v1v2=νi″w1w2=α2w1∧α2w2=α1v1∨α2w1∧α1v2∨α2w2=αv1,w1∧αv2,w2.Case 2:v1=v2,w1w2∈Ri″:(3)νiv1,w1v2,w2=α1v1∨νi″w1w2=νi″w1w2=α2w1∧α2w2=α1v1∨α2w1∧α1v2∨α2w2=αv1,w1∧αv2,w2.
Thus,νiv1,w1v2,w2=αv1,w1∧αv2,w2 for all edges.
Hence,G^1G^2max=α,ν1,ν2,…,νn is an effective FGS.Example 2.
Consider two FGSsG^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″, as shown in Figure 3.
Figure3 shows that ν1′v1v2=α1v1∧α1v2=0.6∧0.6=0.6, ν2′v3v4=α1v3∧α1v4=0.6∧0.6=0.6, ν1″w1w2=α2w1∧α2w2=0.6∧0.6=0.6, and ν2″w2w3=α2w2∧α2w3=0.6∧0.6=0.6. Hence, G^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″ are effective FGSs. Moreover, α1=α2 and νi′ and νi″, i=1,2, are constant functions having the same value, i.e., 0.6. The lexicographic-max product of G^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″ is shown in Figure 4.
It is clear from Figure4 that(4)ν1′v1,w1v2,w1=αv1,w1∧αv2,w1=0.6∧0.6=0.6,ν1′′v1,w1v1,w2=αv1,w1∧αv1,w2=0.6∧0.6=0.6,ν1′v1,w1v2,w2=αv1,w1∧αv2,w2=0.6∧0.6=0.6,ν1′v1,w1v2,w3=αv1,w1∧αv2,w3=0.6∧0.6=0.6,ν1′′v4,w1v4,w2=αv4,w1∧αv4,w2=0.6∧0.6=0.6,ν2′v4,w1v3,w1=αv4,w1∧αv3,w1=0.6∧0.6=0.6,ν2′v4,w1v3,w2=αv4,w1∧αv3,w2=0.6∧0.6=0.6,ν2′v4,w1v3,w3=αv4,w1∧αv3,w3=0.6∧0.6=0.6.
Similarly, membership values of all other edges are calculated. Hence,G^1G^2max shown in Figure 4 is an effective FGS.Figure 3
Two fuzzy graph structures. (a)G1=α1,ν1′,ν2′. (b) G2=α2,ν1′′,ν2′′.
(a)(b)Figure 4
Lexicographic-max product ofG1=α1,ν1′,ν2′ and G2=α2,ν1′′,ν2′′.Theorem 2.
IfG^1 and G^2 are two complete FGSs such that α2≥α1 and νi′ and νi″ are functions having a constant value, then the lexicographic-max product of G^1 and G^2 is a complete FGS.Proof.
The proof is similar to the proof of Theorem1.Theorem 3.
The lexicographic-max productG^1G^2max of two connected FGSs G^1 and G^2 is a connected FGS if and only if G^1 is a connected FGS.Proof.
Assume thatG^1 is a connected FGS. According to the definition of the lexicographic-max product G^1G^2max, the number of copies of G^1 is equal to the number of vertices of G^2; that is, for each vertex of G^2, there exists one copy of GS G^1 in the lexicographic-max product G^1G^2max. Since G^1 is connected, G^1G^2max is a connected FGS. Conversely, assume that G^1 and G^2 are two connected FGSs such that G^1G^2max is a connected FGS. To prove the connectedness of FGS G^1, on contrary, assume that G^1 is not connected. Then, there exist at least two vertices of G^1 having no path connecting them. Since G^1G^2max is connected, then at least one path must exist between any two elements of V1×V2. Thus, there will be at least one path connecting the elements of G^1. This provides a contradiction. Hence, G^1 is a connected FGS.Theorem 4.
The number of connected components in the lexicographic-max productG^1G^2max of two FGSs G^1 and G^2 is equal to the number of components of G^1.Proof.
LetG^1 be a connected FGS and G^2 be FGS. Then, according to Theorem 3, the lexicographic-max product G^1G^2max is a connected FGS. This implies that both G^1 and G^1G^2max are connected. Suppose that G^1 is not a connected FGS having ‘m’ distinct connected components. Vertices of G^1 are renamed such that v1,v2,…,vk1, vk1+1,vk1+2,…,vk2,…,vkm+1,vkm+2,…,vkm+n are “m” distinct connected components of G^1. If w1,w2,…,wn are vertices of G^2, then for an arbitrary vertex of G^2, there must exist one version of each connected component of G^1 in the lexicographic-max product G^1G^2max. There is no edge among all these components. Corresponding to each edge between v1wi and vk1+1wi, there must be an edge between vertex v1 and vk1+1 of G^1 which gives a contradiction. Thus, each component of the lexicographic-max product G^1G^2max is distinct from others.Definition 4.
In the lexicographic-max productG^1G^2max=α,ν1,ν2,…,νn of FGSs G^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″, the degree of a vertex is defined as(5)dG^1G^2maxvi,wj=∑vivk∈Ri′,wl∈V2νi′vivk+∑wjwl∈Rj″,vi=vkνj″wjwl∨α1vi.
νi−the degree of a vertex in the lexicographic-max product G^1G^2max is defined as(6)νi−dG^1G^2maxvi,wj=∑vivk∈Ri′,wl∈V2νi′vivk+∑wjwl∈Ri″,vi=vkνi″wjwl∨α1vi.Example 3.
Consider two FGSsG^1=α1,μ1′,μ2′ and G^2=α2,μ1″,μ2″, which are shown in Figure 5.
The lexicographic-max product of FGSsG^1 and G^2 is shown in Figure 6.
To compute the degrees of all vertices in the lexicographic-max product, the following formula is utilized:(7)dG^1G^2maxvi,wj=∑vivk∈Ri′,wl∈V2νi′vivk+∑wjwl∈Rj″,vi=vkνj″wjwl∨α1vi,(8)dG^1G^2maxv1,w1=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w1w2∨α1v1=0.6+0.6+0.6+0.3∨0.6=1.8+0.6=2.4,(9)dG^1G^2maxv2,w1=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w1w2∨α1v2=0.6+0.6+0.6+0.3∨0.8=1.8+0.8=2.6,(10)dG^1G^2maxv1,w3=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν2″w3w2∨α1v1=0.6+0.6+0.6+0.4∨0.6=1.8+0.6=2.4,(11)dG^1G^2maxv2,w3=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν2″w3w2∨α1v2=0.6+0.6+0.6+0.4∨0.8=1.8+0.8=2.6,(12)dG^1G^2maxv4,w1=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν1″w1w2∨α1v4=0.4+0.4+0.4+0.3∨0.6=1.2+0.6=1.8,(13)dG^1G^2maxv3,w1=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν1″w1w2∨α1v3=0.4+0.4+0.4+0.3∨0.4=1.2+0.4=1.6,(14)dG^1G^2maxv4,w3=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν2″w3w2∨α1v4=0.4+0.4+0.4+0.4∨0.6=1.2+0.6=1.8,(15)dG^1G^2maxv3,w3=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν2″w3w2∨α1v3=0.4+0.4+0.4+0.4∨0.4=1.2+0.4=1.6,(16)dG^1G^2maxv3,w2=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν1″w2w1∨α1v3+ν2″w2w3∨α1v3=0.4+0.4+0.4+0.3∨0.4+0.4∨0.4=1.2+0.4+0.4=2.0,(17)dG^1G^2maxv4,w2=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν1″w2w1∨α1v4+ν2″w2w3∨α1v4=0.4+0.4+0.4+0.3∨0.6+0.4∨0.6=1.2+0.6+0.6=2.4,(18)dG^1G^2maxv1,w2=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w2w1∨α1v1+ν2″w2w3∨α1v1=0.6+0.6+0.6+0.3∨0.6+0.4∨0.6=1.8+0.6+0.6=3.0,(19)dG^1G^2maxv2,w2=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w2w1∨α1v2+ν2″w2w3∨α1v2=0.6+0.6+0.6+0.3∨0.8+0.4∨0.8=1.8+0.8+0.8=3.4.
νi−the degree of the vertex in the lexicographic-max product G^1G^2max is given by(20)νi−dG^1G^2maxvi,wj=∑vivk∈Ri′,wl∈V2νi′vivk+∑wjwl∈R′j′,vi=vkνi″wjwl∨α1vi.
Using the above given formula, we will calculateνi−the degree of the vertices in G^1G^2max as(21)ν1−dG^1G^2maxv1,w1=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w1w2∨α1v1=0.6+0.6+0.6+0.3∨0.6=1.8+0.6=2.4,(22)ν1−dG^1G^2maxv2,w1=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w1w2∨α1v2=0.6+0.6+0.6+0.3∨0.8=1.8+0.8=2.6,(23)ν1−dG^1G^2maxv1,w2=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w2w1∨α1v1=0.6+0.6+0.6+0.3∨0.6=1.8+0.6=2.4,(24)ν1−dG^1G^2maxv2,w2=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w2w1∨α1v2=0.6+0.6+0.6+0.3∨0.8=1.8+0.8=2.6,(25)ν2−dG^1G^2maxv4,w3=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν2″w3w2∨α1v4=0.4+0.4+0.4+0.4∨0.6=1.2+0.6=1.8,(26)ν2−dG^1G^2maxv3,w3=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν2″w3w2∨α1v3=0.4+0.4+0.4+0.4∨0.4=1.2+0.4=1.6,(27)ν2−dG^1G^2maxv3,w2=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν2″w2w3∨α1v3=0.4+0.4+0.4+0.4∨0.4=1.2+0.4=1.6,(28)ν2−dG^1G^2maxv4,w2=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν2″w2w3∨α1v4=0.4+0.4+0.4+0.4∨0.6=1.2+0.6=1.8,(29)ν1−dG^1G^2maxv1,w3=ν1′v1v2+ν1′v1v2+ν1′v1v2=0.6+0.6+0.6=1.8,(30)ν1−dG^1G^2maxv2,w3=ν1′v2v1+ν1′v2v1+ν1′v2v1=0.6+0.6+0.6=1.8,(31)ν2−dG^1G^2maxv4,w1=ν2′v4v3+ν2′v4v3+ν2′v4v3=0.4+0.4+0.4=1.2,(32)ν2−dG^1G^2maxv3,w1=ν2′v3v4+ν2′v3v4+ν2′v3v4=0.4+0.4+0.4=1.2,(33)ν1−dG^1G^2maxv4,w2=ν1″w2w1∨α1v4=0.3∨0.6=0.6,ν1−dG^1G^2maxv3,w2=ν1″w2w1∨α1v3=0.3∨0.4=0.4,ν1−dG^1G^2maxv3,w1=ν1″w1w2∨α1v3=0.3∨0.4=0.4,ν1−dG^1G^2maxv4,w1=ν1″w1w2∨α1v4=0.3∨0.6=0.6,ν2−dG^1G^2maxv2,w2=ν2″w2w3∨α1v2=0.4∨0.8=0.8,ν2−dG^1G^2maxv1,w2=ν2″w2w3∨α1v1=0.4∨0.6=0.6,ν2−dG^1G^2maxv2,w3=ν2″w2w3∨α1v2=0.4∨0.8=0.8,ν2−dG^1G^2maxv1,w3=ν2″w2w3∨α1v1=0.4∨0.6=0.6.Figure 5
Two fuzzy graph structures. (a)G1=α1,v1′,v2′. (b) G2=α2,v1′′,v2′′.
(a)(b)Figure 6
Lexicographic max-product of two FGSs.Theorem 5.
IfG^1 and G^2 are two FGSs such that νi″≥α1, then the degree of the vertex in the lexicographic-max product G^1G^2max of FGSs G^1 and G^2 is given by dG^1G^2maxvi,wj=dG^2wj+dG^1viV2.Proof.
LetG^1 and G^2 be two FGSs such that μi″≥α1, then μi″∨α1=μi″, and then the degree of the vertex in G^1G^2max (lexicographic-max product) is given by(34)dG^1G^2maxvi,vj=∑vivk∈Ri′,wl∈V2μi′vivk+∑wjwl∈Rj″,vi=vkμj″wjwl∨α1vi=V2∑vivk∈Ri′,wj=wlμi′vivk+∑wjwl∈Rj″,vi=vkμj″wjwl=V2dG^1vi+dG^2wj.Theorem 6.
IfG^1 and G^2 are two FGSs such that νi″≥α1, then νi−the degree of any vertex in the lexicographic-max product G^1G^2max of FGSs G^1 and G^2 is given by νi−dG^1G^2maxvi,wj=νi″−dG^2wj+νi′−dG^1viV2.Proof.
LetG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ be two FGSs such that νi″≥α1, then νi″∨α1=νi″, i=1,2,…,n. Then, νi−the degree of the vertex in G^1G^2max (lexicographic-max product) is given as(35)νi−dG^1G^2maxvi,wj=∑wjwl∈Ri″,vi=vkνi″wjwl∨α1vi+∑vivk∈Ri′,wl∈V2νi′vivk=∑wjwl∈Ri″,vi=vkνi″wjwl+V2∑vivk∈Ri′,wj=wlνi′vivk=νi″−dG^2wj+νi′−dG^1viV2.Example 4.
Consider two FGSsG^1=α1,μ1′,μ2′ and G^2=α2,μ1″,μ2″, shown in Figure 7.
In Figure7, μi″≥α1. The lexicographic-max product of G^1=α1,μ1′,μ2′ and G^2=α2,μ1″,μ2″ is shown in Figure 8.
Theorem6 implies the formula to determine the degrees of vertices in the lexicographic-max product as follows:(36)νi−dG^1G^2maxvi,wj=ν′i′−dG^2wj+νi′−dG^1viV2,(37)ν1−dG^1G^2maxv1,w1=V2ν1′−dG^1v1+ν1″−dG^2w1=30.6+0.8=1.8+0.8=2.6,(38)ν2−dG^1G^2maxv1,w1=V2ν2′−dG^1v1+ν2″−dG^2w1=30+0=0+0=0,(39)ν1−dG^1G^2maxv1,w2=V2ν1′−dG^1v1+ν1″−dG^2w2=30.6+0.8=1.8+0.8=2.6,(40)ν2−dG^1G^2maxv1,w2=V2ν2′−dG^1v1+ν2″−dG^2w2=30+0.8=0+0.8=0.8,(41)ν1−dG^1G^2maxv1,w3=V2ν1′−dG^1v1+ν1″−dG^2w3=30.6+0=1.8+0=1.8,(42)ν2−dG^1G^2maxv1,w3=V2ν2′−dG^1v1+ν2″−dG^2w3=30+0.8=0+0.8=0.8,(43)ν1−dG^1G^2maxv2,w1=V2ν1′−dG^1v2+ν1″−dG^2w1=30.6+0.8=1.8+0.8=2.6,(44)ν2−dG^1G^2maxv2,w1=V2ν2′−dG^1v2+ν2″−dG^2w1=30+0=0+0=0,(45)ν1−dG^1G^2maxv2,w2=V2ν1′−dG^1v2+ν1″−dG^2w2=30.6+0.8=1.8+0.8=2.6,(46)ν2−dG^1G^2maxv2,w2=V2ν2′−dG^1v2+ν2″−dG^2w2=30+0.8=0+0.8=0.8,(47)ν1−dG^1G^2maxv2,w3=V2ν1′−dG^1v2+ν1″−dG^2w3=30.6+0=1.8+0=1.8,(48)ν2−dG^1G^2maxv2,w3=V2ν2′−dG^1v2+ν2″−dG^2w3=30+0.8=30+0.8=0.8,(49)ν1−dG^1G^2maxv3,w1=V2ν1′−dG^1v3+ν1″−dG^2w1=30+0.8=0+0.8=0.8,(50)ν2−dG^1G^2maxv3,w1=V2ν2′−dG^1v3+ν2″−dG^2w1=30.4+0=1.2+0=1.2,(51)ν1−dG^1G^2maxv3,w2=V2ν1′−dG^1v3+ν1″−dG^2w2=30+0.8=0+0.8=0,(52)ν2−dG^1G^2maxv3,w2=V2ν2′−dG^1v3+ν2″−dG^2w2=30.4+0.8=1.2+0.8=2.0,(53)ν1−dG^1G^2maxv3,w3=V2ν1′−dG^1v3+ν1″−dG^2w3=30+0=0+0=0,(54)ν2−dG^1G^2maxv3,w3=V2ν2′−dG^1v3+ν2″−dG^2w3=30.4+0.8=1.2+0.8=2.0,(55)ν1−dG^1G^2maxv4,w1=V2ν1′−dG^1v4+ν1″−dG^2w1=30+0.8=0+0.8=0.8,(56)ν2−dG^1G^2maxv4,w1=V2ν2′−dG^1v4+ν2″−dG^2w1=30.4+0=1.2+0=1.2,(57)ν1−dG^1G^2maxv4,w2=V2ν1′−dG^1v4+ν1″−dG^2w2=30+0.8=0+0.8=0.8,(58)ν2−dG^1G^2maxv4,w2=V2ν2′−dG^1v4+ν2″−dG^2w2=30.4+0.8=1.2+0.8=2.0,(59)ν1−dG^1G^2maxv4,w3=V2ν1′−dG^1v4+ν1″−dG^2w3=30+0=0+0=0,(60)ν2−dG^1G^2maxv4,w3=V2ν2′−dG^1v4+ν2″−dG^2w3=30.4+0.8=1.2+0.8=2.0.
Through direct calculations,(61)ν1−dG^1G^2maxv4,w1=0.8,ν2−dG^1G^2maxv2,w3=0.8,ν1−dG^1G^2maxv3,w1=0.8,ν2−dG^1G^2maxv1,w3=0.8,ν1−dG^1G^2maxv3,w2=0.8,ν2−dG^1G^2maxv2,w2=0.8,ν1−dG^1G^2maxv4,w2=0.8,ν2−dG^1G^2maxv1,w2=0.8,ν1−dG^1G^2maxv4,w3=0,ν2−dG^1G^2maxv1,w1=0,ν1−dG^1G^2maxv3,w3=0,ν2−dG^1G^2maxv2,w1=0,ν1−dG^1G^2maxv1,w1=0.6+0.6+0.6+0.8=2.6,ν2−dG^1G^2maxv4,w3=0.4+0.4+0.4+0.8=2.0,ν1−dG^1G^2maxv2,w1=0.6+0.6+0.6+0.8=2.6,ν2−dG^1G^2maxv3,w3=0.4+0.4+0.4+0.8=2.0,ν2−dG^1G^2maxv3,w2=0.4+0.4+0.4+0.8=2.0,ν1−dG^1G^2maxv2,w2=0.6+0.6+0.6+0.8=2.6,ν2−dG^1G^2maxv4,w2=0.4+0.4+0.4+0.8=2.0,ν1−dG^1G^2maxv1,w2=0.6+0.6+0.6+0.8=2.6,ν1−dG^1G^2maxv1,w3=0.6+0.6+0.6=1.8,ν1−dG^1G^2maxv2,w3=0.6+0.6+0.6=1.8,ν2−dG^1G^2maxv3,w1=0.4+0.4+0.4=1.2,ν2−dG^1G^2maxv4,w1=0.4+0.4+0.4=1.2.
This example explains that the degrees of both methods are the same.Figure 7
Two fuzzy graph structures. (a)G1=α1,μ1′,μ2′. (b) G2=α2,μ1′′,μ2′′.
(a)(b)Figure 8
Lexicographic-max product of two FGSs.Theorem 7.
IfG^1 and are two FGSs such that νi″≤α1, then νi−the degree of the vertex in the lexicographic-max product G^1G^2max of FGSs G^1 and G^2 is given by νi−dG^1G^2maxvi,wj=dG^2∗wjα1vi+νi′−dG^1viV2.Proof.
LetG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ be two FGSs such that νi″≤α1, then νi″∨α1=α1, i=1,2,…,n. νi−the degree of the vertex in G^1G^2max (lexicographic-max product) is defined as(62)νi−dG^1G^2maxvi,wj=∑wjwl∈R′i′,vi=vkνi″wjwl∨α1vi+∑vivk∈Ri′,wl∈V2νi′vivk=∑wjwl∈R′i′,vi=vkα1vi+V2νi′−dG^1vi=dG^2∗wjα1vi+V2νi′−dG^1vi.Corollary 1.
IfG^1 and G^2 are two FGSs such that νi″≤α1 and α1 is considered as a constant function having a value c, then νi−the degree of the vertex in the lexicographic-max product G^1G^2max of FGSs G^1 and G^2 is given by νi−dG^1G^1G^2maxvi,wj=dG^2∗wjc+νi−dG^1viV2.Remark 1.
IfG^1 and G^2 are regular FGSs, then their lexicographic-max product G^1G^2max is not a regular FGS.Example 5.
Consider two FGSsG^1=α1,ν1′,ν2′ and G2=α2,ν1″,ν2″ as shown in Figure 9.
It is clear from Figure9 that each vertex of G^1=α1,ν1′,ν2′ has one ν1′ edge with the same degree of membership, that is, 0.7. Hence, G^1 is a 0.7 ν1′-regular FGS. Moreover, each vertex of G^2=α2,ν1″,ν2″ has one ν1″ edge with the same degree of membership, that is, 0.5. Hence, G^2 is a 0.5 ν1″-regular FGS. The lexicographic-max product of G^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″ is shown in Figure 10.
In Figure10, all green-colored edges are ν1′ edges with a membership value 0.7, all yellow-edges are ν2′ edges with a membership value 0.7, all orange-colored edges are ν1′ edges with a membership value 0.7, all purple-colored edges are ν2′ edges with a membership value 0.7, and all blue-colored edges are ν1′ edges with a membership value 0.7. It is clear from Figure 10 that each vertex in the lexicographic-max product G^1G^2max does not have any type of edges with the same membership value. For example, the vertex v1,w1 has five ν1 edges, four ν1 edges with a 0.7 membership value, and one ν1 edge with a membership value 0.8, while the vertex v3,w1 has five ν1 edges, four ν1 edges having a membership value 0.7, and one ν1 edge with a membership value 0.9. Hence, G^1G^2max is not a regular FGS. This example shows that the lexicographic-max product of two regular FGSs is not a regular FGS.Figure 9
ν1′-regular and ν1′′-regular FGSs. (a) G^1=α1,ν1′,ν2′. (b) G^2=α2,ν1′′,ν2′′.
(a)(b)Figure 10
Lexicographic max-product ofG^1 and G^2.Remark 2.
IfG^1 and G^2 are partially regular FGSs, then the lexicographic-max product G^1G^2max is a partially regular FGS since G^1∗ and G2∗ are regular FGSs.Theorem 8.
IfG^1 and G^2 are two FGSs such that νi″≥α1, then the lexicographic-max product G^1G^2max is a νi−regular FGS if and only if G^1 is a νi′−regular FGS and G^2 is a νi″−regular FGS.Proof.
LetG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ be two FGSs such that νi″≥α1. Then, νi−the degree of the vertex in the lexicographic-max product G^1G^2max is given by(63)νi−dG^1G^2maxvi,wj=νi″−dG^2wj+νi′−dG^1viV2.
Assume thatG^1=α1,ν1′,ν2′,…,νn′ is a νi′−regular FGS of degree k1 and G^2=α2,ν1″,ν2″,…,νn″ is a νi″−regular FGS of degree k2, then(64)νi−dG^1G^2maxvi,wj=k2+k1V2.
Sincek1, k2, and V2 are constants, so νi−dG^1G^2max is a constant for all elements of V1×V2. Thus, the lexicographic-max product G^1G^2max is a νi−regular FGS.
Conversely, suppose thatG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ are two FGSs such that νi″≥α1 and the lexicographic-max product G^1G^2max=α,ν1,ν2,…,νn is a νi−regular FGS. Then, νi−the degrees of vertices v1,v2 and w1,w2 in V1×V2 are equal, that is,(65)νi−dG^1G^2maxv1,v2=νi−dG^1G^2maxw1,w2,νi″−dG^2v2+νi′−dG^1v1V2=νi″−dG^2w2+νi′−dG^1w1V2.
Fixingv∈V2 and considering v1,v and w1,v in V1×V2, where v1andw1 are taken as the arbitrary objects of V1, then the above expression is converted as(66)νi″−dG^2v+νi′−dG^1v1V2=νi″−dG^2v+νi′−dG^1w1V2,νi′−dG^1v1V2=νi′−dG^1w1V2,νi−dG^1v1=νi−dG^1w1.
This holds for all verticesv1 and w1 of V1. Thus, G^1=α1,ν1′,ν2′,…,νn′ is a νi′−regular FGS.
Now fixingv′∈V1 and considering v′,v2 and v′,w2 in V1×V2, where v2andw2 are arbitrary elements of the vertex set V1, then the above given equation becomes(67)νi″−dG^2v2+νi′−dG^1v′V2=νi″−dG^2w2+νi′−dG^1v′V2,νi″−dG^2v2=νi″−dG^2w2.
This holds for all elementsv2 and w2 of V2. Thus, G^2=α2,ν1″,ν2″,…,νn″ is a νi″−regular FGS.Theorem 9.
IfG^1 is a νi′−regular FGS and G^2 is a partially regular FGS such that νi″∨α1 is a constant, then the lexicographic-max product G^1G^2max of G^1 and G^2 is a νi-regular FGS.Proof.
LetG^1=α1,ν1′,ν2′,…,νn′ be a νi′−regular FGS of degree k1 and G^2=α2,ν1″,ν2″,…,νn″ be a partially regular FGS such that G^2∗=V2,R1″,R2″,…,Rn″ (underlying GS) is an r1−regular GS.
Moreover,νi″∨α1 is a constant, that is, νi″∨α1=c, where c is a constant.
Then, the degree of the vertex in the lexicographic-max productG^1G^2max of FGSs G^1 and G^2 is given by(68)νi−dG^1G^2maxui,wj=dG^2∗wjν′i′∨α1+νi′−dG^1viV2=dG^2∗wjc+νi′−dG^1viV2=r1c+k1V2.
Sincec, k1, r1, and V2 are constants, so νi−dG^1G^2max is a constant for all elements of V1×V2. Thus, the lexicographic-max product G^1G^2max of FGSs G^1 and G^2 is a νi−regular FGS.Theorem 10.
IfG^1 and G^2 are two FGSs such that νi′≥α2, then the lexicographic-max product G^1G^2max is a νi−regular FGS if and only if G^1 is a −νi′−regular FGS and G^2 is a νi″−regular FGS.Proof.
LetG^1=α1,ν1′,ν2′,…,νn′ and G2=α2,ν1″,ν2″,…,νn″ be two FGSs such that νi′≥α2. Then, νi−the degree of the vertex in the lexicographic-max product G^1G^2max is given by(69)νi−dG^1G^2maxvi,wj=V1νi″−dG^2wj+νi′−dG^1vi.
Assume thatG^1=α1,ν1′,ν2′,…,νn′ is a νi′−regular FGS of degree k1 and G^2=α2,ν1″,ν2″,…,νn″ is a νi″−regular FGS of degree k2:(70)νi−dG^1G^2maxvi,wj=V1k2+k1.
Sincek1, k2, and V1 are constants, so νi−dG^2G^1max is constant for all elements of V1×V2. Thus, the lexicographic-max product G^2G^1max=α,ν1,ν2,…,νn is a νi−regular FGS.
Conversely, suppose that the lexicographic-max productG^2G^1max=α,ν1,ν2,…,νn is a νi−regular FGS. Then, νi−the degrees of two vertices v1,v2 and w1,w2 in V1×V2 are equal, that is,(71)νi−dG^2G^1maxv1,v2=νi−dG^2G^1maxw1,w2,V1νi″−dG^2v2+νi′−dG^1v1=V1νi″−dG^2w2+νi′−dG^1w1.
Fixingv∈V2 and considering v1,v and w1,v in V1×V2, where v1andw1 are arbitrary elements of the vertex set V1, then the above equation becomes(72)V1νi′′−dG^2v+νi′−dG^1v1=V1νi′′−dG^2v+νi′−dG^1w1,νi′−dG^1v1=νi′−dG^1w1.
This holds for all elementsv1 and w1 of V1. Thus, G^1=α1,ν1′,ν2′,…,νn′ is a νi′−regular FGS.
Now fixingv′∈V1 and considering v′,v2 and v′,w2 in V1×V2, where v2andw2 are arbitrary elements of the vertex set V2, then the above equation becomes(73)V1νi″−dG^2v2+νi″−dG^1v′=V1νi″−dG^2w2+νi″−dG^1v′,V1νi″−dG^2v2=V1νi″−dG^2w2,νi″−dG^2v2=νi″−dG^2w2.
This holds for all verticesv2 and w2 of V2. Thus, G^2=α2,ν1″,ν2″,…,νn″ is a νi″−regular FGS.Definition 5.
The total degree of the 5 lexicographic-max productG^1G^2max=α,ν1,ν2,…,νn of FGSs G^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ is defined as(74)tdG^1G^2maxui,vj=∑uiuk∈Ri′,vl∈V2νi′uiuk+∑vjvl∈Ri″,ui=ukνj″vjvl∨α1ui+αui,vj.
νi−the total degree of the vertex in G^1G^2max is defined as(75)νi−tdG^1G^2maxui,vj=∑uiuk∈Ri′,vl∈V2νi′uiuk+∑vjvl∈Ri″,ui=ukνi″vjvl∨α1ui+αui,vjn.Example 6.
Consider two FGSsG^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″, as shown in Figure 11.
The lexicographic-max product ofG^1=α1,ν1′,ν2′ and G^2=α2,ν1″,ν2″ is shown in Figure 12.
The total degrees of vertices in the lexicographic-max product are computed as(76)tdG^1G^2maxvi,wj=∑vivk∈Ri′,wl∈V2νi′vivk+∑wjwl∈Rj″,vi=vkνj″wjwl∨α1vi+αvi,wj,(77)tdG^1G^2maxv1,w1=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w1w2∨α1v1+αv1,w1=0.7+0.7+0.7+0.4∨0.7+0.7=2.1+0.7+0.7=3.5,(78)tdG^1G^2maxv2,w1=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w1w2∨α2v2+αv2,w1=0.7+0.7+0.7+0.4∨0.9+0.9=2.1+0.9+0.9=3.9,(79)tdG^1G^2maxv1,w3=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν2″w3w2∨α1v1+αv1,w3=0.7+0.7+0.7+0.5∨0.7+0.7=2.1+0.7+0.7=3.5,(80)tdG^1G^2maxv2,w3=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν2″w3w2∨α1v2+αv2,w3=0.7+0.7+0.7+0.5∨0.9+0.9=2.1+0.9+0.9=3.9,(81)tdG^1G^2maxv4,w1=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν1″w1w2∨α1v4+αv4,w1=0.5+0.5+0.5+0.4∨0.7+0.7=1.5+0.7+0.7=2.9,(82)tdG^1G^2maxv3,w1=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν1″w1w2∨α1v3+αv3,w1=0.5+0.5+0.5+0.4∨0.5+0.5=1.5+0.5+0.5=2.5,(83)tdG^1G^2maxv4,w3=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν2″w3w2∨α1v4+αv4,w3=0.5+0.5+0.5+0.5∨0.7+0.7=1.5+0.7+0.7=2.9,(84)tdG^1G^2maxv3,w3=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν2″w3w2∨α1v3+αv3,w3=0.5+0.5+0.5+0.5∨0.5+0.6=1.5+0.5+0.6=2.6,(85)tdG^1G^2maxv3,w2=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν1″w2w1∨α1v3+ν2″w2w3∨α1v3+αv3,w2=0.5+0.5+0.5+0.4∨0.5+0.5∨0.5+0.5=1.5+0.5+0.5+0.5=3.0,(86)tdG^1G^2maxv4,w2=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν1″w2w1∨α1v4+ν2″w2w3∨α1v4+αv4,w2=0.5+0.5+0.5+0.4∨0.7+0.5∨0.7+0.7=1.5+0.7+0.7+0.7=3.6,(87)tdG^1G^2maxv1,w2=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w2w1∨α1v1+ν2″w2w3∨α1v1+αv1,w2=0.7+0.7+0.7+0.4∨0.7+0.5∨0.7+0.7=2.1+0.7+0.7+0.7=4.2,(88)tdG^1G^2maxv2,w2=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w2w1∨α1v2+ν2″w2w3∨α1v2+αv2,w2=0.7+0.7+0.7+0.4∨0.9+0.5∨0.9+0.9=2.1+0.9+0.9+0.9=4.8.
To calculateνi−the total degree of a vertex in G^1G^2max, we will use the following formula:(89)νi−tdG^1G^2maxui,vj=∑vivk∈Ri′,wl∈V2νi′vivk+∑wjwl∈Ri″,vi=vkν′i′wjwl∨σ1vi+αvi,wjn,(90)ν1−tdG^1G^2maxv1,w1=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w1w2∨σ1v1+αv1,w12=0.7+0.7+0.7+0.4∨0.7+0.72=2.1+0.7+0.35=3.15,(91)ν1−tdG^1G^2maxv2,w1=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w1w2∨α1v2+σv2,w12=0.7+0.7+0.7+0.4∨0.9+0.92=2.1+0.9+0.45=3.45,(92)ν1−tdG^1G^2maxv1,w2=ν1′v1v2+ν1′v1v2+ν1′v1v2+ν1″w2w1∨α1v1+αv1,w22=0.7+0.7+0.7+0.4∨0.7+0.72=2.1+0.7+0.35=3.15,(93)ν1−tdG^1G^2maxv2,w2=ν1′v2v1+ν1′v2v1+ν1′v2v1+ν1″w2w1∨α1v2+αv2,w22=0.7+0.7+0.7+0.4∨0.9+0.92=2.1+0.9+0.45=3.45,(94)ν2−tdG^1G^2maxv4,w3=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν2″w3w2∨α1v4+αv4,w32=0.5+0.5+0.5+0.5∨0.7+0.72=1.5+0.7+0.35=2.55,(95)ν2−tdG^1G^2maxv3,w3=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν2″w3w2∨α1v3+αv3,w32=0.5+0.5+0.5+0.5∨0.5+0.62=1.5+0.5+0.3=2.3,(96)ν2−tdG^1G^2maxv3,w2=ν2′v3v4+ν2′v3v4+ν2′v3v4+ν2″w2w3∨α1v3+αv3,w22=0.5+0.5+0.5+0.5∨0.5+0.52=1.5+0.5+0.25=2.25,(97)ν2−tdG^1G^2maxv4,w2=ν2′v4v3+ν2′v4v3+ν2′v4v3+ν2″w2w3∨α1v4+αv4,w22=0.5+0.5+0.5+0.5∨0.7+0.72=1.5+0.7+0.35=2.55,(98)ν1−tdG^1G^2maxv1,w3=ν1′v1v2+ν1′v1v2+ν1′v1v2+σv1,w32=0.7+0.7+0.7+0.72=2.1+0.35=2.45,(99)ν1−tdG^1G^2maxv2,w3=ν1′v2v1+ν1′v2v1+ν1′v2v1+αv2,w32=0.7+0.7+0.7+0.92=2.1+0.45=2.55,(100)ν2−tdG^1G^2maxv4,w1=ν2′v4v3+ν2′v4v3+ν2′v4v3+αv4,w12=0.5+0.5+0.5+0.72=1.5+0.35=1.85,(101)ν2−tdG^1G^2maxv3,w1=ν2′v3v4+ν2′v3v4+ν2′v3v4+αv3,w12=0.5+0.5+0.5+0.52=1.5+0.25=1.75,(102)ν1−tdG^1G^2maxv4,w2=ν1″w2w1∨α1v4+αv4,w22=0.4∨0.7+0.72=0.7+0.35=1.05,(103)ν1−tdG^1G^2maxv3,w2=ν1″w2w1∨α1v3+αv3,w22=0.4∨0.5+0.52=0.5+0.25=0.75,(104)ν1−tdG^1G^2maxv3,w1=ν1″w1w2∨α1v3+αv3,w12=0.4∨0.5+0.52=0.5+0.25=0.75,(105)ν1−tdG^1G^2maxv4,w1=ν1″w1w2∨α1v4+αv4,w12=0.4∨0.7+0.72=0.7+0.35=1.05,(106)ν2−tdG^1G^2maxv2,w2=ν2″w2w3∨α1v2+αv2,w22=0.5∨0.9+0.92=0.9+0.45=1.35,(107)ν2−tdG^1G^2maxv1,w2=ν2″w2w3∨α1v1+σv1,w22=0.5∨0.7+0.72=0.7+0.35=1.05,(108)ν2−tdG^1G^2maxv2,w3=ν2″w3w2∨α1v2+αv2,w32=0.5∨0.9+0.92=0.9+0.45=1.35,(109)ν2−tdG^1G^2maxv1,w3=ν2″w3w2∨α1v1+αv1,w32=0.5∨0.7+0.72=0.7+0.35=1.05,(110)ν1−tdG^1G^2maxv4,w3=αv4,w32=0.72=0.35,ν1−tdG^1G^2maxv3,w3=αv3,w32=0.62=0.3,ν2−tdG^1G^2maxv1,w1=αv1,w12=0.72=0.35,ν2−tdG^1G^2maxv2,w1=αv2,w12=0.92=0.45.Figure 11
Two FGSs. (a)G1=α1,ν1′,ν2′. (b) G2=α2,ν1′′,ν2′′.
(a)(b)Figure 12
Lexicographic-max productG^1G^2max.
(a)(b)Remark 3.
The total degree of each vertex inG^1G^2max is equal to the sum of its νi−total degrees, that is,(111)tdG^1G^2maxui,vj=∑i=1nνi−tdG^1G^2maxui,vj.Example 7.
It is clear from Example6 that(112)ν1−tdG^1G^2maxv1,w1+ν2−tdG^1G^2maxv1,w1=3.15+0.35=3.5=tdG^1G^2maxv1,w1,ν1−tdG^1G^2maxv1,w2+ν2−tdG^1G^2maxv1,w2=3.15+1.05=4.2=tdG^1G^2maxv1,w2,ν1−tdG^1G^2maxv1,w3+ν2−tdG^1G^2maxv1,w3=2.45+1.05=3.5=tdG^1G^2maxv1,w3,ν1−tdG^1G^2maxv2,w1+ν2−tdG^1G^2maxv2,w1=3.45+0.45=3.9=tdG^1G^2maxv2,w1,ν1−tdG^1G^2maxv2,w2+ν2−tdG^1G^2maxv2,w2=3.45+1.35=4.8=tdG^1G^2maxv2,w2,ν1−tdG^1G^2maxv2,w3+ν2−tdG^1G^2maxv2,w3=2.55+1.35=3.9=tdG^1G^2maxv2,w3,ν1−tdG^1G^2maxv3,w1+ν2−tdG^1G^2maxv3,w1=0.75+1.75=2.5=tdG^1G^2maxv3,w1,ν1−tdG^1G^2maxv3,w2+ν2−tdG^1G^2maxv3,w2=0.75+2.25=3.0=tdG^1G^2maxv3,w2,ν1−tdG^1G^2maxv4,w1+ν2−tdG^1G^2maxv4,w1=1.05+1.85=2.9=tdG^1G^2maxv4,w1,ν1−tdG^1G^2maxv4,w2+ν2−tdG^1G^2maxv4,w2=1.05+2.55=3.6=tdG^1G^2maxv4,w2,ν1−tdG^1G^2maxv4,w3+ν2−tdG^1G^2maxv4,w3=0.35+2.55=2.9=tdG^1G^2maxv4,w3,ν1−tdG^1G^2maxv3,w3+ν2−tdG^1G^2maxv3,w3=0.3+2.3=2.6=tdG^1G^2maxv3,w3.
This example shows that the total degree of each vertex inG^1G^2max is equal to the sum of its νi−total degrees.Theorem 11.
IfG^1 and G^2 are two FGSs such that νi″≥α1, then the lexicographic-max product G^1G^2max is a totally regular FGS if and only if G^1 and G^2 are totally regular FGSs.Proof.
LetG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ be two FGSs such that νi″≥α1. Then, the total degree of the vertex in the lexicographic-max product G^1G^2max is given by(113)tdG^1G^2maxvi,wj=tdG^2wj+tdG^1viV2.
Assuming thatG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ are two totally regular FGSs of degree k1 and k2, respectively, then(114)tdG^1G^2maxvi,wj=k2+k1V2.
Sincek1, k2, and V2 are constants, so dG^1G^2max is a constant for all elements of V1×V2. Thus, the lexicographic max-product G^1G^2max=α,ν1,ν2,…,νn is a totally regular FGS.
Conversely, suppose thatG^1=α1,ν1′,ν2′,…,νn′ and G^2=α2,ν1″,ν2″,…,νn″ are two FGSs such that νi″≥α1 and the lexicographic-max product G^1G^2max=α,ν1,ν2,…,νn is a totally regular FGS. Then, the total degrees of two vertices v1,v2 and w1,w2 in V1×V2 are equal, that is,(115)tdG^1G^2maxv1,v2=tdG^1G^2maxw1,w2,tdG^2v2+tdG^1v1V2=tdG^2w2+tdG^1w1V2.
Fixingv∈V2 and considering v1,v and w1,v in V1×V2, where v1andw1 are arbitrary elements of the vertex set V1, then the above equation becomes(116)tdG^2v+tdG^1v1V2=tdG^2v+tdG^1w1V2,tdG^1v1V2=tdG^1w1V2,tdG^1v1=tdG^1w1.
This holds for all elementsv1 and w1 of V1. Thus, G^1=α1,ν1′,ν2′,…,νn′ is a totally regular FGS.
Now fixingv′∈V1 and supposing that v′,v2 and v′,w2 in V1×V2, where v2andw2 are taken as the arbitrary vertices of the set V2, then the above expression changes to(117)tdG^2v2+tdG^1v′V2=tdG^2w2+tdG^1v′V2,tdG^2v2=tdG^2w2.
This holds for all verticesv2 and w2 of V2. Thus, G^2=α2,ν1″,ν2″,…,νn″ is a totally regular FGS.Theorem 12.
IfG^1 and G^2 are two FGSs such that νi″≥α1, then the lexicographic-max product G^1G^2max is a regular FGS on complete GS if and only if G^1 and G^2 are regular FGSs on complete GSs G^1∗ and G^2∗, respectively.Proof.
The proof of Theorem11 implies the proof of this theorem.


## 3. Applications

### 3.1. Detection of Marine Crimes

Marine crime is as old as this industry itself. Depending on the times that it prevailed in, the nature of the maritime crime has changed a lot over the years, but its implications remain just as severe. Marine crime is not only a threat to entire maritime security of goods and people in the industry, but also gives major setbacks to the entire marine industry economically. Marine crimes are usually held during the maritime trade. Seaports are very important in the whole world for trade through waters. There are almost four thousand seven hundred and sixty-four seaports, which are frequently used for import and export of various types of goods. Seaports have been very useful and beneficial for hundreds of years. But, they have also been used for the illegal trade due to which seaports always remain sensitive for crimes. Nowadays, crimes occurring on the seaports are in abundance, but some crimes including human trafficking, smuggling of precious metals, illegal carrying of weapons and artillery, smuggling of exotic plants, trade of drugs, and smuggling of exotic animals are crucial. All these crimes pose threat to the maritime security on a daily basis. Security of waters is still suffering because this is a very broad aspect without concrete boundaries. A fuzzy graph structure of marine crimes occurring on seaports can be very helpful to combat these crimes. Using the fuzzy graph structure, we can investigate that between any two seaports which crime is chronic and increasing violently. Moreover, we can decide which seaport is the most sensitive for a particular crime and needed to be focused by security teams. Consider a setV of eight seaports:(118)V=Mongla, Bintulu, Sembawang, Abu Dhabi, Gwadar, Chennai, Malindi, Ciro Marina.Letα be defined as a fuzzy set on V, as given in Table 1.Table 1
Fuzzy setα of seaports.
SeaportDegree of membershipMongla0.8Bintulu0.9Sembawang0.7Abu Dhabi0.8Gwadar0.7Chennai0.9Malindi0.8Ciro Marina0.5In Table1, the membership value of a seaport depicts its importance in the world due to its geographic position.In Tables2–11, we have mentioned membership values of marine crimes among each pair of seaports. The membership value of each pair of seaports is calculated according to νv1v2≤αv1∧αv2, for all v1,v2∈V. By using these membership values, we show the severity level of each marine crime among each pair of seaports.Table 2
Fuzzy set of marine crimes among Gwadar and other seaports.
Type of marine crimes(Gwadar, Chennai)(Gwadar, Malindi)(Gwadar, Ciro Marina)(Gwadar, Mongla)Human trafficking0.70.60.40.1Illegal carrying of weapons and artillery0.20.10.20.1Smuggling of exotic plants0.10.40.20.1Smuggling of precious metals0.10.10.20.3Trade of drugs0.20.70.30.5Smuggling of exotic animals0.20.20.30.1Table 3
Fuzzy set of marine crimes among Malindi and other seaports.
Type of marine crimes(Malindi, Mongla)(Malindi, Bintulu)(Malindi, Sembawang)(Malindi, Chennai)Human trafficking0.80.20.50.7Illegal carrying of weapons and artillery0.60.30.40.6Smuggling of exotic plants0.10.20.30.4Smuggling of precious metals0.60.80.60.6Trade of drugs0.20.50.30.5Smuggling of exotic animals0.30.30.40.4Table 4
Fuzzy set of marine crimes among Ciro Marina and other seaports.
Type of marine crimes(Ciro Marina, Sembawang)(Ciro Marina, Abu Dhabi)(Ciro Marina, Bintulu)Human trafficking0.10.10.1Illegal carrying of weapons and artillery0.10.20.1Smuggling of exotic plants0.20.50.0Smuggling of precious metals0.10.20.2Trade of drugs0.50.20.4Smuggling of exotic animals0.20.10.5Table 5
Fuzzy set of marine crimes among Sembawang and other seaports.
Type of marine crimes(Sembawang, Gwadar)(Sembawang, Chennai)(Sembawang, Mongla)Human trafficking0.10.20.6Illegal carrying of weapons and artillery0.20.30.4Smuggling of exotic plants0.20.20.6Smuggling of precious metals0.10.20.7Trade of drugs0.70.10.4Smuggling of exotic animals0.30.70.6Table 6
Fuzzy set of marine crimes among Mongla and other seaports.
Type of marine crimes(Mongla, Bintulu)(Mongla, Sembawang)(Mongla, Abu Dhabi)Human trafficking0.80.50.8Illegal carrying of weapons and artillery0.60.20.4Smuggling of exotic plants0.10.60.3Smuggling of precious metals0.70.70.4Trade of drugs0.50.50.6Smuggling of exotic animals0.30.60.4Table 7
Fuzzy set of marine crimes among Chennai and other seaports.
Type of marine crimes(Chennai, Mongla)(Chennai, Malindi)(Chennai, Ciro Marina)Human trafficking0.80.50.2Illegal carrying of weapons and artillery0.60.60.2Smuggling of exotic plants0.20.60.1Smuggling of precious metals0.60.50.2Trade of drugs0.50.50.4Smuggling of exotic animals0.60.50.5Table 8
Fuzzy set of marine crimes among Abu Dhabi and other seaports.
Type of marine crimes(Abu Dhabi, Gwadar)(Abu Dhabi, Chennai)(Abu Dhabi, Malindi)Human trafficking0.30.80.5Illegal carrying of weapons and artillery0.20.60.4Smuggling of exotic plants0.50.50.8Smuggling of precious metals0.60.40.7Trade of drugs0.70.50.6Smuggling of exotic animals0.30.30.6Table 9
Fuzzy set of marine crimes among some seaports.
Type of marine crimes(Mongla, Gwadar)(Ciro Marina, Malindi)(Sembawang, Abu Dhabi)Human trafficking0.10.10.5Illegal carrying of weapons and artillery0.20.20.4Smuggling of exotic plants0.20.50.2Smuggling of precious metals0.40.10.4Trade of drugs0.70.30.7Smuggling of exotic animals0.30.20.2Table 10
Fuzzy set of marine crimes among Bintulu and other seaports.
Type of marine crimes(Bintulu, Mongla)(Bintulu, Sembawang)(Bintulu, Abu Dhabi)Human trafficking0.80.50.7Illegal carrying of weapons and artillery0.60.40.8Smuggling of exotic plants0.10.10.3Smuggling of precious metals0.70.20.4Trade of drugs0.50.30.6Smuggling of exotic animals0.30.50.6Table 11
Fuzzy set of marine crimes among some seaports.
Type of marine crimes(Bintulu, Gwadar)(Chennai, Bintulu)(Abu Dhabi, Ciro Marina)Human trafficking0.70.30.3Illegal carrying of weapons and artillery0.60.40.2Smuggling of exotic plants0.20.50.5Smuggling of precious metals0.60.90.1Trade of drugs0.50.30.1Smuggling of exotic animals0.50.40.1On the setV, many relations can be defined; let us define the following relations on V. R1 = human trafficking, R2 = illegal carrying of weapons and artillery, R3 = smuggling of exotic plants, R4 = smuggling of precious metals, R5 = trade of drugs, and R6 = smuggling of exotic animals such that V,R1,R2,R3,R4,R5,R6 is a graph structure. Each element in the relation depicts a particular kind of the most crucial marine crimes among those two seaports.AsV,R1,R2,R3,R4,R5,R6 is the graph structure, therefore a pair of seaports can appear in just one relation. Hence, it would be considered an element of that relation, for which its membership value is comparatively high than that of the other relations. Using the above given data, elements in relations are paired with their membership values, and resulting sets are the fuzzy sets on R1, R2, R3, R4, R5, and R6, respectively. These fuzzy sets are named as ν1, ν2, ν3, ν4, ν5, and ν6, respectively. Let(119)R1=Bintulu, Mongla,Bintulu, Gwadar,Mongla,Chennai,Abu Dhabi, Mongla,R2=Abu Dhabi, Bintulu,R3=Abu Dhabi, Malindi,Ciro Marina, Malindi,R4=Bintulu, Chennai,Sembawang, Mongla,R5=Gwadar, Malindi,Ciro Marina, Sembawang,R6=Bintulu, Ciro Marina.And, their corresponding fuzzy sets are(120)ν1=Bintulu, Mongla,0.8,Bintulu, Gwadar,0.7,Mongla, Chennai,0.8,Abu Dhabi, Mongla,0.8,ν2=Abu Dhabi, Bintulu,0.8,ν3=Abu Dhabi, Malindi,0.8,Ciro Marina, Malindi,0.5,ν4=Bintulu, Chennai,0.9,Sembawang, Mongla,0.7,ν5=Gwadar, Malindi,0.7,Ciro Marina, Sembawang,0.5,ν6=Bintulu, Ciro Marina,0.5.Clearly,α,ν1,ν2,ν3,ν4,ν5,ν6 is a fuzzy graph structure and is shown in Figure 13.Figure 13
Fuzzy graph structure depicting most crucial marine crimes among different seaports.In FGS shown in Figure13, each edge depicts the most crucial marine crime among the corresponding seaports. For example, most crucial marine crime among Bintulu and Mongla is human trafficking, and its membership value is 0.8. It can be noted that the vertex Bintulu has the highest vertex degree for the relation human trafficking, and it means that Bintulu has human trafficking as the most crucial marine crime with other seaports. Moreover, according to this fuzzy graph structure, Abu Dhabi and Malindi have smuggling of exotic plants as the most crucial marine crime at this time with a membership value 0.8. An FGS of all seaports can be very helpful for security agencies of all countries. It highlights those marine crimes which needed to be solved promptly.


### 3.2. Detection of Road Crimes

Road network is very important for any country. Roads connecting cities are the major source of trade, traveling, and many other purposes. But, nowadays, roads have become unsafe due to large amount of crimes happening on the roads. Crimes occurring on roads are very large in number, but some crimes are very crucial including child kidnapping, bike robbery, mobile snatching, women harassment, car robbery, and robbery of animals. Using a fuzzy graph structure, we can investigate that which road is the most crucial for a particular crime. An FGS can also tell us which crime is the most chronic and which one is increasing rapidly. Furthermore, we can judge which road is the most crucial for a particular road crime. An FGS of road crimes can be very helpful for the police department to overcome a particular crime, and it can guide them in making a policy against a crime. For example, if an FIR is written of a crime held in a particular area, an FGS of that area can be very helpful for the police department in taking action against that crime, and it will guide them to which roads are the most sensitive regarding that particular crime. So, police can put check posts on those roads, and police can cordon off the area near to those roads. Consider a setA of eight cities. A = {Islamabad, Karachi, Rawalpindi, Lahore, Multan, Jhang, Faisalabad, Sheikhupura}. Let F be considered as a fuzzy set on A, as given in Table 12.Table 12
Fuzzy set F onA.
CityMembership valueIslamabad0.9Karachi0.5Rawalpindi0.8Lahore0.8Multan0.7Jhang0.8Faisalabad0.7Sheikhupura0.7In Table12, the membership value shows the solidity of that particular city corresponding to economical progress, educational and health facilities, and peace. In Tables 13–20, we have shown membership values of different crimes on roads connecting every pair of cities.Table 13
Fuzzy set of crimes on roads connecting Islamabad and other cities.
Type of crime(Islamabad, Karachi)(Islamabad, Rawalpindi)(Islamabad, Lahore)(Islamabad, Multan)Child kidnapping0.60.50.50.7Bike robbery0.60.60.40.6Mobile snatching0.50.60.30.5Women harassment0.60.50.40.6Car robbery0.50.50.60.5Robbery of animals0.30.50.40.4Table 14
Fuzzy set of crimes on roads connecting Rawalpindi and other cities.
Type of crime(Rawalpindi, Lahore)(Rawalpindi, Multan)(Rawalpindi, Jhang)(Rawalpindi, Islamabad)Child kidnapping0.70.60.70.7Bike robbery0.80.70.40.6Mobile snatching0.60.60.80.7Women harassment0.80.70.60.8Car robbery0.70.50.60.7Robbery of animals0.40.50.50.3Table 15
Fuzzy set of crimes on roads connecting Faisalabad and other cities.
Type of crime(Faisalabad, Islamabad)(Faisalabad, Karachi)(Faisalabad, Rawalpindi)(Faisalabad, Jhang)Child kidnapping0.70.50.60.5Bike robbery0.50.40.40.6Mobile snatching0.70.50.60.6Women harassment0.60.40.70.5Car robbery0.70.50.60.6Robbery of animals0.30.50.60.7Table 16
Fuzzy set of crimes on roads connecting Sheikhupura and other cities.
Type of crime(Sheikhupura, Rawalpindi)(Islamabad, Sheikhupura)(Sheikhupura, Karachi)(Sheikhupura, Faisalabad)Child kidnapping0.70.60.40.5Bike robbery0.60.60.40.6Mobile snatching0.70.50.40.7Women harassment0.60.60.40.5Car robbery0.70.70.50.4Robbery of animals0.60.40.40.6Table 17
Fuzzy set of crimes on roads connecting Lahore and other cities.
Type of crime(Lahore, Multan)(Lahore, Jhang)(Lahore, Faisalabad)(Lahore, Sheikhupura)Child kidnapping0.70.70.60.5Bike robbery0.60.80.40.7Mobile snatching0.60.70.70.6Women harassment0.50.60.40.4Car robbery0.60.50.60.6Robbery of animals0.60.60.50.6Table 18
Fuzzy set of crimes on roads connecting Jhang and other cities.
Type of crime(Jhang, Islamabad)(Jhang, Faisalabad)(Jhang, Sheikhupura)(Jhang, Karachi)Child kidnapping0.70.50.60.6Bike robbery0.50.60.40.2Mobile snatching0.80.60.50.7Women harassment0.80.60.40.7Car robbery0.70.50.60.6Robbery of animals0.60.70.50.7Table 19
Fuzzy set of crimes on roads connecting Multan and other cities.
Type of crime(Multan, Jhang)(Multan, Faisalabad)(Multan, Sheikhupura)(Multan, Islamabad)Child kidnapping0.50.50.50.6Bike robbery0.60.60.60.4Mobile snatching0.60.50.70.6Women harassment0.70.60.50.5Car robbery0.60.50.40.7Robbery of animals0.60.70.70.5Table 20
Fuzzy set of crimes on roads connecting Karachi and other cities.
Type of crime(Islamabad, Multan)(Karachi, Rawalpindi)(Karachi, Lahore)(Karachi, Multan)Child kidnapping0.60.40.50.4Bike robbery0.40.20.40.2Mobile snatching0.60.30.40.5Women harassment0.50.40.40.4Car robbery0.70.50.40.4Robbery of animals0.50.40.40.3Various relations can be defined on the setA, and here we define the following six relations on A. R1 = child kidnapping, R2 = bike robbery, R3 = mobile snatching, R4 = women harassment, R5 = car robbery, and R6 = robbery of animals such that A,R1,R2,R3,R4,R5,R6 is a graph structure. Each object in an arbitrary relation shows a particular type of crime occurring on the roads connecting those two cities.AsA,R1,R2,R3,R4,R5,R6 is a graph structure, so an element cannot appear in more than one relation, and that is why it appears just once. Due to this, it is considered as an element of that particular relation, whose membership degree is higher than the membership values of all remaining relations.In accordance with the above knowledge, elements in these relations are written with their corresponding membership values, and corresponding sets are fuzzy sets onR1, R2, R3, R4, R5, and R6, respectively. These sets are named as μ1, μ2, μ3, μ4, μ5, and μ6, respectively. Let(121)R1=Karachi, Lahore,Lahore, Multan,R2=Jhang, Lahore,Sheikhupura, Lahore,R3=Karachi, Multan,Faisalabad, Lahore,Sheikhupura, Faisalabad,R4=Multan, Jhang,Faisalabad, Rawalpindi,R5=Islamabad, Multan,Karachi, Rawalpindi,Sheikhupura, Karachi,Islamabad, Sheikhupura,R6=Multan, Faisalabad,Jhang, Faisalabad.And, corresponding fuzzy sets are(122)μ1=Karachi, Lahore,0.5,Lahore, Multan,0.7,μ2=Jhang, Lahore,0.8,Sheikhupura, Lahore,0.7,μ3=Karachi, Multan,0.5,Faisalabad, Lahore,0.7,Sheikhupura, Faisalabad,0.7,μ4=Multan, Jhang,0.7,Faisalabad, Rawalpindi,0.7,μ5=Islamabad, Multan,0.7,Karachi, Rawalpindi,0.5,Sheikhupura, Karachi,0.5,Islamabad, Sheikhupura,0.7,μ6=Multan, Faisalabad,0.7,Jhang, Faisalabad,0.7.Clearly,F,μ1,μ2,μ3,μ4,μ5,μ6 is a fuzzy graph structure and is shown in Figure 14.Figure 14
Fuzzy graph structure showing the most crucial crime on roads connecting any two cities.In the fuzzy graph structure shown in Figure14, each edge shows the most frequent crime occurring on roads connecting corresponding cities. For example, most frequent crime on roads connecting Karachi and Rawalpindi is car robbery, and its membership value is 0.5. It can be noted that the vertex Karachi has the highest vertex degree for the relation car robbery, and it means Karachi is the most sensitive city for robbery of cars. Moreover, according to the above fuzzy graph structure, the most frequent road crime is car robbery. Furthermore, we can also observe that the most frequent crime after car robbery is mobile snatching. It means that the police department should take actions to control car robbery and mobile snatching.The general procedure of the adopted method used in the above applications is presented in Algorithm1.Algorithm 1: Computing membership value of an edge between two different vertices.
(1)
Begin(2)
Input membership valuesFci of n number of vertices (seaports or cities) c1,c2,…,cn.(3)
Input the adjacency matrix of vertices (seaports or cities) with respect toR1,R2,…,Rm, mutually disjoint, symmetric, and irreflexive relations.(4)
doi from 1⟶m(5)
doj from 1⟶m(6)
dok from 1⟶n(7)
dol from 1⟶n(8)
ifi≠j,μickcl>μjckcl, then(9)
Labelckcl as Ri(10)
end if(11)
end do(12)
end do(13)
end do(14)
end do(15)
μi membership value of an edge between two different vertices (seaports or cities) ck and cl shows the rate of the most frequent crime occurring between the vertices ck and cl.(16)
End


## 3.1. Detection of Marine Crimes

Marine crime is as old as this industry itself. Depending on the times that it prevailed in, the nature of the maritime crime has changed a lot over the years, but its implications remain just as severe. Marine crime is not only a threat to entire maritime security of goods and people in the industry, but also gives major setbacks to the entire marine industry economically. Marine crimes are usually held during the maritime trade. Seaports are very important in the whole world for trade through waters. There are almost four thousand seven hundred and sixty-four seaports, which are frequently used for import and export of various types of goods. Seaports have been very useful and beneficial for hundreds of years. But, they have also been used for the illegal trade due to which seaports always remain sensitive for crimes. Nowadays, crimes occurring on the seaports are in abundance, but some crimes including human trafficking, smuggling of precious metals, illegal carrying of weapons and artillery, smuggling of exotic plants, trade of drugs, and smuggling of exotic animals are crucial. All these crimes pose threat to the maritime security on a daily basis. Security of waters is still suffering because this is a very broad aspect without concrete boundaries. A fuzzy graph structure of marine crimes occurring on seaports can be very helpful to combat these crimes. Using the fuzzy graph structure, we can investigate that between any two seaports which crime is chronic and increasing violently. Moreover, we can decide which seaport is the most sensitive for a particular crime and needed to be focused by security teams. Consider a setV of eight seaports:(118)V=Mongla, Bintulu, Sembawang, Abu Dhabi, Gwadar, Chennai, Malindi, Ciro Marina.Letα be defined as a fuzzy set on V, as given in Table 1.Table 1
Fuzzy setα of seaports.
SeaportDegree of membershipMongla0.8Bintulu0.9Sembawang0.7Abu Dhabi0.8Gwadar0.7Chennai0.9Malindi0.8Ciro Marina0.5In Table1, the membership value of a seaport depicts its importance in the world due to its geographic position.In Tables2–11, we have mentioned membership values of marine crimes among each pair of seaports. The membership value of each pair of seaports is calculated according to νv1v2≤αv1∧αv2, for all v1,v2∈V. By using these membership values, we show the severity level of each marine crime among each pair of seaports.Table 2
Fuzzy set of marine crimes among Gwadar and other seaports.
Type of marine crimes(Gwadar, Chennai)(Gwadar, Malindi)(Gwadar, Ciro Marina)(Gwadar, Mongla)Human trafficking0.70.60.40.1Illegal carrying of weapons and artillery0.20.10.20.1Smuggling of exotic plants0.10.40.20.1Smuggling of precious metals0.10.10.20.3Trade of drugs0.20.70.30.5Smuggling of exotic animals0.20.20.30.1Table 3
Fuzzy set of marine crimes among Malindi and other seaports.
Type of marine crimes(Malindi, Mongla)(Malindi, Bintulu)(Malindi, Sembawang)(Malindi, Chennai)Human trafficking0.80.20.50.7Illegal carrying of weapons and artillery0.60.30.40.6Smuggling of exotic plants0.10.20.30.4Smuggling of precious metals0.60.80.60.6Trade of drugs0.20.50.30.5Smuggling of exotic animals0.30.30.40.4Table 4
Fuzzy set of marine crimes among Ciro Marina and other seaports.
Type of marine crimes(Ciro Marina, Sembawang)(Ciro Marina, Abu Dhabi)(Ciro Marina, Bintulu)Human trafficking0.10.10.1Illegal carrying of weapons and artillery0.10.20.1Smuggling of exotic plants0.20.50.0Smuggling of precious metals0.10.20.2Trade of drugs0.50.20.4Smuggling of exotic animals0.20.10.5Table 5
Fuzzy set of marine crimes among Sembawang and other seaports.
Type of marine crimes(Sembawang, Gwadar)(Sembawang, Chennai)(Sembawang, Mongla)Human trafficking0.10.20.6Illegal carrying of weapons and artillery0.20.30.4Smuggling of exotic plants0.20.20.6Smuggling of precious metals0.10.20.7Trade of drugs0.70.10.4Smuggling of exotic animals0.30.70.6Table 6
Fuzzy set of marine crimes among Mongla and other seaports.
Type of marine crimes(Mongla, Bintulu)(Mongla, Sembawang)(Mongla, Abu Dhabi)Human trafficking0.80.50.8Illegal carrying of weapons and artillery0.60.20.4Smuggling of exotic plants0.10.60.3Smuggling of precious metals0.70.70.4Trade of drugs0.50.50.6Smuggling of exotic animals0.30.60.4Table 7
Fuzzy set of marine crimes among Chennai and other seaports.
Type of marine crimes(Chennai, Mongla)(Chennai, Malindi)(Chennai, Ciro Marina)Human trafficking0.80.50.2Illegal carrying of weapons and artillery0.60.60.2Smuggling of exotic plants0.20.60.1Smuggling of precious metals0.60.50.2Trade of drugs0.50.50.4Smuggling of exotic animals0.60.50.5Table 8
Fuzzy set of marine crimes among Abu Dhabi and other seaports.
Type of marine crimes(Abu Dhabi, Gwadar)(Abu Dhabi, Chennai)(Abu Dhabi, Malindi)Human trafficking0.30.80.5Illegal carrying of weapons and artillery0.20.60.4Smuggling of exotic plants0.50.50.8Smuggling of precious metals0.60.40.7Trade of drugs0.70.50.6Smuggling of exotic animals0.30.30.6Table 9
Fuzzy set of marine crimes among some seaports.
Type of marine crimes(Mongla, Gwadar)(Ciro Marina, Malindi)(Sembawang, Abu Dhabi)Human trafficking0.10.10.5Illegal carrying of weapons and artillery0.20.20.4Smuggling of exotic plants0.20.50.2Smuggling of precious metals0.40.10.4Trade of drugs0.70.30.7Smuggling of exotic animals0.30.20.2Table 10
Fuzzy set of marine crimes among Bintulu and other seaports.
Type of marine crimes(Bintulu, Mongla)(Bintulu, Sembawang)(Bintulu, Abu Dhabi)Human trafficking0.80.50.7Illegal carrying of weapons and artillery0.60.40.8Smuggling of exotic plants0.10.10.3Smuggling of precious metals0.70.20.4Trade of drugs0.50.30.6Smuggling of exotic animals0.30.50.6Table 11
Fuzzy set of marine crimes among some seaports.
Type of marine crimes(Bintulu, Gwadar)(Chennai, Bintulu)(Abu Dhabi, Ciro Marina)Human trafficking0.70.30.3Illegal carrying of weapons and artillery0.60.40.2Smuggling of exotic plants0.20.50.5Smuggling of precious metals0.60.90.1Trade of drugs0.50.30.1Smuggling of exotic animals0.50.40.1On the setV, many relations can be defined; let us define the following relations on V. R1 = human trafficking, R2 = illegal carrying of weapons and artillery, R3 = smuggling of exotic plants, R4 = smuggling of precious metals, R5 = trade of drugs, and R6 = smuggling of exotic animals such that V,R1,R2,R3,R4,R5,R6 is a graph structure. Each element in the relation depicts a particular kind of the most crucial marine crimes among those two seaports.AsV,R1,R2,R3,R4,R5,R6 is the graph structure, therefore a pair of seaports can appear in just one relation. Hence, it would be considered an element of that relation, for which its membership value is comparatively high than that of the other relations. Using the above given data, elements in relations are paired with their membership values, and resulting sets are the fuzzy sets on R1, R2, R3, R4, R5, and R6, respectively. These fuzzy sets are named as ν1, ν2, ν3, ν4, ν5, and ν6, respectively. Let(119)R1=Bintulu, Mongla,Bintulu, Gwadar,Mongla,Chennai,Abu Dhabi, Mongla,R2=Abu Dhabi, Bintulu,R3=Abu Dhabi, Malindi,Ciro Marina, Malindi,R4=Bintulu, Chennai,Sembawang, Mongla,R5=Gwadar, Malindi,Ciro Marina, Sembawang,R6=Bintulu, Ciro Marina.And, their corresponding fuzzy sets are(120)ν1=Bintulu, Mongla,0.8,Bintulu, Gwadar,0.7,Mongla, Chennai,0.8,Abu Dhabi, Mongla,0.8,ν2=Abu Dhabi, Bintulu,0.8,ν3=Abu Dhabi, Malindi,0.8,Ciro Marina, Malindi,0.5,ν4=Bintulu, Chennai,0.9,Sembawang, Mongla,0.7,ν5=Gwadar, Malindi,0.7,Ciro Marina, Sembawang,0.5,ν6=Bintulu, Ciro Marina,0.5.Clearly,α,ν1,ν2,ν3,ν4,ν5,ν6 is a fuzzy graph structure and is shown in Figure 13.Figure 13
Fuzzy graph structure depicting most crucial marine crimes among different seaports.In FGS shown in Figure13, each edge depicts the most crucial marine crime among the corresponding seaports. For example, most crucial marine crime among Bintulu and Mongla is human trafficking, and its membership value is 0.8. It can be noted that the vertex Bintulu has the highest vertex degree for the relation human trafficking, and it means that Bintulu has human trafficking as the most crucial marine crime with other seaports. Moreover, according to this fuzzy graph structure, Abu Dhabi and Malindi have smuggling of exotic plants as the most crucial marine crime at this time with a membership value 0.8. An FGS of all seaports can be very helpful for security agencies of all countries. It highlights those marine crimes which needed to be solved promptly.


## 3.2. Detection of Road Crimes

Road network is very important for any country. Roads connecting cities are the major source of trade, traveling, and many other purposes. But, nowadays, roads have become unsafe due to large amount of crimes happening on the roads. Crimes occurring on roads are very large in number, but some crimes are very crucial including child kidnapping, bike robbery, mobile snatching, women harassment, car robbery, and robbery of animals. Using a fuzzy graph structure, we can investigate that which road is the most crucial for a particular crime. An FGS can also tell us which crime is the most chronic and which one is increasing rapidly. Furthermore, we can judge which road is the most crucial for a particular road crime. An FGS of road crimes can be very helpful for the police department to overcome a particular crime, and it can guide them in making a policy against a crime. For example, if an FIR is written of a crime held in a particular area, an FGS of that area can be very helpful for the police department in taking action against that crime, and it will guide them to which roads are the most sensitive regarding that particular crime. So, police can put check posts on those roads, and police can cordon off the area near to those roads. Consider a setA of eight cities. A = {Islamabad, Karachi, Rawalpindi, Lahore, Multan, Jhang, Faisalabad, Sheikhupura}. Let F be considered as a fuzzy set on A, as given in Table 12.Table 12
Fuzzy set F onA.
CityMembership valueIslamabad0.9Karachi0.5Rawalpindi0.8Lahore0.8Multan0.7Jhang0.8Faisalabad0.7Sheikhupura0.7In Table12, the membership value shows the solidity of that particular city corresponding to economical progress, educational and health facilities, and peace. In Tables 13–20, we have shown membership values of different crimes on roads connecting every pair of cities.Table 13
Fuzzy set of crimes on roads connecting Islamabad and other cities.
Type of crime(Islamabad, Karachi)(Islamabad, Rawalpindi)(Islamabad, Lahore)(Islamabad, Multan)Child kidnapping0.60.50.50.7Bike robbery0.60.60.40.6Mobile snatching0.50.60.30.5Women harassment0.60.50.40.6Car robbery0.50.50.60.5Robbery of animals0.30.50.40.4Table 14
Fuzzy set of crimes on roads connecting Rawalpindi and other cities.
Type of crime(Rawalpindi, Lahore)(Rawalpindi, Multan)(Rawalpindi, Jhang)(Rawalpindi, Islamabad)Child kidnapping0.70.60.70.7Bike robbery0.80.70.40.6Mobile snatching0.60.60.80.7Women harassment0.80.70.60.8Car robbery0.70.50.60.7Robbery of animals0.40.50.50.3Table 15
Fuzzy set of crimes on roads connecting Faisalabad and other cities.
Type of crime(Faisalabad, Islamabad)(Faisalabad, Karachi)(Faisalabad, Rawalpindi)(Faisalabad, Jhang)Child kidnapping0.70.50.60.5Bike robbery0.50.40.40.6Mobile snatching0.70.50.60.6Women harassment0.60.40.70.5Car robbery0.70.50.60.6Robbery of animals0.30.50.60.7Table 16
Fuzzy set of crimes on roads connecting Sheikhupura and other cities.
Type of crime(Sheikhupura, Rawalpindi)(Islamabad, Sheikhupura)(Sheikhupura, Karachi)(Sheikhupura, Faisalabad)Child kidnapping0.70.60.40.5Bike robbery0.60.60.40.6Mobile snatching0.70.50.40.7Women harassment0.60.60.40.5Car robbery0.70.70.50.4Robbery of animals0.60.40.40.6Table 17
Fuzzy set of crimes on roads connecting Lahore and other cities.
Type of crime(Lahore, Multan)(Lahore, Jhang)(Lahore, Faisalabad)(Lahore, Sheikhupura)Child kidnapping0.70.70.60.5Bike robbery0.60.80.40.7Mobile snatching0.60.70.70.6Women harassment0.50.60.40.4Car robbery0.60.50.60.6Robbery of animals0.60.60.50.6Table 18
Fuzzy set of crimes on roads connecting Jhang and other cities.
Type of crime(Jhang, Islamabad)(Jhang, Faisalabad)(Jhang, Sheikhupura)(Jhang, Karachi)Child kidnapping0.70.50.60.6Bike robbery0.50.60.40.2Mobile snatching0.80.60.50.7Women harassment0.80.60.40.7Car robbery0.70.50.60.6Robbery of animals0.60.70.50.7Table 19
Fuzzy set of crimes on roads connecting Multan and other cities.
Type of crime(Multan, Jhang)(Multan, Faisalabad)(Multan, Sheikhupura)(Multan, Islamabad)Child kidnapping0.50.50.50.6Bike robbery0.60.60.60.4Mobile snatching0.60.50.70.6Women harassment0.70.60.50.5Car robbery0.60.50.40.7Robbery of animals0.60.70.70.5Table 20
Fuzzy set of crimes on roads connecting Karachi and other cities.
Type of crime(Islamabad, Multan)(Karachi, Rawalpindi)(Karachi, Lahore)(Karachi, Multan)Child kidnapping0.60.40.50.4Bike robbery0.40.20.40.2Mobile snatching0.60.30.40.5Women harassment0.50.40.40.4Car robbery0.70.50.40.4Robbery of animals0.50.40.40.3Various relations can be defined on the setA, and here we define the following six relations on A. R1 = child kidnapping, R2 = bike robbery, R3 = mobile snatching, R4 = women harassment, R5 = car robbery, and R6 = robbery of animals such that A,R1,R2,R3,R4,R5,R6 is a graph structure. Each object in an arbitrary relation shows a particular type of crime occurring on the roads connecting those two cities.AsA,R1,R2,R3,R4,R5,R6 is a graph structure, so an element cannot appear in more than one relation, and that is why it appears just once. Due to this, it is considered as an element of that particular relation, whose membership degree is higher than the membership values of all remaining relations.In accordance with the above knowledge, elements in these relations are written with their corresponding membership values, and corresponding sets are fuzzy sets onR1, R2, R3, R4, R5, and R6, respectively. These sets are named as μ1, μ2, μ3, μ4, μ5, and μ6, respectively. Let(121)R1=Karachi, Lahore,Lahore, Multan,R2=Jhang, Lahore,Sheikhupura, Lahore,R3=Karachi, Multan,Faisalabad, Lahore,Sheikhupura, Faisalabad,R4=Multan, Jhang,Faisalabad, Rawalpindi,R5=Islamabad, Multan,Karachi, Rawalpindi,Sheikhupura, Karachi,Islamabad, Sheikhupura,R6=Multan, Faisalabad,Jhang, Faisalabad.And, corresponding fuzzy sets are(122)μ1=Karachi, Lahore,0.5,Lahore, Multan,0.7,μ2=Jhang, Lahore,0.8,Sheikhupura, Lahore,0.7,μ3=Karachi, Multan,0.5,Faisalabad, Lahore,0.7,Sheikhupura, Faisalabad,0.7,μ4=Multan, Jhang,0.7,Faisalabad, Rawalpindi,0.7,μ5=Islamabad, Multan,0.7,Karachi, Rawalpindi,0.5,Sheikhupura, Karachi,0.5,Islamabad, Sheikhupura,0.7,μ6=Multan, Faisalabad,0.7,Jhang, Faisalabad,0.7.Clearly,F,μ1,μ2,μ3,μ4,μ5,μ6 is a fuzzy graph structure and is shown in Figure 14.Figure 14
Fuzzy graph structure showing the most crucial crime on roads connecting any two cities.In the fuzzy graph structure shown in Figure14, each edge shows the most frequent crime occurring on roads connecting corresponding cities. For example, most frequent crime on roads connecting Karachi and Rawalpindi is car robbery, and its membership value is 0.5. It can be noted that the vertex Karachi has the highest vertex degree for the relation car robbery, and it means Karachi is the most sensitive city for robbery of cars. Moreover, according to the above fuzzy graph structure, the most frequent road crime is car robbery. Furthermore, we can also observe that the most frequent crime after car robbery is mobile snatching. It means that the police department should take actions to control car robbery and mobile snatching.The general procedure of the adopted method used in the above applications is presented in Algorithm1.Algorithm 1: Computing membership value of an edge between two different vertices.
(1)
Begin(2)
Input membership valuesFci of n number of vertices (seaports or cities) c1,c2,…,cn.(3)
Input the adjacency matrix of vertices (seaports or cities) with respect toR1,R2,…,Rm, mutually disjoint, symmetric, and irreflexive relations.(4)
doi from 1⟶m(5)
doj from 1⟶m(6)
dok from 1⟶n(7)
dol from 1⟶n(8)
ifi≠j,μickcl>μjckcl, then(9)
Labelckcl as Ri(10)
end if(11)
end do(12)
end do(13)
end do(14)
end do(15)
μi membership value of an edge between two different vertices (seaports or cities) ck and cl shows the rate of the most frequent crime occurring between the vertices ck and cl.(16)
End


## 4. Conclusions

Fuzzy graph structures are more advantageous than graph structures to cope with uncertainty. In this research paper, we have combined the fruitful theories of fuzzy sets and graph structures to introduce certain notions of fuzzy graph structures. We have introduced the lexicographic-max product and computed the total degree andνi−the total degree of a vertex in the lexicographic-max product of fuzzy graph structures. Furthermore, we have presented applications of fuzzy graph structures in decision-making, regarding detection of marine crimes and detection of the road crimes. We aim to broaden our work to (1) rough fuzzy graph structures, (2) rough fuzzy soft graph structures, and (3) fuzzy soft graph structures.



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*Source: 6846257-2020-08-12.xml*