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Article

Vague Graph Structure with Application in Medical Diagnosis

Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(10), 1582; https://doi.org/10.3390/sym12101582
Received: 5 September 2020 / Revised: 20 September 2020 / Accepted: 21 September 2020 / Published: 23 September 2020

Abstract

:
Fuzzy graph models enjoy the ubiquity of being in natural and human-made structures, namely dynamic process in physical, biological and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems which are often uncertain, it is highly difficult for an expert to model those problems based on a fuzzy graph (FG). Vague graph structure (VGS) can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. Likewise, VGSs are very useful tools for the study of different domains of computer science such as networking, capturing the image, clustering, and also other issues like bioscience, medical science, and traffic plan. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGSs. Operations are conveniently used in many combinatorial applications. In various situations, they present a suitable construction means; therefore, in this research, three new operations on VGSs, namely, maximal product, rejection, residue product were presented, and some results concerning their degrees and total degrees were introduced. Irregularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology and economy; so special concepts of irregular VGSs with several key properties were explained. Today one of the most important applications of decision making is in medical science for diagnosing the patient’s disease. Hence, we recommend an application of VGS in medical diagnosis.

1. Introduction

Graph theory serves as an exceptionally useful tool in solving combinatorial problems in different areas including geometry, algebra, number theory, topology, and social systems. A graph is basically a model of relations, and it is used to embody the real-life problems entailing the relationships between objects. The vertices and edges of the graph are employed to represent the objects and the relations between objects, respectively. In several optimization problems, the available information is inexact or imprecise for various reasons such as the loss of information, a lack of evidence, imperfect statistical data and insufficient information. Generally, the uncertainty in real-life problems may be present in the information that defines the problem. Fuzzy graph (FG) models are advantageous mathematical tools for treating the combinatorial problems in various domains encompassing research, optimization, algebra, computing, environmental science and topology. Fuzzy graphical models are obviously better than graphical models because of the natural existence of vagueness and ambiguity. Originally, fuzzy set theory is required to deal with many complex issues having incomplete information. In 1965, Zadeh [1] at first suggested the fuzzy set theory. Fuzzy set theory is a very powerful mathematical tool for solving approximate reasoning related problems. These notions effectively illustrate complex phenomena, which are not accurately described using classical mathematics. In a fuzzy set, each object is assigned a single value in the interval [0,1] reflecting its grade of membership. This single value does not allow a separation of evidence for membership and evidence against membership. Hence, Gau and Buehrer [2] organized the fuzzy set theory by presenting the VS notion through changing the value of an element in a set with a sub-interval of [ 0 , 1 ] . Specifically, a true membership function of t v ( x ) and false membership function of f v ( x ) are used to define the boundaries of the membership degree. This kind of reasoning is also called interval membership, as opposed to point membership in the context of fuzzy sets. The VSs describe more possibilities than fuzzy sets. A VS is more initiative and helpful due to the existence of false membership degree. VSs are higher order fuzzy sets. The application of higher order fuzzy sets complicates the solution procedure, but if the complexity on computation time, computation volume, or memory space is not a matter of concern, then better results can be achieved. There are some interesting features for handling vague data that are unique to vague sets. For example, VSs allow for a more intuitive graphical representation of vague data, which significantly facilitates better analysis in data relationships, incompleteness, and similarity measures.
Many real-word phenomena provided motivation to describe the FGs. Kauffman [3] define fuzzy graphs using Zadeh’s fuzzy relation [4,5]. Akram et al. [6,7,8,9,10,11] introduced several concepts in FGs. Rashmanlou et al. [12,13,14,15,16] introduced some properties in FGs. Product operations on graphs play a very important role in graph theory. They have applications in different branches, such as coding theory, network designs, chemical graph theory and others. Many scholars discussed product operations on various generalized FGs. Mordeson and Peng [17] defined some of these product operations on FGs. Subsequently, utilizing these operations, the degree of the vertices is obtained from two FGs in [18]. Sahoo and Pal [19] presented some operations on intuitionistic fuzzy graphs. FG theory is growing rapidly, with numerous applications in many domains, including networking, communication, data mining, planning, and scheduling. However, Rosenfeld [20] described another detailed definition including fuzzy vertex and fuzzy edges and various fuzzy analogs of graph theoretic concepts such as paths, cycles, connectedness etc. A vague graph (VG) is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In addition, a VG can focus on resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may not bring about satisfactory results. Ramakrishna [21] recommended the VG notion and evaluated some of their features. Borzooei and Rashmanlou [22,23,24,25] investigated different concepts of VG. Irregularity definitions have been of high significance in the study of network heterogeneity, which has implications in networks found across biology, ecology, technology, and economy. Several graph statistics have existed that have been proposed, many of which are based on the number of vertices in a graph and their degrees. The concepts of the irregularity play a significant role in both graph theory and application in the vague environment. The characterization of highly irregular graphs has also been applied to the question of heterogeneity, yet all of these fail to shed enough light on real-world situations. Efforts continue to be made to find appropriate ways to quantify network heterogeneity; so, Gani and Radha [26,27,28,29] introduced irregular FGs, some properties of conjunction of FGs, and new concepts in regular FGs. Samanta and Pal [30] described irregular bipolar fuzzy graphs. A graph structure (GS) is a generalization of simple graphs. GSs are very useful in the study of different domains of computer science and computational intelligence. GS was introduced first by Sampathkumar [31]. Fuzzy graph structures (FGSs) are more useful than GSs because they deal with the uncertainty and ambiguity of many real-world phenomena, and they are widely useful in the study of some structures, like graphs, signed graphs, semi graphs, edge-colored graphs and edge-labeled graphs. FGSs have many applications in solving various problems of several domains, including communication, data mining, image capturing, and image segmentation. Dinesh [32] defined the concept of FGSs and investigated some related concepts. Ramakrishna and Dinesh [33,34] studied generalized FGSs. Shao et al. [35] introduced new concepts in intuitionistic fuzzy graphs.
VGSs are the generalization of FGSs and extremely useful in the study of some structures, like graphs, colored graphs, signed graphs, and edge-labeled graphs. VGSs are more useful than GSs because they deal with the uncertainty and ambiguity of many real-word phenomena. Hence, in this paper we introduced three operations on VGSs, namely, maximal product, rejection, residue product, and represented some results concerning their degrees and total degrees. As the concept of regularity led to many developments in the structural theory of graphs, meanwhile, the irregular graphs have also been significant while dealing with network heterogeneity, which as many applications across biology, economy, and technology. So, in this research, we explained special concepts of irregular-VGSs with several key properties. Decision making in medical diagnosis is a vast research field in medical science. Hence, we described an application of VGS by applying the new distance measure for medical diagnosis.

2. Preliminaries

A graph is a pair G = ( V , E ) which satisfies E V × V The elements of V ( G ) and E ( G ) are the nodes and edges of the graph G, respectively.
An FG is of the form G = ( ψ ; ϕ ) which is a pair of mappings ψ : V [ 0 , 1 ] and ϕ : V × V [ 0 , 1 ] as is defined as ϕ ( m , n ) ψ ( m ) ψ ( n ) , m , n V , and ϕ is a symmetric fuzzy relation on ψ and ∧ denotes the minimum.
A (VS) A is a pair ( t A , f A ) on set V where t A and f A are taken as real valued functions which can be defined on V [ 0 , 1 ] , so that t A ( m ) + f A ( m ) 1 , for all m belongs V. The interval [ t A ( m ) , 1 f A ( m ) ] is known as the vague value of m in A. t A ( m ) , in this definition, is taken for degree of membership as the lower bound when m in A and f A ( m ) is the lower bound for negative of membership of m in A.
Definition 1
([21]). A pair G = ( A , B ) is said to be a VG on a crisp graph G = ( V , E ) , where A = ( t A , f A ) is a VS on V and B = ( t B , f B ) is a VS on E V × V such that t B ( m n ) min ( t A ( m ) , t A ( n ) ) and f B ( m n ) max ( f A ( m ) , f A ( n ) ) , for each edge m n E .
Definition 2
([31]). A graph structure (GS) G = ( V , E 1 , E 2 , , E n ) contains of a non-empty set V with relations E 1 , E 2 , , E n on set V that are separated such that each relation E i , 1 i n , is symmetric and irreflexive. Graph structure G = ( V , E 1 , E 2 , , E n ) can be described as similar as a graph where each edge is labeled as E i , 1 i n .
Definition 3
([32]). Let ψ be the fuzzy set on V and ϕ 1 , ϕ 2 , , ϕ n be fuzzy sets on E 1 , E 2 , , E n , respectively. If 0 ϕ i ( m n ) ϕ ( m ) ϕ ( n ) , for all m , n V , i = 1 , 2 , , n , then G = ( ψ , ϕ 1 , ϕ 2 , , ϕ n ) is called FGS of graph structure G .
Example 1
([32]). Consider a GS G = ( V , E 1 , E 2 ) as shown in Figure 1. Now we define fuzzy set ψ : V [ 0 , 1 ] by
ψ ( m ) = 0.2 , ψ ( n ) = 0.3 , ψ ( z ) = 0.4 .
We define fuzzy sets ϕ 1 , ϕ 2 on relations E 1 , E 2 , respectively, as follows:
ϕ 1 ( m n ) = 0.2 , ϕ 1 ( n z ) = 0.3 , ϕ 2 ( m z ) = 0.2 .
By simple computation, it is easy to show that G = ( ψ , ϕ 1 , ϕ 2 ) is a FGS.

3. Vague Graph Structures

Product operations on graphs play a very important role in graph theory. In various situations they present a suitable construction means. The first definition of product in fuzzy graphs was introduced in [17]. However, for the first time in this section, two types of products called maximal product, and residue product between the two VGSs were being introduced, which are widely employed in computer networks, and accordingly their properties were evaluated. In addition, another new operation called rejection is introduced, too. Note that the degree and the total degree of vertices that were calculated in the maximal product, rejection and residue product; are presented in (Appendix A).
Definition 4.
G = ( A , B 1 , B 2 , , B n ) is called a VGS of a GS G = ( V , E 1 , E 2 , , E n ) , if A = ( t A , f A ) is a vague set on V and for each i = 1 , 2 , , n ; B i = ( t B i , f B i ) is a vague set on E i such that:
t B i ( m n ) t A ( m ) t A ( n ) , f B i ( m n ) f A ( m ) f A ( n ) ,
m n E i V × V . Note that t B i ( m n ) = 0 = f B i ( m n ) , for all m n V × V E i and 0 t B i ( m n ) 1 , 0 f B i ( m n ) 1 , m n E i , where V and E i ( i = 1 , 2 , , n ) are called underlying vertex set and underlying i-edge set of G, respectively.
Example 2.
Let ( V , E 1 , E 2 ) be a graph structure such that V = { m , n , x , w } , E 1 = { m n , n w } and E 2 = { x w , m x } . Let A, B 1 , and B 2 be vague subsets of V, E 1 , and E 2 respectively such that:
A = { ( m , 0.3 , 0.4 ) , ( n , 0.3 , 0.5 ) , ( w , 0.2 , 0.3 ) , ( x , 0.3 , 0.3 ) } , B 1 = { ( m n , 0.3 , 0.5 ) , ( n w , 0.2 , 0.5 ) } , a n d B 2 = { ( x w , 0.2 , 0.3 ) , ( m x , 0.2 , 0.4 ) } .
Then G = ( A , B 1 , B 2 ) is a VGS of G as shown in Figure 2.
Definition 5.
Let G 1 = ( A 1 , B 1 , B 2 , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) be two vague graph structures with underlying crisp graph structures G 1 = ( V 1 , E 1 , E 2 , , E n ) and G 2 = ( V 2 , E 1 , E 2 , , E n ) , respectively. G 1 G 2 = ( A , B 1 , B 2 , , B n ) is called maximal vague graph structure with underlying crisp graph structure G = ( V , E 1 , E 2 , , E n ) , where V = V 1 × V 2 and E i = { ( m 1 , n 1 ) ( m 2 , n 2 ) | m 1 = m 2 , n 1 n 2 E i o r n 1 = n 2 , m 1 m 2 E i } . Vague vertex set A and vague relation B i in maximal product G 1 G 2 = ( A , B 1 , B 2 , , B n ) are defined as: A = A 1 A 2 ,
( i ) ( t A 1 t A 2 ) ( m , n ) = max { t A 1 ( m ) , t A 2 ( n ) } , ( f A 1 f A 2 ) ( m , n ) = min { f A 1 ( m ) , f A 2 ( n ) } , f o r a l l ( m , n ) V = V 1 × V 2 ( i i ) ( t B i t B i ) ( ( m 1 , n 1 ) ( m 2 , n 2 ) ) = max t A 1 ( m 1 ) , t B i ( n 1 n 2 ) ( f B i f B i ) ( ( m 1 , n 1 ) ( m 2 , n 2 ) ) = min f A 1 ( m 1 ) , f B i ( n 1 n 2 ) , m 1 = m 2 , n 1 n 2 E i ( i i i ) ( t B i t B i ) ( ( m 1 , n 1 ) ( m 2 , n 2 ) ) = max t A 2 ( n 1 ) , t B i ( m 1 m 2 ) ( f B i f B i ) ( ( m 1 , n 1 ) ( m 2 , n 2 ) ) = min f A 2 ( n 1 ) , f B i ( m 1 m 2 ) , m 1 m 2 E i , n 1 = n 2
i = 1 , 2 , , n .
Example 3.
Consider two VGSs G 1 and G 2 as shown in Figure 3. Their maximal product G 1 G 2 is shown in Figure 4.
For vertex ( m , w ) , we find both true membership and false membership value as follows:
( t A 1 t A 2 ) ( m , w ) = max { t A 1 ( m ) , t A 2 ( w ) } = max { 0.1 , 0.1 } = 0.1 , ( f A 1 f A 2 ) ( m , w ) = min { f A 1 ( m ) , f A 2 ( w ) } = min { 0.2 , 0.2 } = 0.2 ,
for m V 1 and w V 2 .
For edge ( m , z ) ( x , z ) , we find both true membership value and false membership value.
( t B 1 t B 2 ) ( m , z ) ( x , z ) = max { t A 2 ( z ) , t B 1 ( m x ) } = max { 0.2 , 0.1 } = 0.2 , ( f B 1 f B 2 ) ( m , z ) ( x , z ) = min { f A 2 ( z ) , f B 1 ( m x ) } = min { 0.3 , 0.6 } = 0.3 ,
for m x E 3 and z V 2 .
Similarly, we can find both true membership and false membership value for all remaining vertices and edges.
Theorem 1.
The maximal product of two VGSs G 1 and G 2 , is a VGS, too.
Proof. 
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) be two VGSs and ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E i , i = 1 , 2 , , n . Then by Definition 4, we have two cases:
(i) if m 1 = n 1 = m
( t B i t B i ) ( m , m 2 ) ( m , n 2 ) = max { t A 1 ( m ) , t B i ( m 2 n 2 ) } max t A 1 ( m ) , min { t A 2 ( m 2 ) , t A 2 ( n 2 ) } = min max { t A 1 ( m ) , t A 2 ( m 2 ) } , max { t A 1 ( m ) , t A 2 ( n 2 ) } = min ( t A 1 t A 2 ) ( m , m 2 ) , ( t A 1 t A 2 ) ( m , n 2 ) , ( f B i f B i ) ( m , m 2 ) ( m , n 2 ) = min { f A 1 ( m ) , f B i ( m 2 n 2 ) } min f A 1 ( m ) , max { f A 2 ( m 2 ) , f A 2 ( n 2 ) } = max min { f A 1 ( m ) , f A 2 ( m 2 ) } , min { f A 1 ( m ) , f A 2 ( n 2 ) } = max ( f A 1 f A 2 ) ( m , m 2 ) , ( f A 1 f A 2 ) ( m , n 2 ) .
(ii) if m 2 = n 2 = z
( t B i t B i ) ( m 1 , z ) ( n 1 , z ) = max { t B i ( m 1 n 1 ) , t A 2 ( z ) } max min { t A 1 ( m 1 ) , t A 1 ( n 1 ) } , t A 2 ( z ) = min max { t A 1 ( m 1 ) , t A 2 ( z ) } , max { t A 1 ( n 1 ) , t A 2 ( z ) } = min ( t A 1 t A 2 ) ( m 1 , z ) , ( t A 1 t A 2 ) ( n 1 , z ) , ( f B i f B i ) ( m 1 , z ) ( n 1 , z ) = min { f B i ( m 1 n 1 ) , f A 2 ( z ) } min max { f A 1 ( m 1 ) , f A 1 ( n 1 ) } , f A 2 ( z ) = max min { f A 1 ( m 1 ) , f A 2 ( z ) } , min { f A 1 ( n 1 ) , f A 2 ( z ) } = max ( f A 1 f A 2 ) ( m 1 , z ) , ( f A 1 f A 2 ) ( n 1 , z ) .
Hence, G 1 G 2 is a VGS. □
Definition 6.
The rejection G 1 | G 2 of two VGSs G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) is defined as:
( i ) ( t A 1 | t A 2 ) ( m , n ) = min { t A 1 ( m ) , t A 2 ( n ) } ( f A 1 | f A 2 ) ( m , n ) = max { f A 1 ( m ) , f A 2 ( n ) } , f o r a l l ( m , n ) V 1 × V 2 , ( i i ) ( t B i | t B i ) ( m , m 2 ) ( m , n 2 ) = min { t A 1 ( m ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } ( f B i | f B i ) ( m , m 2 ) ( m , n 2 ) = max { f A 1 ( m ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } , f o r a l l m V 1 a n d m 2 n 2 E i , ( i i i ) ( t B i | t B i ) ( m 1 , m ) ( n 1 , m ) = min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m ) } ( f B i | f B i ) ( m 1 , m ) ( n 1 , m ) = max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m ) } , f o r a l l m V 2 a n d m 1 n 1 E i , ( i v ) ( t B i | t B i ) ( m 1 , m 2 ) ( n 1 , n 2 ) = min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } ( f B i | f B i ) ( m 1 , m 2 ) ( n 1 , n 2 ) = max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } , f o r a l l m 1 n 1 E i a n d m 2 n 2 E i .
Example 4.
Consider the VGSs G 1 and G 2 as in Figure 5. We can see that the rejection of two VGSs G 1 and G 2 , that is G 1 | G 2 in Figure 6.
For vertex ( x , k ) , we find both membership value and non-membership value as follows:
( t A 1 | t A 2 ) ( ( x , k ) ) = min { t A 1 ( x ) , t A 2 ( k ) } = min { 0.3 , 0.2 } = 0.2 , ( f A 1 | f A 2 ) ( ( x , k ) ) = max { f A 1 ( x ) , f A 2 ( k ) } = max { 0.5 , 0.4 } = 0.5 ,
for x V 1 and k V 2 .
For edge ( x , k ) ( x , z ) , we find both membership value and non-membership value.
( t B i | t B i ) ( ( x , k ) ( x , z ) ) = min { t A 1 ( x ) , t A 2 ( k ) , t A 2 ( z ) } = min { 0.3 , 0.2 , 0.2 } = 0.2 , ( f B i | f B i ) ( ( x , k ) ( x , z ) ) = max { f A 1 ( x ) , f A 2 ( k ) , f A 2 ( z ) } = max { 0.5 , 0.4 , 0.3 } = 0.5 ,
for x V 1 and k z E i .
Now, for edge ( x , z ) ( m , z ) we have:
( t B i | t B i ) ( ( x , z ) ( m , z ) ) = min { t A 1 ( x ) , t A 1 ( m ) , t A 2 ( z ) } = min { 0.3 , 0.1 , 0.2 } = 0.1 , ( f B i | f B i ) ( ( x , z ) ( m , z ) ) = max { f A 1 ( x ) , f A 1 ( m ) , f A 2 ( z ) } = max { 0.5 , 0.2 , 0.3 } = 0.5 ,
for z V 2 and m x E i .
Similarly, we can find both membership value and non-membership value for all remaining vertices and edges.
Proposition 1.
The rejection of two VGSs G 1 and G 2 , is a VGS.
Proof. 
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) be two VGSs and ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) E i × E i , i = 1 , 2 , , n . Then by Definition 6 we have:
(i) If m 1 = n 1 and m 2 n 2 E i ,
( t B i | t B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = min { t A 1 ( m 1 ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } = min min { t A 1 ( m 1 ) , t A 2 ( m 2 ) } , min { t A 1 ( n 1 ) , t A 2 ( n 2 ) } = min ( t A 1 | t A 2 ) ( m 1 , m 2 ) , ( t A 1 | t A 2 ) ( n 1 , n 2 ) ,
( f B i | f B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = max { f A 1 ( m 1 ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } = max max { f A 1 ( m 1 ) , f A 2 ( m 2 ) } , max { f A 1 ( n 1 ) , f A 2 ( n 2 ) } = max ( f A 1 | f A 2 ) ( m 1 , m 2 ) , ( f A 1 | f A 2 ) ( n 1 , n 2 ) .
(ii) If m 2 = n 2 and m 1 n 1 E i
( t B i | t B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m 2 ) } = min min { t A 1 ( m 1 ) , t A 2 ( m 2 ) } , min { t A 1 ( n 1 ) , t A 2 ( n 2 ) } = min ( t A 1 | t A 2 ) ( m 1 , m 2 ) , ( t A 1 | t A 2 ) ( n 1 , n 2 ) ,
( f B i | f B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m 2 ) } = max max { f A 1 ( m 1 ) , f A 2 ( m 2 ) } , max { f A 1 ( n 1 ) , f A 2 ( n 2 ) } = max ( f A 1 | f A 2 ) ( m 1 , m 2 ) , ( f A 1 | f A 2 ) ( n 1 , n 2 ) .
(iii) If m 1 n 1 E i and m 2 n 2 E i
( t B i | t B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } = min min { t A 1 ( m 1 ) , t A 2 ( m 2 ) } , min { t A 1 ( n 1 ) , t A 2 ( n 2 ) } = min ( t A 1 | t A 2 ) ( m 1 , m 2 ) , ( t A 1 | t A 2 ) ( n 1 , n 2 ) ,
( f B i | f B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } = max max { f A 1 ( m 1 ) , f A 2 ( m 2 ) } , max { f A 1 ( n 1 ) , f A 2 ( n 2 ) } = max ( f A 1 | f A 2 ) ( m 1 , m 2 ) , ( f A 1 | f A 2 ) ( n 1 , n 2 ) .
Therefore, G 1 | G 2 is a VGS. □
Definition 7.
The residue product G 1 G 2 of two VGSs G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) is defined as:
( i ) ( t A 1 t A 2 ) ( ( m 1 , m 2 ) ) = max { t A 1 ( m 1 ) , t A 2 ( m 2 ) } , ( f A 1 f A 2 ) ( ( m 1 , m 2 ) ) = min { f A 1 ( m 1 ) , f A 2 ( m 2 ) } ,
for all ( m 1 , m 2 ) V 1 × V 2 .
( i i ) ( t B i t B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = t B i ( m 1 n 1 ) , ( f B i f B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = f B i ( m 1 n 1 ) ,
for all m 1 n 1 E i , m 2 n 2 .
Example 5.
Consider the VGSs G 1 and G 2 as in Figure 7. The residue product of G 1 and G 2 ( G 1 G 2 ) shown in Figure 8.
For vertex ( u , w ) , we find both membership value and non-membership value as follows:
( t A 1 t A 2 ) ( u , w ) = max { t A 1 ( u ) , t A 2 ( w ) } = max { 0.1 , 0.2 } = 0.2 , ( f A 1 f A 2 ) ( u , w ) = min { f A 1 ( u ) , f A 2 ( w ) } = min { 0.2 , 0.4 } = 0.2 ,
for u V 1 and w V 2 .
Now, for edge ( u , z ) ( x , w ) we have:
( t B i t B i ) ( ( u , z ) ( x , w ) ) = t B i ( u x ) = 0.1 , ( f B i f B i ) ( ( u , z ) ( x , w ) ) = f B i ( u x ) = 0.5 ,
for u x E 1 and z w .
In the same way, we can find both membership value and non-membership value for all remaining vertices and edges.
Proposition 2.
The residue product of two VGSs G 1 and G 2 is a VGS.
Proof. 
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) be two VGSs and ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) E i × E i , i = 1 , 2 , , n . If m 1 n 1 E i and m 2 n 2 , then we have:
( t B i t B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = t B i ( m 1 n 1 ) min { t A 1 ( m 1 ) , t A 1 ( n 1 ) } max min { t A 1 ( m 1 ) , t A 1 ( n 1 ) } , min { t A 2 ( m 2 ) , t A 2 ( n 2 ) } = min max { t A 1 ( m 1 ) , t A 1 ( n 1 ) } , max { t A 2 ( m 2 ) , t A 2 ( n 2 ) } = min ( t A 1 t A 2 ) ( m 1 , m 2 ) , ( t A 1 t A 2 ) ( n 1 , n 2 ) ,
( f B i f B i ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = f B i ( m 1 n 1 ) max { f A 1 ( m 1 ) , f A 1 ( n 1 ) } min max { f A 1 ( m 1 ) , f A 1 ( n 1 ) } , max { f A 2 ( m 2 ) , f A 2 ( n 2 ) } = max min { f A 1 ( m 1 ) , f A 1 ( n 1 ) } , min { f A 2 ( m 2 ) , f A 2 ( n 2 ) } = max ( f A 1 f A 2 ) ( m 1 , m 2 ) , ( f A 1 f A 2 ) ( n 1 , n 2 ) .
 □

4. Irregularity in VGSs

The irregularity concepts play an important role in both the graph theory application and theory in the vague environment. The characterizations of highly irregular and neighborly irregular graphs have also been applied to the question of heterogeneity. One of the most broadly studied classes of FGs is IFGs. They are being applied in many contexts, for example, the r-irregular (FGs) with connectivity and edge-connectivity equal to r play a key role in designing reliable communication networks. This idea inspires us to present different types of IVGSs such as totally irregular-VGS (TI-VGS), highly irregular-VGS (HI-VGS), strongly irregular-VGS (SI-VGS), strongly totally irregular-VGS (STI-VGS), highly totally irregular-VGS (HTI-VGS), neighborly edge irregular-VGS (NEI-VGS), neighborly edge totally irregular-VGS (NETI-VGS), and strongly edge totally irregular-VGS (SETI-VGS) and their related theorems.
Definition 8.
An VGS G is said to be an irregular-VGS if there is a vertex which is adjacent to vertices with distinct degrees.
Example 6.
Consider a VGS G = ( A , B 1 , B 2 ) as shown in Figure 9.
By direct calculation, we have d G ( v ) = ( 0.2 , 1.1 ) , d G ( u ) = ( 0.2 , 1 ) , d G ( w ) = ( 0.3 , 1 ) , d G ( z ) = ( 0.3 , 1.1 ) . From this, we see that v is adjacent to vertices of different degrees. Hence, G is an I-VGS.
Definition 9.
A VGS G is said to be a totally irregular-VGS if ∃ a vertex which is adjacent to vertices with different total degrees.
Example 7.
Consider a VGS G = ( A , B 1 , B 2 ) as shown in Figure 10.
By direct calculation, we have t d G ( u ) = ( 0.3 , 1.1 ) , t d G ( v ) = ( 0.5 , 1.6 ) , t d G ( w ) = ( 0.7 , 1.8 ) , and t d G ( z ) = ( 0.5 , 1.3 ) . From this, we see that u is adjacent to vertices of different degrees. Hence, G is a TI-VGS.
Definition 10.
A VGS G is said to be a strongly I-VGS if every vertex has a different degree.
Example 8.
Consider a VGS G = ( A , B 1 , B 2 ) as shown in Figure 11.
It is easy to see that d G ( u ) = ( 0.1 , 0.5 ) , d G ( v ) = ( 0.3 , 1.3 ) , d G ( w ) = ( 0.2 , 0.6 ) , d G ( x ) = ( 0.1 , 0.6 ) , d G ( y ) = ( 0.3 , 1.1 ) , d G ( z ) = ( 0.2 , 0.7 ) . We can see that every vertex has a different degree. Hence, G is strongly I-VGS.
The first definition of neighborly irregular in fuzzy graph was introduced in [29]. Neighborly irregular has not been much discussed although the totally irregular and edge irregular concepts are very important and they can be useful in computer science and the problem of finding the shortest path in a computer network. Therefore, the following definitions are provided to highlight the issue.
Definition 11.
A VGS G is said to be a strongly TI-VGS if every vertex has a different total degree.
Example 9.
Consider a VGS G as shown in Figure 11.
By direct calculation we have t d G ( u ) = ( 0.2 , 0.7 ) , t d G ( v ) = ( 0.5 , 1.6 ) , t d G ( w ) = ( 0.5 , 1 ) , t d G ( x ) = ( 0.3 , 1.1 ) , t d G ( y ) = ( 0.6 , 1.6 ) , t d G ( z ) = ( 0.6 , 1.2 ) . From this, we see that every vertex has a different total degree. Hence, G is strongly TI-VGS.
Definition 12.
A VGS G is said to be highly I-VGS if every vertex in G is adjacent to vertices of different degrees.
Example 10.
Consider the VGS G as shown in Figure 9. It is easy to see that every vertex is adjacent to vertices of different degrees. Hence, G is highly I-VGS.
Definition 13.
A VGS G is said to be highly TI-VGS if every vertex in G is adjacent to vertices of different total degrees.
Example 11.
Consider VGS G = ( A , B 1 , B 2 ) as shown in Figure 12.
By simple calculation, we have t d G ( u ) = ( 0.5 , 1.6 ) , t d G ( v ) = ( 0.3 , 1.5 ) , t d G ( w ) = ( 0.6 , 1.8 ) , and t d G ( z ) = ( 0.7 , 1.9 ) . So, every vertex is adjacent to vertices of different degrees. Hence, G is a highly TI-VGS.
Definition 14.
The degree and the total degree of an edge u v of a VGS G are defined by d G ( u v ) = ( d t ( u v ) , d f ( u v ) ) , and t d G ( u v ) = ( t d t ( u v ) , t d f ( u v ) ) , respectively, and we have:
d t ( u v ) = d t ( u ) + d t ( v ) 2 t B ( u v ) , d f ( u v ) = d f ( u ) + d f ( v ) 2 f B ( u v ) , t d t ( u v ) = d t ( u v ) + t B ( u v ) , t d f ( u v ) = d f ( u v ) + f B ( u v ) .
Example 12.
Consider the VGS G = ( A , B 1 , B 2 , B 3 ) as shown in Figure 13.
By direct calculation, we have d G ( u ) = ( 0.2 , 0.6 ) , d G ( v ) = ( 0.3 , 1.3 ) , d G ( w ) = ( 0.4 , 1.2 ) , d G ( z ) = ( 0.3 , 0.5 ) . The degree of every edge is given as:
d t ( u v ) = d t ( u ) + d t ( v ) 2 t B ( u v ) = 0.2 + 0.3 2 ( 0.2 ) = 0.1 , d f ( u v ) = d f ( u ) + d f ( v ) 2 f B ( u v ) = 0.6 + 1.3 2 ( 0.6 ) = 0.7 ,
d t ( v w ) = d t ( v ) + d t ( w ) 2 t B ( v w ) = 0.3 + 0.4 0.2 ( 0.1 ) = 0.5 , d f ( v w ) = d f ( v ) + d f ( w ) 2 f B ( v w ) = 1.3 + 1.2 2 ( 0.7 ) = 1.1 ,
d t ( w z ) = d t ( w ) + d t ( z ) 2 t B ( w z ) = 0.4 + 0.3 2 ( 0.3 ) = 0.1 , d f ( w z ) = d f ( w ) + d f ( z ) 2 f B ( w z ) = 1.2 + 0.5 1 = 0.7 .
Hence, d G ( u v ) = ( 0.1 , 0.7 ) , d G ( v w ) = ( 0.5 , 1.1 ) , d G ( z , w ) = ( 0.1 , 0.7 ) . The total degree of every edge is given as:
t d t ( u v ) = d t ( u v ) + t B ( u v ) = 0.1 + 0.2 = 0.3 , t d f ( u v ) = d f ( u v ) + f B ( u v ) = 0.7 + 0.6 = 1.3 , t d t ( v w ) = d t ( v w ) + t B ( v w ) = 0.5 + 0.1 = 0.6 , t d f ( v w ) = d f ( v w ) + f B ( v w ) = 1.1 + 0.7 = 1.8 , t d t ( w z ) = d t ( w z ) + t B ( w z ) = 0.1 + 0.3 = 0.4 , t d f ( w z ) = d f ( w z ) + f B ( w z ) = 0.7 + 0.5 = 1.2 .
So, t d G ( u v ) = ( 0.3 , 1.3 ) , t d G ( v w ) = ( 0.6 , 1.8 ) , and t d G ( w z ) = ( 0.4 , 1.2 ) .
Definition 15.
A connected VGS G is said to be a neighborly edge irregular-VGS (NEI-VGS), if every pair of adjacent edges in G has different degrees.
Example 13.
Consider the VGS G as shown in Figure 13. From this, it is easy to see that every pair of adjacent edges has different degrees. Hence, G is an NEI-VGS.
Definition 16.
A connected VGS G is said to be a neighborly edge totally irregular-VGS (NETI-VGS), if every pair of adjacent edges has different total degrees.
Example 14.
Consider the VGS G as shown in Figure 13. From it, we see that every pair of adjacent edges has different total degrees. Therefore, G is an NETI-VGS.
Definition 17.
A VGS G is said to be a strongly edge irregular-VGS (SEI-VGS), if every edge in G has a different degree.
Example 15.
Consider the VGS G = ( A , B 1 , B 2 , B 3 ) as shown in Figure 14.
By a simple calculation, we have d G ( u ) = ( 0.3 , 1.1 ) , d G ( v ) = ( 0.3 , 0.9 ) , and d G ( w ) = ( 0.2 , 1 ) . The degree of each edge is given as:
d G ( u v ) = ( 0.2 , 1 ) , d G ( v w ) = ( 0.3 , 1.1 ) , d G ( u w ) = ( 0.3 , 0.9 ) .
Since that each edge has different degree, so G is an SEI-VGS.
Definition 18.
A VGS G is said to be a strongly edge totally irregular-VGS (SETI-VGS), if every edge in G has a different total degree.
Example 16.
Consider the VGS G = ( A , B 1 , B 2 ) as shown in Figure 15.
By a simple calculation, we have d G ( u ) = ( 0.3 , 1.2 ) , d G ( v ) = ( 0.3 , 1.1 ) , d G ( z ) = ( 0.4 , 1 ) , d G ( w ) = ( 0.4 , 1.1 ) .
The degree of every edge is given as:
d G ( u v ) = ( 0.4 , 1.3 ) , d G ( v z ) = ( 0.2 , 0.9 ) , d G ( z w ) = ( 0.4 , 1.3 ) , d G ( u w ) = ( 0.3 , 0.9 ) .
The total degree of each edges is given as:
t d G ( u v ) = ( 0.5 , 1.8 ) , t d G ( v z ) = ( 0.5 , 1.5 ) , t d G ( z w ) = ( 0.6 , 1.7 ) , t d G ( u w ) = ( 0.5 , 1.6 ) .
It shows that every edge in G has a different total degree. Hence, G is an SETI-VGS.
Remark 1.
The SEI-VGS G may not be SETI-VGS.
Example 17.
Consider a VGS G = ( A , B 1 , B 2 , B 3 ) as shown in Figure 16.
By direct calculation, we have:
d G ( u ) = ( 0.5 , 1.5 ) , d G ( v ) = ( 0.3 , 1.7 ) , d G ( w ) = ( 0.4 , 1.6 ) , d G ( u v ) = ( 0.4 , 1.6 ) , d G ( v w ) = ( 0.5 , 1.5 ) , d G ( u w ) = ( 0.3 , 1.7 ) .
Since every edge in G has a different degree, so G is an SEI-VGS. Now, we calculate the total degree of every edge as follows:
t d G ( u v ) = ( 0.6 , 2.4 ) = t d G ( v w ) = t d G ( u w ) .
Hence, G is not an SETI-VGS.
Theorem 2.
If G = ( A , B 1 , B 2 , , B n ) is a strongly edge irregular connected VGS (SEIC-VGS), where B i ( i = 1 , 2 , , n ) , is a constant function. Then G is an SETI-VGS.
Proof. 
Let G be an SEIC-VGS. Consider B i as a constant function. Then t B i ( u v ) = m 1 and f B i ( u v ) = m 2 , for all u v E i , where m 1 , and m 2 are constants. Consider two edges u v and z w in E i . Since G is an SEI-VGS so d G ( u v ) d G ( z w ) , where u v and z w are two edges in E i . This shows that d G ( u v ) + ( m 1 , m 2 ) d G ( z w ) + ( m 1 , m 2 ) . This implies that d G ( u v ) + ( t B i ( u v ) , f B i ( u v ) ) d G ( z w ) + ( t B i ( z w ) , f B i ( z w ) ) . Hence, t d G ( u v ) t d G ( z w ) , where, u v and z w are two edges in E i . Since the edges u v and z w were taken to be arbitrary, this shows every two edge in G have different total degrees. Therefore, G is an SETI-VGS. □
Theorem 3.
If G = ( A , B 1 , B 2 , , B n ) is a SETIC-VGS, where B i is a constant function. Then, G is an SEI-VGS.
Proof. 
Let G be an SETIC-VGS. Consider B i , ( i = 1 , 2 , , n ) is a constant function. Then, t B i ( u v ) = m 1 , f B i ( u v ) = m 2 , for all u v E i , where m 1 and m 2 are constants. Consider the edges u v and z w in E i . Since G is an SETI-VGS, so t d G ( u v ) t d G ( z w ) , where u v and z w are edges in E i . This results that d G ( u v ) + ( t B i ( u v ) , f B i ( u v ) ) d G ( z w ) + ( t B i ( z w ) , f B i ( z w ) ) . This shows that d G ( u v ) + ( m 1 , m 2 ) d G ( z w ) + ( m 1 , m 2 ) . So, d G ( u v ) d G ( z w ) , where u v and z w are two edges in E i . Since that u v and z w were arbitrary edges, so every two edges in G have different degrees. Therefore, G is an SEI-VGS. □
Theorem 4.
Let G = ( A , B 1 , B 2 , , B n ) be an SEI-VGS. Then G is an NEI-VGS.
Proof. 
Let G be an SEI-VGS. Then, every edge in G has a different degree. This shows that every two neighbor edges have different degrees. Hence, G is an NEI-VGS. □
Theorem 5.
Let G = ( A , B 1 , B 2 , , B n ) be an SETI-VGS. Then, G is an NETI-VGS.
Proof. 
Let G be an SETI-VGS. Then, each edge in G has a different total degree. This shows that every two neighbor edges in G have different total degrees. Hence, G is an NETI-VGS. □
Remark 2.
If G is an NEI-VGS, then, it is not compulsory that G is an SEI-VGS.
Example 18.
Consider the VGS G = ( A , B 1 , B 2 , B 3 ) as shown in Figure 17.
By simple calculation we have:
d G ( u ) = ( 0.1 , 0.4 ) , d G ( v ) = ( 0.2 , 0.8 ) , d G ( w ) = ( 0.2 , 0.8 ) , a n d d G ( z ) = ( 0.1 , 0.4 ) .
The degree of each edge is d G ( u v ) = ( 0.1 , 0.4 ) , d G ( v w ) = ( 0.2 , 0.8 ) , and d G ( w z ) = ( 0.1 , 0.4 ) . Since each two neighbor edges have different degrees, i.e., d G ( u v ) d G ( v w ) and d G ( v w ) d G ( w z ) . Hence, G is an NEI-VGS. Here, we can see that d G ( u v ) = d G ( w z ) . Hence, G is not an SEI-VGS.
Theorem 6.
Let G = ( A , B 1 , B 2 , , B n ) be an SEIC-VGS, with B i ( i = 1 , 2 , , n ) as constant function. Then, G is an I-VGS.
Proof. 
Let G be an SEIC-VGS, with B i as constant function. Then, t B i ( u v ) = m 1 and f B i ( u v ) = m 2 , for every edge u v E i , where m 1 and m 2 are constants. Likewise, every edge in G has a different degree, so G is an SEI-VGS. Suppose that u v and v w be any two neighbor edges in G such that d G ( u v ) d G ( v w ) . This shows that d G ( u ) + d G ( v ) 2 ( t B ( u v ) , f B ( u v ) ) d G ( v ) + d G ( w ) 2 ( t B ( v w ) , f B ( v w ) ) . This results that d G ( u ) + d G ( v ) 2 ( m 1 , m 2 ) d G ( v ) + d G ( w ) 2 ( m 1 , m 2 ) . This implies that d G ( u ) d G ( w ) . Hence, there is a vertex v in G which is neighbor to the vertices with different degrees. This shows that G is an I-VGS. □
Theorem 7.
Let G = ( A , B 1 , B 2 , , B n ) be an SETIC-VGS, with B i ( i = 1 , 2 , , B n ) , as constant function. Then, G is an I-VGS.
Proof. 
Let G = ( A , B 1 , B 2 , , B n ) be an SETIC-VGS, with B i as constant function. Then, t B i ( u v ) = m 1 and f B i ( u v ) = m 2 , for every edge u v E i , where m 1 and m 2 are constants and every edge in G has a different total degree. Hence G is an SETI-VGS. Suppose that u v and v w be any two neighbor edges in G such that t d G ( u v ) t d G ( v w ) . This shows that d G ( u v ) + ( t B i ( u v ) , f B i ( u v ) ) d G ( v w ) + ( t B i ( v w ) , f B i ( v w ) ) . This results that d G ( u ) + d G ( v ) ( t B i ( u v ) , f B i ( u v ) ) d G ( v ) + d G ( w ) ( t B i ( v w ) , f B i ( v w ) ) . This represents that d G ( u ) + d G ( v ) 2 ( m 1 , m 2 ) d G ( v ) + d G ( w ) 2 ( m 1 , m 2 ) . This shows that d G ( u ) d G ( w ) . Hence, there is a vertex v in G which is neighbor to the vertices with different degrees. This indicates that G is an I-VGS. □

5. Application of VGS in Medical Diagnosis

In this section, an attempt was made to run through a fuzzy decision-making approach using (VS). In addition, an interesting distance measure on VSs has been introduced.
Definition 19.
Let X = { m 1 , m 2 , , m n } be the universe of discourse. Let A = { ( m i , t A ( m i ) , f A ( m i ) : m i X } and B = { ( m i , t B ( m i ) , f B ( m i ) : m i X } be two VSs. The the new distance measure can be defined as:
d ( A , B ) = 2 n i = 1 n sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | }
To be a distance measure, it is easy to show that all four conditions are satisfied (See Appendix A, Remark A5).
Suppose that { D 1 , D 2 , , D n } be a set of number of possible diseases and { P 1 , P 2 , , P n } be a set of n number of patients. Let, { S 1 ( t 1 D i , f 1 D i ) , S 2 ( t 2 D i , f 2 D i ) , , S l ( t l D i , f l D i ) } be the symptoms the disease D i and { S 1 ( t 1 P j , f 1 P j ) , S 2 ( t 2 P j , f 2 P j ) , , S l ( t l P j , f l P j ) } be the symptoms of patient P j expressed in VSs. Hence, the distance between the symptoms disease D i and symptoms of patient P j can be evaluated as follows using the proposed distance measure:
d ( D i , P j ) = 2 l k = 1 l sin { π 6 | t k D i t k P j | } + sin { π 6 | f k D i f k P j | } 1 + sin { π 6 | t k D i t k P j | } + sin { π 6 | f k D i f k P j | }
where = 1 , 2 , , m and j = 1 , 2 , , n .
We can describe the distances between every pair of disease and patients using the following matrix:
P 1 P 2 P n D 1 D 2 D m d ( D 1 , P 1 ) d ( D 1 , P 2 ) d ( D 1 , P n ) d ( D 2 , P 1 ) d ( D 2 , P 2 ) d ( D 2 , P n ) d ( D m , P 1 ) d ( D m , P 2 ) d ( D m , P n )
Applying the fact that, less distance between two VSs shows more similarity between them, therefore, it can be described that for the patient P j the disease most possibly suffered by him or her is the disease corresponding to min i ( D i , P j ) .
Here, we consider a set of symptoms S, a set of diagnosis D, and a set of patients P. Consider P = { T e d , J o h n , J o s e p h , J a c k } , S = { D y s p n e a , u r i n a r y s y m p t o m s , A b d o m i n a l p a i n , C h e s t p a i n , F e v e r } , and D = { P n e u m o n i a ( P n ) , P y e l o n e p h r i t i s ( P y ) , A p p e n d i c i t i s ( A p ) , M I , G a s t r o e n t e r i t i s ( G a ) } . Our aim is to make the right diagnosis decision for each patient, from the set of symptoms, for each disease.
Now, in Figure 18, we assume G 1 = ( A 1 , B 1 , B 2 , B 3 , B 4 ) is the VGS of the set of patients, and G 2 = ( A 2 , B 1 , B 2 , B 3 , B 4 , B 5 ) is the VGS of the set of diagnosis where A 1 = { T e d , J o h n , J o s e p h , J a c k } and A 2 = { P n , P y , A p , M I , G a } .
The relation between symptom and disease, and the relation between pain and symptom is given in Table 1 and Table 2, respectively.
The five diagnoses are now described as vague sets.
P n = { D y , ( 0.8 , 0 ) , U s , ( 0.2 , 0.5 ) , A p , ( 0.1 , 0.2 ) , C h p , ( 0.7 , 0.2 ) , F e , ( 0.1 , 0.4 ) } P y = { D y , ( 0.3 , 0.3 ) , U s , ( 0.8 , 0.2 ) , A p , ( 0.2 , 0.3 ) , C h p , ( 0.1 , 0.5 ) , F e , ( 0.2 , 0.6 ) } A p = { D y , ( 0.1 , 0.6 ) , U s , ( 0.1 , 0.3 ) , A p , ( 0.8 , 0.2 ) , C h p , ( 0.2 , 0.5 ) , F e , ( 0.3 , 0.5 ) } M I = { D y , ( 0.7 , 0.1 ) , U s , ( 0.3 , 0.4 ) , A p , ( 0.2 , 0.4 ) , C h p , ( 0.8 , 0.3 ) , F e , ( 0.4 , 0.3 ) } G a = { D y , ( 0.2 , 0.4 ) , U s , ( 0.4 , 0.3 ) , A p , ( 0.6 , 0.3 ) , C h p , ( 0.3 , 0.5 ) , F e , ( 0.8 , 0.2 ) } .
Likewise, the patients as vague sets can be described as follows:
T e d = { D y , ( 0.2 , 0.5 ) , U s , ( 0.8 , 0.1 ) , A p , ( 0.3 , 0.4 ) , C h p , ( 0.4 , 0.2 ) , F e , ( 0.1 , 0.3 ) } J o h n = { D y , ( 0.2 , 0.3 ) , U s , ( 0.1 , 0.4 ) , A p , ( 0.3 , 0.3 ) , C h p , ( 0.2 , 0.4 ) , F e , ( 0.8 , 0 ) } J o s e p h = { D y , ( 0.7 , 0.2 ) , U s , ( 0.3 , 0.2 ) , A p , ( 0.4 , 0.3 ) , C h p , ( 0.5 , 0.1 ) , F e , ( 0.2 , 0.3 ) } J a c k = { D y , ( 0.1 , 0.2 ) , U s , ( 0.2 , 0.4 ) , A p , ( 0.9 , 0.1 ) , C h p , ( 0.2 , 0.3 ) , F e , ( 0.2 , 0.5 ) } .
Now the vague distance is calculated between the disease and the patients based on their symptoms.
d ( P n , T e d ) = 2 5 { sin π 6 | 0.8 0.2 | + sin π 6 | 0 0.5 | 1 + sin π 6 | 0.8 0.2 | + sin π 6 | 0 0.5 | + sin π 6 | 0.2 0.8 | + sin π 6 | 0.5 1 | 1 + sin π 6 | 0.2 0.8 | + sin π 6 | 0.5 1 | + sin π 6 | 0.1 0.3 | + sin π 6 | 0.2 0.4 | 1 + sin π 6 | 0.1 0.3 | + sin π 6 | 0.2 0.4 | + sin π 6 | 0.7 0.4 | + sin π 6 | 0.2 0.2 | 1 + sin π 6 | 0.7 0.4 | + sin π 6 | 0.2 0.2 | + sin π 6 | 0.1 0.1 | + sin π 6 | 0.4 0.3 | 1 + sin π 6 | 0.1 0.1 | + sin π 6 | 0.4 0.3 | } = 2 5 0.3548 + 0.3333 + 0.1666 + 0.1304 + 0.0476 = 0.41308 .
d ( P n , J o h n ) = 2 5 { sin π 6 | 0.8 0.2 | + sin π 6 | 0 0.3 | 1 + sin π 6 | 0.8 0.2 | + sin π 6 | 0 0.3 | + sin π 6 | 0.2 0.1 | + sin π 6 | 0.5 0.4 | 1 + sin π 6 | 0.2 0.1 | + sin π 6 | 0.5 0.4 | + sin π 6 | 0.1 0.3 | + sin π 6 | 0.2 0.4 | 1 + sin π 6 | 0.1 0.3 | + sin π 6 | 0.2 0.4 | + sin π 6 | 0.7 0.2 | + sin π 6 | 0.2 0.4 | 1 + sin π 6 | 0.7 0.2 | + sin π 6 | 0.2 0.4 | + sin π 6 | 0.1 0.8 | + sin π 6 | 0.4 0 | 1 + sin π 6 | 0.1 0.8 | + sin π 6 | 0.4 0 | } = 2 5 0.31 . 3 + 0.0909 + 0.1666 + 0.2592 + 0.3548 = 0.4727 .
d ( P n , J o s e p h ) = 2 5 { sin π 6 | 0.8 0.7 | + sin π 6 | 0 0.2 | 1 + sin π 6 | 0.8 0.7 | + sin π 6 | 0 0.2 | + sin π 6 | 0.2 0.3 | + sin π 6 | 0.5 0.2 | 1 + sin π 6 | 0.2 0.3 | + sin π 6 | 0.5 0.2 | + sin π 6 | 0.1 0.4 | + sin π 6 | 0.2 0.3 | 1 + sin π 6 | 0.1 0.4 | + sin π 6 | 0.2 0.3 | + sin π 6 | 0.7 0.5 | + sin π 6 | 0.2 0.1 | 1 + sin π 6 | 0.7 0.5 | + sin π 6 | 0.2 0.1 | + sin π 6 | 0.1 0.2 | + sin π 6 | 0.4 0.3 | 1 + sin π 6 | 0.1 0.2 | + sin π 6 | 0.4 0.3 | } = 2 5 0.1304 + 0.1666 + 0.1666 + 0.1304 + 0.0909 = 0.2739 .
d ( P n , J a c k ) = 2 5 { sin π 6 | 0.8 0.1 | + sin π 6 | 0 0.2 | 1 + sin π 6 | 0.8 0.1 | + sin π 6 | 0 0.2 | + sin π 6 | 0.2 0.2 | + sin π 6 | 0.5 0.4 | 1 + sin π 6 | 0.2 0.2 | + sin π 6 | 0.5 0.4 | + sin π 6 | 0.1 0.9 | + sin π 6 | 0.2 0.1 | 1 + sin π 6 | 0.1 0.9 | + sin π 6 | 0.2 0.1 | + sin π 6 | 0.7 0.2 | + sin π 6 | 0.2 0.3 | 1 + sin π 6 | 0.7 0.2 | + sin π 6 | 0.2 0.3 | + sin π 6 | 0.1 0.2 | + sin π 6 | 0.4 0.5 | 1 + sin π 6 | 0.1 0.2 | + sin π 6 | 0.4 0.5 | } = 2 5 0.3103 + 0.0476 + 0.3103 + 0.2307 + 0.0909 = 0.3959 .
Similarly, the vague distance measure between the patients and the diseases can be found as:
d ( P y , T e d ) = 0.2491 , d ( P y , J o h n ) = 0.6758 , d ( P y , J o s e p h ) = 0.7747 , d ( P y , J a c k ) = 0.3617 , d ( A p , T e d ) = 0.7380 , d ( A p , J o h n ) = 0.3634 , d ( A p , J o s e p h ) = 0.6904 , d ( A p , J a c k ) = 1.1764 , d ( M I , T e d ) = 0.3896 , d ( M I , J o h n ) = 1.0382 , d ( M I , J o s e p h ) = 0.8784 , d ( M I , J a c k ) = 0.4149 , d ( G a , T e d ) = 0.3588 , d ( G a , J o h n ) = 0.8650 , d ( G a , J o s e p h ) = 0.3723 , d ( G a , J a c k ) = 0.3606 .
Vague graph structure G 1 G 2 is shown in Figure 19.
In this figure, we clearly see that people who have at least one similar symptom, are closer to each other in terms of diagnosis, and this is also acceptable in medical science. Therefore, with the help of maximized multiplication graph, we can find the right diagnosis for different people, with the same symptoms.
Now by applying the following distance matrix, it can be shown to what extent the symptoms of disease differ by the symptoms of patients.
Ted John Joseph Jack Pneumonia Pyelonephritis Appendicitis MI Gastroenteritis 0.4138 0.4727 0.2739 0.3959 0.2491 0.6758 0.7747 0.3617 0.7380 0.3634 0.6904 1.1764 0.3896 1.0382 0.8784 0.4149 0.3588 0.8650 0.3723 0.3606
Less distance between patient and disease implies more possibility of having the disease; we can predict which disease is suffered by the four people. Considering the distance matrix, one can see that, if the doctor agrees, Ted suffers from pyelonephritis, John suffers from appendicitis, Joseph suffers from pneumonia, and Jack suffers from gastroenteritis.

6. Conclusions

Graph theory has many applications in solving various problems of several domains, including networking, communication, data mining, clustering, image capturing, image segmentation, planning, and scheduling. However, in some situations, certain aspects of a graph-theoretical system may be uncertain. Applying the fuzzy-graphical methods in meeting the ambiguity and vague notions is very natural. Fuzzy-graph theory has an extensive number of applications in modeling various real-time systems where the inherent information level in the system varies with different levels of precision. Vague graph structures (VGSs) are very useful tools for the study of different domains of computational intelligence and computer science. VGSs have many applications in different sciences such as optimization, topology, neural networks, and operations research. Operations are conveniently used in many combinatorial applications. In various situations they present a suitable construction means. The concepts of the irregularity play a significant role in both graph theory and application in the vague environment. So, in this paper, we described three new operations on VGSs, namely, maximal product, rejection, residue product, and introduced special concepts of irregularity in VGSs. Finally, an application of VGS in medical diagnosis is presented. In our future work we will investigate the concepts of energy, Laplacian matrix, adjacency matrix, spectrum, and density in VGSs and give some applications of energy in VGSs and other sciences.

Author Contributions

All authors have contributed equally to this work. All authors have read and agreed for the possible publication of the manuscript.

Funding

This work was supported by the National Key R & D Program of China (Grant No. 2018YFB1005104), the Guanzhou Academician and Expert Workstation (No. 20200115-9).

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Definition A1.
A VGS G = ( A , B 1 , B 2 , , B n ) is ( t B i , f B i ) -strong if:
t B i ( m n ) = t A ( m ) t A ( n ) , f B i ( m n ) = f A ( m ) f A ( n ) ,
for all m n E i , i { 1 , 2 , , n } .
If G is ( t B i , f B i ) -strong, i { 1 , 2 , , n } , then G is called strong-VGS.
Theorem A1.
The maximal product of two strong VGSs G 1 and G 2 , is a strong VGS.
Proof. 
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) be two strong VGSs. Then t B i ( m 1 m 2 ) = min ( t A 1 ( m 1 ) , t A 1 ( m 2 ) ) , f B i ( m 1 m 2 ) = max ( f A 1 ( m 1 ) , f A 1 ( m 2 ) ) , for any m 1 m 2 E i and t B i ( n 1 n 2 ) = min { t A 1 ( n 1 ) , t A 2 ( n 2 ) } , f B i ( n 1 n 2 ) = max { f A 1 ( n 1 ) , f A 2 ( n 2 ) } , for any n 1 n 2 E i , i = 1 , 2 , , n . Then, proceeding according to the definition of maximal product,
(i) if n 1 = n 2 and m 1 m 2 E i . Then
( t B i t B i ) ( n 1 , m 1 ) ( n 2 , m 2 ) = max { t A 1 ( n 1 ) , t B i ( m 1 m 2 ) } = max t A 1 ( n 1 ) , min { t A 2 ( m 1 ) , t A 2 ( m 2 ) } = min max { t A 1 ( n 1 ) , t A 2 ( m 1 ) } , max { t A 1 ( n 1 ) , t A 2 ( m 2 ) } = min ( t A 1 t A 2 ) ( n 1 , m 1 ) , ( t A 1 t A 2 ) ( n 1 , m 2 ) , ( f B i f B i ) ( n 1 , m 1 ) ( n 2 , m 2 ) = min { f A 1 ( n 1 ) , f B i ( m 1 m 2 ) } = min f A 1 ( n 1 ) , max { f A 2 ( m 1 ) , f A 2 ( m 2 ) } = max min { f A 1 ( n 1 ) , f A 2 ( m 1 ) } , min { f A 1 ( n 1 ) , f A 2 ( m 2 ) } = max ( f A 1 f A 2 ) ( n 1 , m 1 ) , ( f A 1 f A 2 ) ( n 1 , m 2 ) .
(ii) if m 1 = m 2 and n 1 n 2 E i . Then
( t B i t B i ) ( n 1 , m 1 ) ( n 2 , m 2 ) = max { t B i ( n 1 n 2 ) , t A 2 ( m 2 ) } = max min { t A 1 ( n 1 ) , t A 1 ( n 2 ) } , t A 2 ( m 2 ) = min max { t A 1 ( n 1 ) , t A 2 ( m 2 ) } , max { t A 1 ( n 2 ) , t A 2 ( m 2 ) } = min ( t A 1 t A 2 ) ( n 1 , m 2 ) , ( t A 1 t A 2 ) ( n 2 , m 2 ) , ( f B i f B i ) ( n 1 , m 1 ) ( n 2 , m 2 ) = min { f B i ( n 1 n 2 ) , f A 2 ( m 2 ) } = min max { f A 1 ( n 1 ) , f A 1 ( n 2 ) } , f A 2 ( m 2 ) = max min { f A 1 ( n 1 ) , f A 2 ( m 2 ) } , min { f A 1 ( n 2 ) , f A 2 ( m 2 ) } = max ( f A 1 f A 2 ) ( n 1 , m 2 ) , ( f A 1 f A 2 ) ( n 2 , m 2 ) .
Therefore, G 1 G 2 is a strong VGS. □
Figure A1. VGSs G 1 , G 2 , and G 1 G 2 .
Figure A1. VGSs G 1 , G 2 , and G 1 G 2 .
Symmetry 12 01582 g0a1
Example A1.
Consider the strong VGSs G 1 = ( A , B 1 ) and G 2 = ( A , B 1 ) as in Figure A1. It is easy to see that G 1 G 2 = ( A , B 1 ) is a strong VGS, too.
Remark A1.
If the maximal product of two VGSs ( G 1 G 2 ) is a strong, then G 1 and G 2 need not to be strong, in general.
Figure A2. VGSs G 1 , G 2 , and G 1 G 2 .
Figure A2. VGSs G 1 , G 2 , and G 1 G 2 .
Symmetry 12 01582 g0a2
Example A2.
Consider the VGSs G 1 , G 2 , and G 1 G 2 as in Figure A2.
Hence, G 1 and G 1 G 2 are strong, but G 2 is not strong. Since t B 1 ( z w ) = 0.1 but min { t A 2 ( z ) , t A 2 ( w ) } = min { 0.2 , 0.2 } = 0.2 . Hence, t B 1 ( z w ) min { t A 2 ( z ) , t A 2 ( w ) } .
Theorem A2.
The maximal product of two connected VGSs is a connected VGS.
Proof. 
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) be two connected VGSs with underlying crisp graph structures G 1 = ( V 1 , E 1 , E 2 , , E n ) and G 2 = ( V 2 , E 1 , E 2 , , E n ) , respectively. Let V 1 = { m 1 , m 2 , , m k } and V 2 = { n 1 , n 2 , , n s } . Then t B i ( m i m j ) > 0 , for all m i , m j V 1 and t B i ( n i n j ) > 0 , for all n i , n j V 2 (or f B i ( m i m j ) < 1 , for all m i , m j V 1 and f B i ( n i n j ) < 1 , for all n i , n j V 2 ). The maximal product G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) is written as G = ( A , B 1 , B 2 , , B n ) . Now consider the k subgraphs of G with the vertex set { ( m i , n 1 ) , ( m i , n 2 ) , , ( m i , n s ) } , for i = 1 , 2 , , k . Each of these subgraphs of G is connected, since the m i s are the same and since G 2 is connected, each n i is adjacent to at least one of the vertices in V 2 . In addition, since G 1 is connected, each x i is adjacent to at least one of the vertices in V 1 . Hence, there exists at least one edge between any pair of the above k subgraphs. Thus, we have t B i ( m i , n j ) ( m m , n n ) > 0 (or f B i ( m i , n j ) ( m m , n n ) < 1 ), for all ( m i , n j ) ( m m , n n ) E i . So, G is a connected VGS. □
Remark A2.
The maximal product of two complete VGSs is not a complete VGS, in general. Because we do not include the case ( m 1 , m 2 ) E i and ( n 1 , n 2 ) E i in the definition of the maximal product of two VGSs.
Remark A3.
The maximal product of two complete VGSs is a strong VGS.
Example A3.
Consider the complete VGSs G 1 and G 2 as in Figure A1. A simple calculation concludes that G 1 G 2 is a strong VGS.
Definition A2.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) be two VGSs. For any vertex ( m 1 , m 2 ) V 1 × V 2 we have:
( d t ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j t B i t B j ( m 1 , m 2 ) ( n 1 , n 2 ) = m 1 = n 1 , m 2 n 2 E j max { t A 1 ( m 1 ) , t B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 max { t B i ( m 1 n 1 ) , t A 2 ( m 2 ) } ,
( d f ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j f B i f B j ( m 1 , m 2 ) ( n 1 , n 2 ) = m 1 = n 1 , m 2 n 2 E j min { f A 1 ( m 1 ) , f B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 min { f B i ( m 1 n 1 ) , f A 2 ( m 2 ) } .
Theorem A3.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) be two VGSs. If t A 1 t B j , f A 1 f B j , and t A 2 t B i , f A 2 f B i , i , j = 1 , 2 , , n , then
( d t ) G 1 G 2 ( m 1 , m 2 ) = ( d ) G 2 ( m 2 ) t A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) t A 2 ( m 2 ) , ( d f ) G 1 G 2 ( m 1 , m 2 ) = ( d ) G 2 ( m 2 ) f A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) f A 2 ( m 2 ) .
Proof. 
From the definition of a vertex in Cartesian product, we have:
( d t ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j t B i t B j ( m 1 , m 2 ) ( n 1 , n 2 ) = m 1 = n 1 , m 2 n 2 E j max { t A 1 ( m 1 ) , t B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 max { t B i ( m 1 n 1 ) , t A 2 ( m 2 ) } = m 2 n 2 E j , m 1 = n 1 t B j ( m 2 n 2 ) + m 1 n 1 E i , m 2 = n 2 t B i ( m 1 n 1 ) = ( d ) G 2 ( m 2 ) t A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) t A 2 ( m 2 ) ( d f ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j f B i f B j ( m 1 , m 2 ) ( n 1 , n 2 ) = m 1 = n 1 , m 2 n 2 E j min { f A 1 ( m 1 ) , f B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 min { f B i ( m 1 n 1 ) , f A 2 ( m 2 ) } = m 2 n 2 E j , m 1 = n 1 f B j ( m 2 n 2 ) + m 1 n 1 E i , m 2 = n 2 f B i ( m 1 n 1 ) = ( d ) G 2 ( m 2 ) f A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) f A 2 ( m 2 )
 □
Figure A3. VGSs G 1 , G 2 , and G 1 G 2 .
Figure A3. VGSs G 1 , G 2 , and G 1 G 2 .
Symmetry 12 01582 g0a3
Example A4.
Consider the VGSs G 1 , G 2 and G 1 G 2 as in Figure A3. Since t A 1 t B 1 , f A 1 f B 1 , t A 2 t B 1 and f A 2 f B 1 by Theorem A3 we have:
( d t ) G 1 G 2 ( m , z ) = ( d t ) G 2 ( z ) t A 1 ( m ) + ( d t ) G 1 ( m ) t A 2 ( z ) = 1 · ( 0.4 ) + 1 · ( 0.3 ) = 0.7 , ( d f ) G 1 G 2 ( m , z ) = ( d f ) G 2 ( z ) f A 1 ( m ) + ( d f ) G 1 ( m ) f A 2 ( z ) = 1 · ( 0.5 ) + 1 · ( 0.4 ) = 0.9 , ( d t ) G 1 G 2 ( m , w ) = ( d t ) G 2 ( w ) t A 1 ( m ) + ( d t ) G 1 ( m ) t A 2 ( w ) = 1 · ( 0.4 ) + 1 · ( 0.4 ) = 0.8 , ( d f ) G 1 G 2 ( m , w ) = ( d f ) G 2 ( w ) f A 1 ( m ) + ( d f ) G 1 ( m ) f A 2 ( w ) = 1 · ( 0.5 ) + 1 · ( 0.5 ) = 1 , ( d t ) G 1 G 2 ( n , z ) = ( d t ) G 2 ( z ) t A 1 ( n ) + ( d t ) G 1 ( n ) t A 2 ( z ) = 1 · ( 0.3 ) + 1 · ( 0.3 ) = 0.6 , ( d f ) G 1 G 2 ( n , z ) = ( d f ) G 2 ( z ) f A 1 ( n ) + ( d f ) G 1 ( n ) f A 2 ( z ) = 1 · ( 0.4 ) + 1 · ( 0.4 ) = 0.8 , ( d t ) G 1 G 2 ( n , w ) = ( d t ) G 2 ( w ) t A 1 ( n ) + ( d t ) G 1 ( n ) t A 2 ( w ) = 1 · ( 0.3 ) + 1 · ( 0.4 ) = 0.7 , ( d f ) G 1 G 2 ( n , w ) = ( d f ) G 2 ( w ) f A 1 ( n ) + ( d f ) G 1 ( n ) f A 2 ( w ) = 1 · ( 0.4 ) + 1 · ( 0.5 ) = 0.9 .
By direct calculations:
( d t ) G 1 G 2 ( m , z ) = 0.4 + 0.3 = 0.7 , ( d f ) G 1 G 2 ( m , z ) = 0.5 + 0.4 = 0.9 , ( d t ) G 1 G 2 ( m , w ) = 0.4 + 0.4 = 0.8 , ( d f ) G 1 G 2 ( m , w ) = 0.5 + 0.5 = 1 , ( d t ) G 1 G 2 ( n , z ) = 0.3 + 0.3 = 0.6 , ( d f ) G 1 G 2 ( n , z ) = 0.4 + 0.4 = 0.8 , ( d t ) G 1 G 2 ( n , w ) = 0.3 + 0.4 = 0.7 , ( d f ) G 1 G 2 ( n , w ) = 0.4 + 0.5 = 0.9 .
It is clear from the above calculations that the degrees of vertices calculated by using the formula of the above theorem and by directed method are the same.
Definition A3.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) be two VGSs. For any vertex ( m 1 , m 2 ) V 1 × V 2 we have:
( t d t ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( t B i t B j ) ( m 1 , m 2 ) ( n 1 , n 2 ) + ( t A 1 t A 2 ) ( m 1 , m 2 ) = m 1 = n 1 , m 2 n 2 E j max { t A 1 ( m 1 ) + t B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 max { t B i ( m 1 n 1 ) , t A 2 ( m 2 ) } + max { t A 1 ( m 1 ) , t A 2 ( m 2 ) } ,
( t d f ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( f B i f B j ) ( m 1 , m 2 ) ( n 1 , n 2 ) + ( f A 1 f A 2 ) ( m 1 , m 2 ) = m 1 = n 1 , m 2 n 2 E j min { f A 1 ( m 1 ) + f B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 min { f B i ( m 1 n 1 ) , f A 2 ( m 2 ) } + min { f A 1 ( m 1 ) , f A 2 ( m 2 ) } .
Example A5.
In this example we find the degree and the total degree of vertices ( m , z ) and ( m , w ) in Example 3.
( d t ) G 1 G 2 ( m , z ) = ( d ) G 2 ( m ) t A 1 ( z ) + ( d ) G 1 ( z ) t A 2 ( m ) = 2 · ( 0.2 ) + 1 · ( 0.1 ) = 0.4 + 0.1 = 0.5 , ( d f ) G 1 G 2 ( m , z ) = ( d ) G 2 ( m ) f A 1 ( z ) + ( d ) G 1 ( z ) f A 2 ( m ) = 2 · ( 0.3 ) + 1 · ( 0.2 ) = 0.6 + 0.2 = 0.8 .
Therefore, d G 1 G 2 ( m , z ) = ( 0.5 , 0.8 ) .
In addition, by definition of total vertex degree in maximal product,
( t d t ) G 1 G 2 ( m , z ) = ( d ) G 2 ( m ) t A 1 ( z ) + ( d ) G 1 ( z ) t A 2 ( m ) + max { t A 1 ( m ) , t A 2 ( z ) } = 2 · ( 0.2 ) + 1 · ( 0.1 ) + max { 0.1 , 0.2 } = 0.7 , ( t d f ) G 1 G 2 ( m , z ) = ( d ) G 2 ( m ) f A 1 ( z ) + ( d ) G 1 ( z ) f A 2 ( m ) + min { f A 1 ( m ) , f A 2 ( z ) } = 2 · ( 0.3 ) + 1 · ( 0.2 ) + min { 0.2 , 0.3 } = 1 .
Hence, t d G 1 G 2 ( m , z ) = ( 0.7 , 1 ) .
( d t ) G 1 G 2 ( m , w ) = ( d ) G 2 ( m ) t A 1 ( w ) + ( d ) G 1 ( w ) t A 2 ( m ) = 2 · ( 0.1 ) + 1 · ( 0.1 ) = 0.2 + 0.1 = 0.3 , ( d f ) G 1 G 2 ( m , w ) = ( d ) G 2 ( m ) f A 1 ( w ) + ( d ) G 1 ( w ) f A 2 ( m ) = 2 · ( 0.2 ) + 1 · ( 0.2 ) = 0.4 + 0.2 = 0.6 ,
( t d t ) G 1 G 2 ( m , w ) = ( d ) G 2 ( m ) t A 1 ( w ) + ( d ) G 1 ( w ) t A 2 ( m ) + max { t A 1 ( m ) , t A 2 ( w ) } = 2 · ( 0.1 ) + 1 · ( 0.1 ) + max { 0.1 , 0.1 } = 0.4 , ( t d f ) G 1 G 2 ( m , w ) = ( d ) G 2 ( m ) f A 1 ( w ) + ( d ) G 1 ( w ) f A 2 ( m ) + min { f A 1 ( m ) , f A 2 ( w ) } = 2 · ( 0.2 ) + 1 · ( 0.2 ) + min { 0.2 , 0.2 } = 0.8 .
Therefore, d G 1 G 2 ( m , w ) = ( 0.3 , 0.6 ) and t d G 1 G 2 ( m , w ) = ( 0.4 , 0.8 ) .
Similarly, we can find the degree and the total degree of all vertices in G 1 G 2 .
Theorem A4.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , , B n ) be two VGSs. If t A 1 t B j , f A 1 f B j , and t A 2 t B i , f A 2 f B i , then ( t d t ) G 1 G 2 ( m 1 , m 2 ) = ( d ) G 2 ( m 2 ) t A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) t A 2 ( m 2 ) + max { t A 1 ( m 1 ) , t A 2 ( m 2 ) } , ( t d f ) G 1 G 2 ( m 1 , m 2 ) = ( d ) G 2 ( m 2 ) f A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) f A 2 ( m 2 ) + min { f A 1 ( m 1 ) , f A 2 ( m 2 ) } .
Proof. 
From the Definition A3 we have:
( t d t ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j t B i t B j ( m 1 , m 2 ) ( n 1 , n 2 ) + t A 1 t A 2 ( m 1 , m 2 ) = m 1 = n 1 , m 2 n 2 E j max { t A 1 ( m 1 ) , t B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 max { t B i ( m 1 n 1 ) , t A 2 ( m 2 ) } + max { t A 1 ( m 1 ) , t A 2 ( m 2 ) } = m 2 n 2 E j , m 1 = n 1 t B j ( m 2 n 2 ) + m 1 n 1 E i , m 2 = n 2 t B i ( m 1 n 1 ) + max { t A 1 ( m 1 ) , t A 2 ( m 2 ) } = ( d ) G 2 ( m 2 ) t A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) t A 2 ( m 2 ) + max { t A 1 ( m 1 ) , t A 2 ( m 2 ) } ,
( t d f ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j f B i f B j ( m 1 , m 2 ) ( n 1 , n 2 ) + f A 1 f A 2 ( m 1 , m 2 ) = m 1 = n 1 , m 2 n 2 E j min { f A 1 ( m 1 ) , f B j ( m 2 n 2 ) } + m 1 n 1 E i , m 2 = n 2 min { f B i ( m 1 n 1 ) , f A 2 ( m 2 ) } + min { f A 1 ( m 1 ) , f A 2 ( m 2 ) } = m 2 n 2 E j , m 1 = n 1 f B j ( m 2 n 2 ) + m 1 n 1 E i , m 2 = n 2 f B i ( m 1 n 1 ) + min { f A 1 ( m 1 ) , f A 2 ( m 2 ) } = ( d ) G 2 ( m 2 ) f A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) f A 2 ( m 2 ) + min { f A 1 ( m 1 ) , f A 2 ( m 2 ) } .
 □
Example A6.
Consider the VGSs G 1 , G 2 and G 1 G 2 as in Figure A3. The total degree of vertex in maximal product is calculated by using the following formula:
( t d t ) G 1 G 2 ( m 1 , m 2 ) = ( d ) G 2 ( m 2 ) t A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) t A 2 ( m 2 ) + max { t A 1 ( m 1 ) , t A 2 ( m 2 ) } , ( t d f ) G 1 G 2 ( m 1 , m 2 ) = ( d ) G 2 ( m 2 ) f A 1 ( m 1 ) + ( d ) G 1 ( m 1 ) f A 2 ( m 2 ) + min { f A 1 ( m 1 ) , f A 2 ( m 2 ) } .
( t d t ) G 1 G 2 ( m , z ) = ( d ) G 2 ( z ) t A 1 ( m ) + ( d ) G 1 ( m ) t A 2 ( z ) + max { t A 1 ( m ) , t A 2 ( z ) } = 1 · ( 0.4 ) + 1 · ( 0.3 ) + max { 0.4 , 0.3 } = 1.1 , ( t d f ) G 1 G 2 ( m , z ) = ( d ) G 2 ( z ) f A 1 ( m ) + ( d ) G 1 ( m ) f A 2 ( z ) + min { f A 1 ( m ) , f A 2 ( z ) } = 1 · ( 0.5 ) + 1 · ( 0.4 ) + min { 0.5 , 0.4 } = 1.3 , ( t d t ) G 1 G 2 ( m , w ) = ( d ) G 2 ( w ) t A 1 ( m ) + ( d ) G 1 ( m ) t A 2 ( w ) + max { t A 1 ( m ) , t A 2 ( w ) } = 1 · ( 0.4 ) + 1 · ( 0.4 ) + max { 0.4 , 0.4 } = 1.2 , ( t d f ) G 1 G 2 ( m , w ) = ( d ) G 2 ( w ) f A 1 ( m ) + ( d ) G 1 ( m ) f A 2 ( w ) + min { f A 1 ( m ) , f A 2 ( w ) } = 1 · ( 0.5 ) + 1 · ( 0.5 ) + min { 0.5 , 0.5 } = 1.5 , ( t d t ) G 1 G 2 ( n , z ) = ( d ) G 2 ( z ) t A 1 ( n ) + ( d ) G 1 ( n ) t A 2 ( z ) + max { t A 1 ( n ) , t A 2 ( z ) } = 1 · ( 0.3 ) + 1 · ( 0.3 ) + max { 0.3 , 0.3 } = 0.9 ,
( t d f ) G 1 G 2 ( n , z ) = ( d ) G 2 ( z ) f A 1 ( n ) + ( d ) G 1 ( n ) f A 2 ( z ) + min { f A 1 ( n ) , f A 2 ( z ) } = 1 · ( 0.4 ) + 1 · ( 0.4 ) + min { 0.4 , 0.4 } = 1.2 , ( t d t ) G 1 G 2 ( n , w ) = ( d ) G 2 ( w ) t A 1 ( n ) + ( d ) G 1 ( n ) t A 2 ( w ) + max { t A 1 ( n ) , t A 2 ( w ) } = 1 · ( 0.3 ) + 1 · ( 0.4 ) + max { 0.4 , 0.3 } = 1.1 , ( t d f ) G 1 G 2 ( n , w ) = ( d ) G 2 ( w ) f A 1 ( n ) + ( d ) G 1 ( n ) f A 2 ( w ) + min { f A 1 ( n ) , f A 2 ( w ) } = 1 · ( 0.4 ) + 1 · ( 0.5 ) + min { 0.4 , 0.5 } = 1.3 .
By direct calculations:
( t d t ) G 1 G 2 ( m , z ) = 0.4 + 0.3 + 0.4 = 1.1 , ( t d f ) G 1 G 2 ( m , z ) = 0.5 + 0.4 + 0.4 = 1.3 , ( t d t ) G 1 G 2 ( m , w ) = 0.4 + 0.4 + 0.4 = 1.2 , ( t d f ) G 1 G 2 ( m , w ) = 0.5 + 0.5 + 0.5 = 1.5 , ( t d t ) G 1 G 2 ( n , z ) = 0.3 + 0.3 + 0.3 = 0.9 , ( t d f ) G 1 G 2 ( n , z ) = 0.4 + 0.4 + 0.4 = 1.2 , ( t d t ) G 1 G 2 ( n , w ) = 0.3 + 0.4 + 0.4 = 1.1 , ( t d f ) G 1 G 2 ( n , w ) = 0.4 + 0.5 + 0.4 = 1.3 .
It is clear from the above calculations that the total degree of vertices calculated by using the formula of the above theorem and by directed method are the same.
Remark A4.
The rejection of two complete VGSs G 1 and G 2 is a complete VGS.
Definition A4.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) be two VGSs. For any vertex ( m 1 , m 2 ) V 1 × V 2 we have:
( d t ) G 1 | G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( t B i | t B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = m 1 = n 1 , m 2 n 2 E j min { t A 1 ( m 1 ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } + m 2 = n 2 , m 1 n 1 E i min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m 2 ) } + m 1 n 1 E i , m 2 n 2 E j min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } ,
( d f ) G 1 | G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( f B i | f B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = m 1 = n 1 , m 2 n 2 E j max { f A 1 ( m 1 ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } + m 2 = n 2 , m 1 n 1 E i max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m 2 ) } + m 1 n 1 E i , m 2 n 2 E j max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } .
Definition A5.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) be two VGSs. For any vertex ( m 1 , m 2 ) V 1 × V 2 we have:
( t d t ) G 1 | G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( t B i | t B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) + ( t A 1 | t A 2 ) ( m 1 , m 2 ) = m 1 = n 1 , m 2 n 2 E j min { t A 1 ( m 1 ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } + m 2 = n 2 , m 1 n 1 E i min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m 2 ) } + m 1 n 1 E i , m 2 n 2 E j min { t A 1 ( m 1 ) , t A 1 ( n 1 ) , t A 2 ( m 2 ) , t A 2 ( n 2 ) } ,
( t d f ) G 1 | G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( f B i | f B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) + ( f A 1 | f A 2 ) ( m 1 , m 2 ) = m 1 = n 1 , m 2 n 2 E j max { f A 1 ( m 1 ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } + m 2 = n 2 , m 1 n 1 E i max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m 2 ) } + m 1 n 1 E i , m 2 n 2 E j max { f A 1 ( m 1 ) , f A 1 ( n 1 ) , f A 2 ( m 2 ) , f A 2 ( n 2 ) } .
Definition A6.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) be two VGSs. For any vertex ( m 1 , m 2 ) V 1 × V 2 we have:
( d t ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( t B i t B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = m 1 n 1 E i , m 2 n 2 t B i ( m 1 n 1 ) = ( d t ) G 1 ( m 1 ) ,
( d f ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( f B i f B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) = m 1 n 1 E i , m 2 n 2 f B i ( m 1 n 1 ) = ( d f ) G 1 ( m 1 ) .
Definition A7.
Let G 1 = ( A 1 , B 1 , B 2 , , B n ) and G 2 = ( A 2 , B 1 , B 2 , B n ) be two VGSs. For any vertex ( m 1 , m 2 ) V 1 × V 2 we have:
( t d t ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( t B i t B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) + ( t A 1 t A 2 ) ( m 1 , m 2 ) = m 1 n 1 E i , m 2 n 2 t B i ( m 1 n 1 ) + min { t A 1 ( m 1 ) , t A 2 ( m 2 ) } = m 1 n 1 E i , m 2 n 2 t B i ( m 1 n 1 ) + t A 1 ( m 1 ) + t A 2 ( m 2 ) min { t A 1 ( m 1 ) , t A 2 ( m 2 ) } = ( t d t ) G 1 ( m 1 ) + t A 2 ( m 2 ) min { t A 1 ( m 1 ) , t A 2 ( m 2 ) } ,
( t d f ) G 1 G 2 ( m 1 , m 2 ) = ( m 1 , m 2 ) ( n 1 , n 2 ) E i × E j ( f B i f B j ) ( ( m 1 , m 2 ) ( n 1 , n 2 ) ) + ( f A 1 f A 2 ) ( m 1 , m 2 ) = m 1 n 1 E i , m 2 n 2 f B i ( m 1 n 1 ) + max { f A 1 ( m 1 ) , f A 2 ( m 2 ) } = m 1 n 1 E i , m 2 n 2 f B i ( m 1 n 1 ) + f A 1 ( m 1 ) + f A 2 ( m 2 ) max { f A 1 ( m 1 ) , f A 2 ( m 2 ) } = ( t d f ) G 1 ( m 1 ) + f A 2 ( m 2 ) max { f A 1 ( m 1 ) , f A 2 ( m 2 ) } .
Example A7.
In this example we find the degree and the total degree of vertex ( u , z ) in Example 5.
( d t ) G 1 G 2 ( u , z ) = ( d t ) G 1 ( u ) = 0.1 + 0.1 = 0.2 , ( d f ) G 1 G 2 ( u , z ) = ( d f ) G 1 ( u ) = 0.4 + 0.5 = 0.9 .
Therefore, ( d ) G 1 G 2 ( u , z ) = ( 0.2 , 0.9 ) . In addition, total degree of vertex ( u , z ) is given by:
( t d t ) G 1 G 2 ( u , z ) = ( t d t ) G 1 ( u ) + t A 2 ( z ) min { t A 1 ( u ) , t A 2 ( z ) } = ( 0.3 + 0.3 ) min { 0.1 , 0.3 } = 0.6 0.1 = 0.5 , ( t d f ) G 1 G 2 ( u , z ) = ( t d f ) G 1 ( u ) + f A 2 ( z ) max { f A 1 ( u ) , f A 2 ( z ) } = ( 1.1 + 0.5 ) max { 0.2 , 0.5 } = 1.6 0.5 = 1.1 .
Hence, ( t d ) G 1 G 2 ( u , z ) = ( 0.5 , 1.1 ) .
Similarly, we can find the degree and the total degree of all vertices in G 1 G 2 .
Remark A5.
d ( A , B ) = 2 n i = 1 n sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | }
is a distance measure.
Proof. 
(i) 0 d ( A , B ) 1 .
As we know, A = { ( m , t A ( m ) , f A ( m ) : m X } for true membership 0 t A ( m ) 1 and false membership 0 f A ( m ) 1 . This shows for A = { ( m i , t A ( m i ) , f A ( m i ) ) : m i X } and B = { ( m i , t B ( m i ) , f B ( m i ) ) : m i X } , 0 | t A ( m i ) t B ( m i ) | 1 and 0 | f A ( m i ) f B ( m i ) | 1 . Hence, 0 sin { π 6 | t A ( m i ) t B ( m i ) | } 1 2 and 0 sin { π 6 | f A ( m i ) f B ( m i ) | } 1 2 . So,
0 sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 .
0 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 2 .
Therefore, from Equations (A1) and (A2) we have:
0 2 · sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 0 2 n i = 1 n sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 .
Hence, 0 d ( A , B ) 1 .
(ii) d ( A , B ) = 0 if and only if A = B . □
Proof. 
d ( A , B ) = 0
2 n i = 1 n sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } = 0 sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } = 0 sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } = 0 .
For, this reason
sin { π 6 | t A ( m i ) t B ( m i ) | } = 0 a n d sin { π 6 | f A ( m i ) f B ( m i ) | } = 0 | t A ( m i ) t B ( m i ) | = 0 a n d | f A ( m i ) f B ( m i ) | = 0 t A ( m i ) = t B ( m i ) a n d f A ( m i ) = f B ( m i ) , f o r a l l m i X A = B .
So, d ( A , B ) = 0 if and only if A = B .
(iii) d ( A , B ) = d ( B , A )  □
Proof. 
d ( A , B ) = 2 n i = 1 n sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } = 2 n i = 1 n sin { π 6 | t B ( m i ) t A ( m i ) | } + sin { π 6 | f B ( m i ) f A ( m i ) | } 1 + sin { π 6 | t B ( m i ) t A ( m i ) | } + sin { π 6 | f B ( m i ) f A ( m i ) | } = d ( B , A ) .
(iv) If A, B, and C are three vague sets, then the distance measure satisfies the triangular inequality. I.e., d ( A , C ) d ( A , B ) + d ( B , C ) . □
Consider A = { ( m i , t A ( m i ) , f A ( m i ) : m i X } , B = { ( m i , t B ( m i ) , f B ( m i ) : m i X } , and C = { ( m i , t C ( m i ) , f C ( m i ) : m i X } as three vague sets. Now from the basic inequality of real numbers we know that:
| t A ( m i ) t C ( m i ) | | t A ( m i ) t B ( m i ) | + | t B ( m i ) t C ( m i ) | .
So, sin { π 6 | t A ( m i ) t C ( m i ) | } sin π 6 { | t A ( m i ) t B ( m i ) | + | t B ( m i ) t C ( m i ) | } .
Hence,
sin { π 6 | t A ( m i ) t C ( m i ) | } sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } .
(Since sin ( A + B ) sin A + sin B when A , B π 2 ). In the same way, we have:
sin { π 6 | f A ( m i ) f C ( m i ) | } sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } .
Adding Equations (A3) and (A4) we get:
sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | } sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } 1 + sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } 1 1 + sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | }
1 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | }
1 1 1 + sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | }
1 1 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | }
sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | } 1 + sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | }
sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | }
sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } 1 + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } 2 n i = 1 n sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | } 1 + sin { π 6 | t A ( m i ) t C ( m i ) | } + sin { π 6 | f A ( m i ) f C ( m i ) | } 2 n i = 1 n sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } 1 + sin { π 6 | t A ( m i ) t B ( m i ) | } + sin { π 6 | f A ( m i ) f B ( m i ) | } + 2 n i = 1 n sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } 1 + sin { π 6 | t B ( m i ) t C ( m i ) | } + sin { π 6 | f B ( m i ) f C ( m i ) | } .
So, d ( A , C ) d ( A , B ) + d ( B , C ) .
Thus, it is clear that the proposed distance measure satisfies all the metric properties. References

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Figure 1. Graph structure (GS)- G and fuzzy graph structure (FGS)-G.
Figure 1. Graph structure (GS)- G and fuzzy graph structure (FGS)-G.
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Figure 2. Vague graph structure G.
Figure 2. Vague graph structure G.
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Figure 3. VGSs G 1 and G 2 .
Figure 3. VGSs G 1 and G 2 .
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Figure 4. Maximal product of G 1 and G 2 .
Figure 4. Maximal product of G 1 and G 2 .
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Figure 5. Vague graph structures (VGSs) G 1 and G 2 .
Figure 5. Vague graph structures (VGSs) G 1 and G 2 .
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Figure 6. Rejection of two VGSs.
Figure 6. Rejection of two VGSs.
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Figure 7. VGSs G 1 and G 2 .
Figure 7. VGSs G 1 and G 2 .
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Figure 8. Residue product of two VGSs.
Figure 8. Residue product of two VGSs.
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Figure 9. Irregular VGS (I-VGS) G.
Figure 9. Irregular VGS (I-VGS) G.
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Figure 10. Totally irregular VGS (TI-VGS) G.
Figure 10. Totally irregular VGS (TI-VGS) G.
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Figure 11. Strongly I-VGS G.
Figure 11. Strongly I-VGS G.
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Figure 12. Highly TI-VGS G.
Figure 12. Highly TI-VGS G.
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Figure 13. VGS G.
Figure 13. VGS G.
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Figure 14. SEI-VGS G.
Figure 14. SEI-VGS G.
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Figure 15. SETI-VGS G.
Figure 15. SETI-VGS G.
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Figure 16. (SEI-VGS) G.
Figure 16. (SEI-VGS) G.
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Figure 17. VGS G.
Figure 17. VGS G.
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Figure 18. VGSs G 1 and G 2 .
Figure 18. VGSs G 1 and G 2 .
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Figure 19. Maximal product of G 1 and G 2 .
Figure 19. Maximal product of G 1 and G 2 .
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Table 1. Symptoms–diseases vague relation.
Table 1. Symptoms–diseases vague relation.
Disease
Symptoms
Pneumonia
(Pn)
Pyelonephritis
(Py)
Appendicitis
(Ap)
MIGastroenteritis
(Ga)
Dyspnea (Dy)(0.8, 1)(0.3, 0.3)(0.1, 0.6)(0.7, 0.1)(0.2, 0.4)
Urinary symptoms (Us)(0.2, 0.5)(0.8, 0.2)(0.1, 0.3)(0.3, 0.4)(0.4, 0.3)
Abdominal pain (Ap)(0.1, 0.2)(0.2, 0.3)(0.8, 0.2)(0.2, 0.4)(0.6, 0.3)
Chest pain (Ch-p)(0.7, 0.2)(0.1, 0.5)(0.2, 0.5)(0.8, 0.3)(0.3, 0.5)
Fever (Fe)(0.1, 0.4)(0.2, 0.6)(0.3, 0.5)(0.4, 0.3)(0.8, 0.2)
Table 2. Patients–symptoms vague relation.
Table 2. Patients–symptoms vague relation.
Dyspnea
(Dy)
Urinary Symptoms
(Us)
Abdominal Pain
(Ap)
Chest Pain
(Ch-p)
Fever
(Fe)
Ted(0.2, 0.5)(0.8, 0.1)(0.3, 0.4)(0.4, 0.2)(0.1, 0.3)
John(0.2, 0.3)(0.1, 0.4)(0.3, 0.3)(0.2, 0.4)(0.8, 0)
Joseph(0.7, 0.2)(0.3, 0.2)(0.4, 0.3)(0.5, 0.1)(0.2, 0.3)
Jack(0.1, 0.2)(0.2, 0.4)(0.9, 0.1)(0.2, 0.3)(0.2, 0.5)

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Kosari S, Rao Y, Jiang H, Liu X, Wu P, Shao Z. Vague Graph Structure with Application in Medical Diagnosis. Symmetry. 2020; 12(10):1582. https://doi.org/10.3390/sym12101582

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Kosari, Saeed, Yongsheng Rao, Huiqin Jiang, Xinyue Liu, Pu Wu, and Zehui Shao. 2020. "Vague Graph Structure with Application in Medical Diagnosis" Symmetry 12, no. 10: 1582. https://doi.org/10.3390/sym12101582

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