#### 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.

#### 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 [1434].

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 [3941] 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 Section 2, 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) consists of a nonempty set together with relations on which are mutually disjoint such that each , , is symmetric and irreflexive. One can represent a graph structure in the plane just like a graph where each edge is labeled as , .

Definition 2 (see [38]). Let be the fuzzy set on set and be fuzzy sets on , respectively. If , , then is called the fuzzy graph structure (FGS) of the graph structure . If , then is named as the -edge of FGS .

Definition 3. Let and be two fuzzy graph structures (FGSs) having underlying crisp graph structures (GSs) and , respectively. A fuzzy graph structure with an underlying crisp graph structure , where and , is defined.
The fuzzy vertex set is defined as , for all .
Fuzzy relations are defined as is called the lexicographic-max product of and and is denoted by .

Example 1. Consider two FGSs and with underlying crisp GSs and , respectively, which are shown in Figure 1, where , , , and .
The lexicographic-max product of the above FGSs and is shown in Figure 2.
In the lexicographic-max product, and edges belong to the fuzzy set.

Theorem 1. If and are two effective FGSs such that and and are constant functions having similar values, then the lexicographic-max product of and is an effective FGS.

Proof. Let and be two effective FGSs such that and and be constant functions having similar values, then the definition of the lexicographic-max product providesCase 1: :Case 2: :Thus, for all edges.
Hence, is an effective FGS.

Example 2. Consider two FGSs and , as shown in Figure 3.
Figure 3 shows that , , , and . Hence, and are effective FGSs. Moreover, and and , , are constant functions having the same value, i.e., 0.6. The lexicographic-max product of and is shown in Figure 4.
It is clear from Figure 4 thatSimilarly, membership values of all other edges are calculated. Hence, shown in Figure 4 is an effective FGS.

Theorem 2. If and are two complete FGSs such that and and are functions having a constant value, then the lexicographic-max product of and is a complete FGS.

Proof. The proof is similar to the proof of Theorem 1.

Theorem 3. The lexicographic-max product of two connected FGSs and is a connected FGS if and only if is a connected FGS.

Proof. Assume that is a connected FGS. According to the definition of the lexicographic-max product , the number of copies of is equal to the number of vertices of ; that is, for each vertex of , there exists one copy of GS in the lexicographic-max product . Since is connected, is a connected FGS. Conversely, assume that and are two connected FGSs such that is a connected FGS. To prove the connectedness of FGS , on contrary, assume that is not connected. Then, there exist at least two vertices of having no path connecting them. Since is connected, then at least one path must exist between any two elements of . Thus, there will be at least one path connecting the elements of . This provides a contradiction. Hence, is a connected FGS.

Theorem 4. The number of connected components in the lexicographic-max product of two FGSs and is equal to the number of components of .

Proof. Let be a connected FGS and be FGS. Then, according to Theorem 3, the lexicographic-max product is a connected FGS. This implies that both and are connected. Suppose that is not a connected FGS having ‘m’ distinct connected components. Vertices of are renamed such that , are “m” distinct connected components of . If are vertices of , then for an arbitrary vertex of , there must exist one version of each connected component of in the lexicographic-max product . There is no edge among all these components. Corresponding to each edge between and , there must be an edge between vertex and of which gives a contradiction. Thus, each component of the lexicographic-max product is distinct from others.

Definition 4. In the lexicographic-max product of FGSs and , the degree of a vertex is defined asthe degree of a vertex in the lexicographic-max product is defined as

Example 3. Consider two FGSs and , which are shown in Figure 5.
The lexicographic-max product of FGSs and is shown in Figure 6.
To compute the degrees of all vertices in the lexicographic-max product, the following formula is utilized:the degree of the vertex in the lexicographic-max product is given byUsing the above given formula, we will calculate the degree of the vertices in as

Theorem 5. If and are two FGSs such that , then the degree of the vertex in the lexicographic-max product of FGSs and is given by .

Proof. Let and be two FGSs such that , then , and then the degree of the vertex in (lexicographic-max product) is given by

Theorem 6. If and are two FGSs such that , then the degree of any vertex in the lexicographic-max product of FGSs and is given by .

Proof. Let and be two FGSs such that , then , . Then, the degree of the vertex in (lexicographic-max product) is given as

Example 4. Consider two FGSs and , shown in Figure 7.
In Figure 7, . The lexicographic-max product of and is shown in Figure 8.
Theorem 6 implies the formula to determine the degrees of vertices in the lexicographic-max product as follows: