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Original Research ARTICLE

Front. Phys., 11 November 2020 | https://doi.org/10.3389/fphy.2020.00357

Certain Concepts of Vague Graphs With Applications to Medical Diagnosis

Zehui Shao1, Saeed Kosari1*, Muhammad Shoaib2 and Hossein Rashmanlou3
  • 1Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
  • 2Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan
  • 3Mazandaran Adib Institute of Higher Education, Sari, Iran

The purpose of this research study is to present and explore the key properties of some new operations on vague graphs, including rejection, maximal product, symmetric difference, and residue product. This article introduces the notions of degree of a vertex and total degree of a vertex in a vague graph. As well, this study outlines the specific conditions required for obtaining the degrees of vertices in vague graphs under the operations of maximal product, symmetric difference, and rejection. The article also discusses applications of vague sets in medical diagnosis.

1. Introduction

Graph theory is an extremely useful tool for solving combinatorial problems in a wide range of fields, including geometry, algebra, number theory, topology, operations research, biology, and social systems. Graph theory also has many applications of great scope, such as in networking, image capture, clustering, handling uncertainty, image segmentation, finding communities in networks, bioscience, information technology, operations research, and social science networks consisting of points connected by lines. In fact, graph theory studies connections between objects, such as vertices and edges and the various relations between them. Fuzzy graph theory is finding an increasing number of applications in modeling real-time systems, where the amount of information inherent in the system varies with different levels of precision. In 1965, Zadeh [1] first proposed the theory of fuzzy sets. The fuzzy graph, with the approximate reasoning, enables many combinatorial problems in fields, such as topology and algebra to be solved more easily. The concept of fuzzy graphs is discussed by Rosenfeld [2] as well as by Bhattacharya [3, 4]. Fuzzy graphs date back to the nineteenth century, and their use has grown tremendously in recent years [5, 6]. Gau and Buehrer [7] proposed the concept of vague set in 1993, which replaces the value of an element in a set with a subinterval of [0, 1]. Specifically, a true-membership function tv(x) and a false-membership function fv(x) are used to describe the boundaries of the membership degree. Descriptions of real-world problems can be improved by using the theory of vague sets. Researchers have applied this theory to several real-world situations, such as decision-making and fuzzy control. The theory of vague sets is also helpful for fault diagnosis and knowledge discovery. Interval-valued fuzzy sets have a case vague set, which has been applied in different fields of mathematics. Ramakrishna [8] introduced the concept of vague graph and also studied related properties. Vague graphs have numerous applications in geometry and operations research and are also useful in many areas of computer science. Rashmanlou and Borzooei [9] studied new concepts relating to vague graphs, product vague graphs [10], regularity of vague graphs [11], and vague competition graphs [12]. Krishna and Lavanya [13] developed new concepts of coloring in vague graphs. Besides the membership degree, the non-membership degree has been introduced as well, which is presented by Atanassove [14] in an intuitionistic fuzzy set, a type of extension of a fuzzy set. Parvathi and Karunambigai [15] discussed intuitionistic fuzzy graphs. Devi et al. [16] presented new concepts regarding intuitionistic fuzzy labeling graphs.

In this study we outline and explore the key properties of some new operations on vague graphs, including rejection, maximal product, symmetric difference, and residue product. We introduce new notions, such as degree of a vertex and total degree of a vertex in a vague graph. We also outline specific conditions for obtaining the degrees of vertices in vague graphs under the operations of maximal product, symmetric difference, and rejection. Furthermore, we explore applications of vague sets in medical diagnosis.

2. Preliminaries

In this section we introduce the key preliminary notions and definitions that are used in this study.

Definition 2.1 ([17]). A graph is an ordered pair G = (V, E), where V is the set of vertices of G and E is the set of all edges, arcs, or lines, which are two-element subsets of V (that is, an edge is related to two vertices and the relation is represented as an unordered pair {m, n} of those vertices).

Note that for an edge {m, n}, graph theorists usually use the somewhat shorter notation mn. Two vertices m and n in an undirected graph G are said to be adjacent in G if mn is an edge of G. An edge whose endpoints are the same is called a loop. A graph without loops is called a simple graph.

Definition 2.2 ([7]). A vague set M is a pair (TM; FM) of functions on a set V, where TM and FM are real-valued V → [0, 1] functions such that TM(m) + FM(m) < 1 for all mV. The interval [TM(m), 1 − FM(m)] is known as the vague value of m in M.

In this definition, for m in M, TM(m) is the lower bound for the degree of membership and FM(m) is the lower bound for the negative of the degree of membership. Therefore, the degree of membership of mM is given by the interval [TM(m), 1 − FM(m)].

Definition 2.3 ([8]). Let G = (V, E) be a crisp graph. A pair G = (M, N) is called a vague graph defined on the crisp graph G = (V, E) if M = (TM, FM) is a vague set on V and N = (TN, FN) is vague set on EV × V such that TN(mn) ≤ min(TM(m), TM(n)) and FN(mn) ≥ max(FM(m), FM(n)) for each edge mn in E.

Definition 2.4 ([9]). A vague graph G is said to be strong if TN(mn) = min(TM(m), TM(n)) and FN(mn) = max(FM(m), FM(n)) for all m, nV.

Definition 2.5 ([9]). A vague graph G is said to be complete if TN(mn) = min(TM(m), TM(n)) and FN(mn) = max(FM(m), FM(n)) for all mnE.

Definition 2.6 ([11]). A vague graph G is said to be connected if TN(mimj)>0 and FN(mimj)<1 for all mi, mjV. Also, we have

TN(mn)=sup{TN(mn1)TN(n1n2)TN(n2n3)TN(nk1n)m,n1,n2,,nk1,nV}

and

FN(mn)=inf{FN(mn1)FN(n1n2)FN(n2n3)FN(nk1n)m,n1,n2,,nk1,nV}.

Example 2.7. Consider a vague graph G such that V = {a, b, c}, E = {ab, bc, cd, ad}, M=(a0.3,b0.4,c0.3,d0.6),(a0.6,b0.4,c0.5,a0.2), and N=(ab0.2,bc0.2,cd0.2,ad0.2),(ab0.7,bc0.6,cd0.6,ad0.7).

By routine computations, it is easy to show that G is a vague graph (Figure 1).

FIGURE 1
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Figure 1. The vague graph G in Example 2.7.

3. Operations on Vague Graphs

In this section we define four new kinds of operations on vague graphs: the maximal product, residue product, rejection, and symmetric difference. We show that the maximal product, residue product, or rejection of two vague graphs is again a vague graph.

Definition 3.1. The maximal product G1 * G2 = (M1*M2, N1*N2) of two vague graphs G1 = (M1, N1) and G2 = (M2, N2) is defined by

(i)                 (TM1*TM2)((m1,m2))=max{TM1(m1),TM2(m2)},                      (FM1*FM2)((m1,m2))=min{FM1(m1),FM2(m2)}                                                                                  (m1,m2)(V1×V2);
(ii)    (TM1*TM2)((m,m2)(m,n2))=max{TM1(m),TN2(m2n2)},          (FM1*FM2)((m,m2)(m,n2))=min{FM1(m),FN2(m2n2)}                                                                                  mV1  and m2n2E2;
(iii)     (TM1*TM2)((m1,z)(n1,z))=max{TN1(m1n1),TM2(z)},             (FM1*FM2)((m1,z)(n1,z))=min{FN1(m1n1),FM2(z)}                                                                                 zV2 and m1n1E1.

Example 3.2. Consider the two vague graphs G1 and G2 shown in Figures 2, 3. Their maximal product G1 * G2 is shown in Figure 4.

FIGURE 2
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Figure 2. G1.

FIGURE 3
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Figure 3. G2.

FIGURE 4
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Figure 4. G1 * G2.

For the vertex (a, d), we find the membership and non-membership values as follows:

(TM1*TM2)((a,d))=max{TM1(a),TM2(d)}                                          =max{0.4,0.1}=0.4,(FM1*FM2)((a,d))=min{FM1(a),FM2(d)}                                          =min{0.5,0.3}=0.3,                                                for  aV1 and dV2.

For the edge (a, d)(a, e), we find the following membership and non-membership values:

(TM1*TM2)((a,d)(a,e))=max{TM1(a),TN2(de)}                                                    =max{0.4,0.1}=0.4,(FM1*FM2)((a,d)(a,e))=min{FM1(a),FN2(de)}                                                    =min{0.5,0.6}=0.5,                                                          for  aV1 and deE2.

Now, for edge (a, g)(b, g) we have

(TM1*TM2)((a,g)(b,g))=max{TN1(ab),TM2(g)}                                                     =max{0.2,0.3}=0.3,(FM1*FM2)((a,g)(b,g))=min{FN1(ab),FM2(g)}                                                     =min{0.7,0.4}=0.4,                                                           for gV2 and abE1.

Similarly, we can find the membership and non-membership values for all the remaining vertices and edges.

Proposition 3.3. The maximal product of two vague graphs G1 and G2 is a vague graph.

Proof: Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs on crisp graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, and let ((m1, m2)(n1, n2)) ∈ E1 × E2. Then by Definition 3.1 we have two cases:

(i) If m1 = n1 = m, then

(TN1TN2)((m,m2)(m,n2))=max{TM1(m),TN2(m2n2)}  max{TM1(m),min{TM2(m2),TM2(n2)}}  =min{max{{TM1(m),TM2(m2)},max{{TM1(m),TM2(n2)}}  =min{(TM1TM2)(m,m2),(TM1TM2)(m,n2)},(FN1FN2)((m,m2)(m,n2))=min{FM1(m),FN2(m2n2)}  min{FM1(m),max{FM2(m2),FM2(n2)}}  =max{min{{FM1(m),FM2(m2)},min{{FM1(m),FM2(n2)}}  =max{(FM1FM2)(m,m2),(FM1FM2)(m,n2)}.

(ii) If m2 = n2 = z, then

(TN1TN2)((m1,z)(n1,z))=max{TN1(m1n1),TM2(z)}  max{min{TN1(m1n1),TM2(z)}  =min{max{{TN1(m1),TM2(z)},max{{TM1(n1),TM2(z)}}  =min{(TM1TM2)(m1,z),(TM1TM2)(n1,z)},(FN1FN2)((m1,z)(n1,z))=min{FN1(m1n1),FM2(z)}  min{max{FN1(m1n1),FM2(z)}  =max{min{{FM1(m1),FM2(z)},min{{FM1(n1),FM2(z)}}  =max{(FM1FM2)(m1,z),(FM1FM2)(n1,z)}.

Therefore, G1 * G2 is a vague graph.

Theorem 3.4. The maximal product of two strong vague graphs G1 and G2 is a strong vague graph.

Proof: Let G1 = (M1, N1) and G2 = (M2, N2) be two strong vague graphs on crisp graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, and let ((m1, m2)(n1, n2)) ∈ E1 × E2. Then, by Proposition 3.3, G1 * G2 is a vague graph. Now we have two cases:

(i) If m1 = n1 = m, then

(TN1TN2)((m,m2)(m,n2))=max{TM1(m),TN2(m2n2)}  =max{TM1(m),min{TM2(m2),TM2(n2)}}  =min{max{{TM1(m),TM2(m2)},max{{TM1(m),TM2(n2)}}  =min{(TM1TM2)(m,m2),(TM1TM2)(m,n2)},(FN1FN2)((m,m2)(m,n2))=min{FM1(m),FN2(m2n2)}  =min{FM1(m),max{FM2(m2),FM2(n2)}}  =max{min{{FM1(m),FM2(m2)},min{{FM1(m),FM2(n2)}}  =max{(FM1FM2)(m,m2),(FM1FM2)(m,n2)}.

(ii) If m2 = n2 = z, then

(TN1TN2)((m1,z)(n1,z))=max{TN1(m1n1),TM2(z)}  =max{min{TN1(m1n1),TM2(z)}  =min{max{{TN1(m1),TM2(z)},max{{TM1(n1),TM2(z)}}  =min{(TM1TM2)(m1,z),(TM1TM2)(n1,z)},(FN1FN2)((m1,z)(n1,z))=min{FN1(m1n1),FM2(z)}  =min{max{FN1(m1n1),FM2(z)}  =max{min{{FM1(m1),FM2(z)},min{{FM1(n1),FM2(z)}}  =max{(FM1FM2)(m1,z),(FM1FM2)(n1,z)}.

Therefore, G1 * G2 is a strong vague graph.

Example 3.5. Consider the strong vague graphs G1 and G2 as in Figure 5.

FIGURE 5
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Figure 5. Vague graphs G1, G2, and G1 * G2.

It is easy to see that G1 * G2 is a strong vague graph too.

Remark 3.1. If the maximal product of two vague graphs G1 = (M1, N1) and G2 = (M2, N2) is a strong vague graph, G1 and G2 need not be strong in general.

Example 3.6. Consider the vague graphs G1 and G2 as in Figures 6, 7. The maximal product of G1 and G2 is G1 * G2 shown in Figure 8.

FIGURE 6
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Figure 6. G1.

FIGURE 7
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Figure 7. G2.

FIGURE 8
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Figure 8. G1 * G2.

We can see that G1 and G1 * G2 are strong vague graphs, but G2 is not strong: since TN2(m2, n2) = 0.1 but min{TM2(m2), TM2(n2} = min{0.2, 0.2} = 0.2, we have TN2(m2, n2) ≠ min{TM2(m2), TM2(n2}.

Theorem 3.7. The maximal product of two connected vague graphs is a connected vague graph.

Proof: Let G1 = (M1, N1) and G2 = (M2, N2) be two connected vague graphs on crisp graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, where V1 = {m1, m2, …, mk} and V2 = {n1, n2, …, ns}. Then TN1(mimj)>0 for all mi, mjV1 and TN2(ninj)>0 for all ni, njV2 (or FN1(mimj)<1 for all mi, mjV1 and FN2(ninj)<1 for all ni, njV2). The maximal product of G1 = (M1, N1) and G2 = (M2, N2) can be taken as G = (M, N). Now, consider the k subgraphs of G with the vertex set {(mi, n1), (mi, n2), …, (mi, ns)} for i = 1, 2, …, k. Each of these subgraphs of G is connected, since the mi's are the same and G2 is connected, so that each ni is adjacent to at least one of the vertices in V2. Also, since G1 is connected, each xi is adjacent to at least one of the vertices in V1.

Hence, there exists at least one edge between any pair of the above k subgraphs. Thus, we have TN((mi,nj)(mm,nn))>0 (or FN((mi,nj)(mm,nn))<1) for all ((mi, nj)(mm, nn)) ∈ E. Therefore, G is a connected vague graph.

Remark 3.2. The maximal product of two complete vague graphs is not a complete vague graph in general. This is because we do not include the case where (m1, m2) ∈ E1 and (n1, n2) ∈ E2 in the definition of the maximal product of two vague graphs.

Remark 3.3. The maximal product of two complete vague graphs is a strong vague graph.

Example 3.8. Consider the complete vague graphs G1 and G2 in Figure 5. A simple calculation yields that G1 * G2 is a strong vague graph.

Definition 3.9. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. For any vertex (m1, m2) ∈ V1 × V2 we define

(dT)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1*TN2)((m1,m2)(n1,n2))                                           =m1=n1,m2n2E2max{TM1(m1),TN2(m2n2)}                                            +m1n1E1,m2=n2max{TN1(m1n1),TM2(m2)}, (dF)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1*FN2)((m1,m2)(n1,n2))                                            =m1=n1,m2n2E2min{FM1(m1),FN2(m2n2)}                                            +m1n1E1,m2=n2min{FN1(m1n1),FM2(m2)}.

Theorem 3.10. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. If TM1TN2, FM1FN2, TM2TN1, and FM2FN1, then (dT)G1 * G2(m1, m2) = (d)G2(m2)TM1(m1) + (d)G1(m1)TM2(m2) and (dF)G1 * G2(m1, m2) = (d)G2(m2)FM1(m1) + (d)G1(m1)FM2(m2).

Proof: From the definition of a vertex in the cartesian product, we have

(dT)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1*TN2)((m1,m2)(n1,n2))                                           =m1=n1,m2n2E2max{TM1(m1),TN2(m2n2)}                                           +m1n1E1,m2=n2max{TN1(m1n1),TM2(m2)}                                           =m2n2E2,m1=n1TN2(m2n2)+m1n1E1,m2=n2TN1(m1n1)                                           =(d)G2(m2)TM1(m1)+(d)G1(m1)TM2(m2),
(dF)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1*FN2)((m1,m2)(n1,n2))                                           =m1=n1,m2n2E2min{FM1(m1),FN2(m2n2)}                                           +m1n1E1,m2=n2min{FN1(m1n1),FM2(m2)}                                           =m2n2E2,m1=n1FN2(m2n2)+m1n1E1,m2=n2FN1(m1n1)                                           =(d)G2(m2)FM1(m1)+(d)G1(m1)FM2(m2),

as claimed.

Example 3.11. Consider the vague graphs G1, G2, and G1 * G2 as in Figure 9. Since TM1TN2, FM1FN2, TM2TN1, and FM2FN1, by Theorem 3.10 we have

(dT)G1*G2(a,c)=(d)G2(c)TM1(a)+(d)G1(a)TM2(c)=1·(0.3) +1·(0.2)=0.5,(dF)G1*G2(a,c)=(d)G2(c)FM1(a)+(d)G1(a)FM2(c)=1·(0.4) +1·(0.3)=0.7.(dT)G1*G2(a,d)=(d)G2(d)TM1(a)+(d)G1(a)TM2(d)=1·(0.3) +1·(0.3)=0.6,(dF)G1*G2(a,d)=(d)G2(d)FM1(a)+(d)G1(a)FM2(d)=1·(0.4) +1·(0.4)=0.8.(dT)G1*G2(b,c)=(d)G2(c)TM1(b)+(d)G1(b)TM2(c)=1·(0.2) +1·(0.2)=0.4,(dF)G1*G2(b,c)=(d)G2(c)FM1(b)+(d)G1(b)FM2(c)=1·(0.3) +1·(0.3)=0.6.(dT)G1*G2(b,d)=(d)G2(d)TM1(b)+(d)G1(b)TM2(d)=1·(0.2) +1·(0.3)=0.5,(dF)G1*G2(b,d)=(d)G2(d)FM1(b)+(d)G1(b)FM2(d)=1·(0.3) +1·(0.4)=0.7.
FIGURE 9
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Figure 9. Vague graphs G1, G2, and G1 * G2.

By direct calculations we obtain

(dT)G1*G2(a,c)=0.3+0.2=0.5,(dF)G1*G2(a,c)=0.4+0.3=0.7,(dT)G1*G2(a,d)=0.3+0.3=0.6,(dF)G1*G2(a,d)=0.4+0.4=0.8,(dT)G1*G2(b,c)=0.2+0.2=0.4,(dF)G1*G2(b,c)=0.3+0.3=0.6,(dT)G1*G2(b,d)=0.3+0.2=0.5,(dF)G1*G2(b,d)=0.3+0.4=0.7.

It is clear that the degrees of vertices calculated using the formula in Theorem 3.10 and by the direct method are the same.

Definition 3.12. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. For any vertex (m1, m2) ∈ V1 × V2 we define

(tdT)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1*TN2)((m1,m2)(n1,n2))                                              +(TM1*TM2)(m1,m2)                                             =m1=n1,m2n2E2max{TM1(m1),TN2(m2n2)}                                             +m1n1E1,m2=n2max{TN1(m1n1),TM2(m2)}                                                  +max{TM1(m1),TM2(m2)},(tdF)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1*FN2)((m1,m2)(n1,n2))                                              +(FM1*FM2)(m1,m2)                                             =m1=n1,m2n2E2min{FM1(m1),FN2(m2n2)}                                             +m1n1E1,m2=n2min{FN1(m1n1),FM2(m2)}                                                 +min{FM1(m1),FM2(m2)}.

Example 3.13. In this example we find the degree and the total degree of vertices (a, c) and (a, d) in Example 3.2:

(dT)G1*G2(a,c)=(d)G2(a)TM1(c)+(d)G1(c)TM2(a)                                   =1(0.2)+4(0.4)=0.2+1.6=1.8,(dF)G1*G2(a,c)=(d)G2(a)FM1(c)+(d)G1(c)FM2(a)                                   =1(0.5)+4(0.5)=0.3+1.2=1.5.

Therefore, dG1 * G2(a, c) = (1.8, 1.5). In addition, by the definition of the total vertex degree in the maximal product,

(tdT)G1*G2(a,c)=(d)G2(a)TM1(c)+(d)G1(c)TM2(a)                                     +max{TM1(a),TM2(c)}                                     =1(0.2)+4(0.4)+max(0.2,0.4)=2.2,(tdF)G1*G2(a,c)=(d)G2(a)FM1(c)+(d)G1(c)FM2(a)                                     +min{FM1(a),FM2(c)}                                    =1(0.5)+4(0.5)+min(0.3,0.4)=1.8.

Therefore, tdG1 * G2(a, c) = (2.2, 1.8).

We also have

(dT)G1*G2(a,d)=(d)G2(a)TM1(d)+(d)G1(d)TM2(a)                                   =1(0.1)+4(0.4)=0.1+1.6=1.7,(dF)G1*G2(a,d)=(d)G2(a)FM1(d)+(d)G1(d)FM2(a)                                    =1(0.3)+4(0.5)=0.3+2=2.3,
(tdT)G1*G2(a,d)=(d)G2(a)TM1(d)+(d)G1(d)TM2(a)                                     +max{TM1(a),TM2(d)}                                     =1(0.1)+4(0.4)+max(0.4,0.1)=2.1,(tdF)G1*G2(a,d)=(d)G2(a)FM1(d)+(d)G1(d)FM2(a)                                     +min{FM1(a),FM2(d)}                                     =1(0.3)+4(0.5)+min(0.5,0.3)=2.6.

Hence, dG1 * G2(a, d) = (1.7, 2.3) and tdG1 * G2(a, d) = (2.1, 2.6).

Similarly, we can find the degree and the total degree of all vertices in G1 * G2.

Theorem 3.14. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. If TM1TN2, FM1FN2, TM2TN1, and FM2FN1, then (tdT)G1 * G2(m1, m2) = (d)G2(m2)TM1(m1) + (d)G1(m1)TM2(m2) + max{TM1(m1), TM2(m2)} and (tdF)G1 * G2(m1, m2) = (d)G2(m2)FM1(m1) + (d)G1(m1)FM2(m2) + min{FM1(m1), FM2(m2)}.

Proof: From Definition 3.12 we have

(tdT)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1*TN2)((m1,m2)(n1,n2))                                             +(TM1*TM2)(m1,m2)                                             =m1=n1,m2n2E2max{TM1(m1),TN2(m2n2)}                                             +m1n1E1,m2=n2max{TN1(m1n1),TM2(m2)}                                                  +max{TM1(m1),TM2(m2)}                                             =m2n2E2,m1=n1TN2(m2n2)+m1n1E1,m2=n2TN1(m1n1)                                                  +max{TM1(m1),TM2(m2)}                                             =(d)G2(m2)TM1(m1)+(d)G1(m1)TM2(m2)                                             +max{TM1(m1),TM2(m2)}

and

(tdF)G1*G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1*FN2)((m1,m2)(n1,n2))                                             +(FM1*FM2)(m1,m2)                                             =m1=n1,m2n2E2min{FM1(m1),FN2(m2n2)}                                             +m1n1E1,m2=n2min{FN1(m1n1),FM2(m2)}                                             +min{FM1(m1),FM2(m2)}                                             =m2n2E2,m1=n1FN2(m2n2)+m1n1E1,m2=n2FN1(m1n1)                                                  +min{FM1(m1),FM2(m2)}                                             =(d)G2(m2)FM1(m1)+(d)G1(m1)FM2(m2)                                                 +min{FM1(m1),FM2(m2)},

as asserted.

Example 3.15. Consider the vague graphs G1, G2, and G1 * G2 in Figure 9. The total degree of the vertex in the maximal product is calculated by the following formula:

(tdT)G1*G2(m1,m2)=(d)G2(m2)TM1(m1)+(d)G1(m1)TM2(m2)                                                  +max{TM1(m1),TM2(m2)},(tdF)G1*G2(m1,m2)=(d)G2(m2)FM1(m1)+(d)G1(m1)FM2(m2)                                                  +min{FM1(m1),FM2(m2)}.

Using the formula we find that

(tdT)G1*G2(a,c)=(d)G2(c)TM1(a)+(d)G1(a)TM2(c)                                     +max{TM1(a),TM2(c)}                                     =1·(0.3)+1·(0.2)+max{0.2,0.3}                                     =0.3+0.2+0.3=0.8,(tdF)G1*G2(a,c)=(d)G2(c)FM1(a)+(d)G1(a)FM2(c)                                     +min{FM1(a),FM2(c)}                                     =1·(0.4)+1·(0.3)+min{0.3,0.4}                                     =0.4+0.3+0.3=1.(tdT)G1*G2(a,d)=(d)G2(d)TM1(a)+(d)G1(a)TM2(d)                                     +max{TM1(a),TM2(d)}                                     =1·(0.3)+1·(0.3)+max{0.3,0.3}                                     =0.3+0.3+0.3=0.9,(tdF)G1*G2(a,d)=(d)G2(d)FM1(a)+(d)G1(a)FM2(d)                                     +min{FM1(a),FM2(d)}                                     =1·(0.4)+1·(0.4)+min{0.4,0.4}                                     =0.4+0.4+0.4=1.2.(tdT)G1*G2(b,c)=(d)G2(c)TM1(b)+(d)G1(b)TM2(c)                                     +max{TM1(b),TM2(c)}                                     =1·(0.2)+1·(0.2)+max{0.2,0.2}                                     =0.2+0.2+0.2=0.6,(tdF)G1*G2(b,c)=(d)G2(c)FM1(b)+(d)G1(b)FM2(c)                                     +min{FM1(b),FM2(c)}                                     =1·(0.3)+1·(0.3)+min{0.3,0.3}                                     =0.3+0.3+0.3=0.9.(tdT)G1*G2(b,d)=(d)G2(d)TM1(b)+(d)G1(b)TM2(d)                                     +max{TM1(b),TM2(d)}                                     =1·(0.2)+1·(0.3)+max{0.2,0.3}                                     =0.2+0.3+0.3=0.8,(tdF)G1*G2(b,d)=(d)G2(d)FM1(b)+(d)G1(b)FM2(d)                                     +min{FM1(b),FM2(d)}                                     =1·(0.3)+1·(0.4)+min{0.3,0.4}                                     =0.3+0.4+0.3=1.

On the other hand, by direct calculations we obtain

(tdT)G1*G2(a,c)=0.3+0.2+0.3=0.8,(tdF)G1*G2(a,c)=0.4+0.3+0.3=1,(tdT)G1*G2(a,d)=0.3+0.3+0.3=0.9,(tdF)G1*G2(a,d)=0.4+0.4+0.4=1.2,(tdT)G1*G2(b,c)=0.2+0.2+0.2=0.6,(tdF)G1*G2(b,c)=0.3+0.3+0.3=0.9,(tdT)G1*G2(b,d)=0.3+0.2+0.3=0.8,(tdF)G1*G2(b,d)=0.4+0.3+0.3=1.

It is thus clear that the total degrees of vertices calculated using the formula and by the direct method are the same.

Definition 3.16. The rejection G1|G2 = (M1|M2, N1|N2) of two vague graphs G1 = (M1, N1) and G2 = (M2, N2) is defined as follows:

(i)                 (TM1|TM2)((m1,m2))=min{TM1(m1),TM2(m2)},                      (FM1|FM2)((m1,m2))=max{FM1(m1),FM2(m2)}                                                                               (m1,m2)(V1×V2);
(ii)     (TN1|TN2)((m,m2)(m,n2))=min{TM1(m),TM2(m2),TM2(n2)},           (FN1|FN2)((m,m2)(m,n2))=max{FM1(m),FM2(m2),FM2(n2)}                                                                              mV2 and m2n2E2;
(iii)   (TN1|TN2)((m,m2)(m,n2))=min{TM1(m),TM2(m2),TM2(n2)}          (FN1|FN2)((m,m2)(m,n2))=max{FM1(m),FM2(m2),FM2(n2)}                                                                              mV2 and m1n1E1;
(iv)(TN1|TN2)((m1,m2)(n1,n2))=min{TM1(m1),TM1(n1),TM2(m2),TM2(n2)},       (FN1|FN2)((m1,m2)(n1,n2))=max{FM1(m1),FM1(n1),FM2(m2),FN2(n2)}                                                                              m1n1E1 and m2n2E2.

Example 3.17. Consider the vague graphs G1 and G2 in Figures 10, 11. The rejection of G1 and G2, i.e., G1|G2, is shown in Figure 12.

FIGURE 10
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Figure 10. G1.

FIGURE 11
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Figure 11. G2.

FIGURE 12
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Figure 12. G1|G2.

For the vertex (a, e), we find the membership and non-membership values as follows:

(TM1|TM2)((a,e))=min{TM1(a),TM2(e)}                                         =min{0.4,0.3}=0.3,(FM1|FM2)((a,e))=max{FM1(a),FM2(e)}                                         =max{0.4,0.4}=0.4

for aV1 and eV2.

For the edge (e, c)(e, a), the membership and non-membership values are given by

(TN1|TN2)((e,c)(e,a))=min{TM1(e),TM2(c),TM2(a)}                                                 =min{0.3,0.3,0.4}=0.3,(FN1|FN2)((e,c)(e,a))=max{FM1(e),FM2(c),FM2(a)}                                                 =max{0.4,0.4,0.4}=0.4

for eV2 and acE1.

For the edge (e, c)(e, g) we have

(TN1|TN2)((e,c)(e,g))=min{TM1(e),TM2(c),TM2(g)}                                                 =min{0.3,0.3,0.3}=0.3,(FN1|FN2)((e,c)(e,g))=max{FM1(e),FM2(c),FM2(g)}                                                  =max{0.4,0.4,0.2}=0.4

for eV2 and cgE2.

Similarly, we can find the membership and non-membership values for all the remaining vertices and edges.

Proposition 3.18. The rejection of two vague graphs G1 and G2 is a vague graph.

Proof: Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs on crisp graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, and let ((m1, m2)(n1, n2)) ∈ E1 × E2. Then by Definition 3.16 we have the following:

(i) If m1 = n1 and m2n2E2, then

(TN1|TN2)((m1,m2)(n1,n2))=min{TM1(m1),TM2(m2),TM2(n2)}=min{min{TM1(m1),TM2(m2)},min{TM1(n1),TM2(n2)}}=min{(TM1|TM2)(m1,m2),(TM1|TM2)(n1,n2)},(FN1|FN2)((m1,m2)(n1,n2))=max{FM1(m1),FM2(m2),FM2(n2)}=max{max{FM1(m1),FM2(m2)},max{FM1(n1),FM2(n2)}}=max{(FM1|FM2)(m1,m2),(FM1|FM2)(n1,n2)}.

(ii) If m2 = n2 and m1n1E1, then

(TN1|TN2)((m1,m2)(n1,n2))=min{TM1(m1),TM1(n1),TM2(m2)}=min{min{TM1(m1),TM2(m2)},min{TM1(n1),TM2(n2)}}=min{(TM1|TM2)(m1,m2),(TM1|TM2)(n1,n2)},(FN1|FN2)((m1,m2)(n1,n2))=max{FM1(m1),FM1(n1),FM2(m2)}=max{max{FM1(m1),FM2(m2)},max{FM1(n1),FM2(n2)}}=max{(FM1|FM2)(m1,m2),(FM1|FM2)(n1,n2)}.

(iii) If m1n1E1 and m2n2E2, then

(TN1|TN2)((m1,m2)(n1,n2))=min{TM1(m1),TM1(n1),TM2(m2),TM2(n2)}=min{min{TM1(m1),TM2(m2)},min{TM1(n1),TM2(n2)}}=min{(TM1|TM2)(m1,m2),(TM1|TM2)(n1,n2)},(FN1|FN2)((m1,m2)(n1,n2))=max{FM1(m1),FM1(n1),FM2(m2),TM2(n2)}=max{max{FM1(m1),FM2(m2)},max{FM1(n1),FM2(n2)}}=max{(FM1|FM2)(m1,m2),(FM1|FM2)(n1,n2)}.

Therefore, G1|G2 = (M1|M2, N1|N2) is a vague graph.

Remark 3.4. The rejection of two complete vague graphs G1 = (M1, N1) and G2 = (M2, N2) is a complete vague graph.

Definition 3.19. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. For any vertex (m1, m2) ∈ V1 × V2 we define

(dT)G1|G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1|TN2)((m1,m2)(n1,n2))      =m1=n1,m2n2E2min{TM1(m1),TM2(m2),TM2(n2)}      +m2=n2,m1n1E1min{TM1(m1),TM1(n1),TM2(m2)}      +m1n1E1 and m2n2E2min{TM1(m1),TM1(n1),TM2(m2),TM2(n2)},
(dF)G1|G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1|FN2)((m1,m2)(n1,n2))=m1=n1,m2n2E2max{FM1(m1),FM2(m2),FM2(n2)}     +m2=n2,m1n1E1max{FM1(m1),FM1(n1),FM2(m2)}     +m1n1E1 and m2n2E2max{FM1(m1),FM1(n1),FM2(m2),FM2(n2)}.

Definition 3.20. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. For any vertex (m1, m2) ∈ V1 × V2 we define

(tdT)G1|G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1|TN2)((m1,m2)(n1,n2)) +(TM1|TM2)(m1,m2) =m1=n1,m2n2E2min{TM1(m1),TM2(m2),TM2(n2)} +m2=n2,m1n1E1min{TM1(m1),TM1(n1),TM2(m2)}      +m1n1E1 and m2n2E2min{TM1(m1),TM1(n1),TM2(m2),TM2(n2)},
(tdF)G1|G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1|FN2)((m1,m2)(n1,n2))+(FM1|FM2)(m1,m2)=m1=n1,m2n2E2max{FM1(m1),FM2(m2),FM2(n2)}     +m2=n2,m1n1E1max{FM1(m1),FM1(n1),FM2(m2)}+m1n1E1 and m2n2E2max{FM1(m1),FM1(n1),FM2(m2),FM2(n2)}.

Example 3.21. In this example we find the degree and total degree of the vertex (e, a) in Example 3.17:

(dT)G1|G2(e,a)=min{TM2(e),TM1(a),TM1(c)}                                  +min{TM2(e),TM1(a),TM2(g)}                                  =min{0.3,0.4,0.3}+min{0.3,0.4,0.3}                                  =0.3+0.3                                  =0.6,(dF)G1|G2(e,a)=max{FM2(e),FM1(a),FM1(c)}                                   +max{FM2(e),FM1(a),FM2(g)                                   =max{0.4,0.4,0.4}+max{0.4,0.4,0.2}                                   =0.4+0.4                                   =0.8.

Therefore, dG1|G2(a, c)= (0.6,0.8).

In addition, by the definition of the total vertex degree in the maximal product,

(tdT)G1|G2(e,a)=min{TM2(e),TM1(a),TM1(c)}                                    +min{TM2(e),TM1(a),TM2(g)}                                        +min{TM2(e),TM1(a)}                                    =min{0.3,0.4,0.3}+min{0.3,0.4,0.3}                                    +min{0.3,0.4}                                    =0.3+0.3+0.3                                    =0.9,(tdF)G1|G2(e,a)=max{FM2(e),FM1(a),FM1(c)}                                    +max{FM2(e),FM1(a),FM2(g)}                                        +max{FM2(e),FM1(a)}                                    =max{0.4,0.4,0.4}+max{0.4,0.4,0.2}                                    +max{0.3,0.4}                                    =0.4+0.4+0.4                                    =1.2.

Therefore, tdG1|G2(a, c)= (0.9,1.2).

Similarly, we can find the degree and the total degree of all vertices in G1|G2.

Definition 3.22. The symmetric difference G1G2 = (M1M2, N1N2) of two vague graphs G1 = (M1, N1) and G2 = (M2, N2) is defined as follows:

(i)                  (TM1TM2)((m1,m2))=min{TM1(m1),TM2(m2)},                       (FM1FM2)((m1,m2))=max{FM1(m1),FM2(m2)}                                                                           (m1,m2)(V1×V2);
(ii)      (TN1TN2)((m,m2)(m,n2))=min{TM1(m),TN2(m2n2)},            (FN1FN2)((m,m2)(m,n2))=max{FM1(m),FN2(m2n2)}                                                                             mV1 and m2n2E2;
(iii)       (TN1TN2)((m1,z)(n1,z))=min{TN1(m1n1),TM2(z)},              (FN1FN2)((m1,z)(n1,z))=max{FN1(m1n1),FM2(z)}                                                                            zV2 and m1n1E1;
(iv)(TN1TN2)((m1,m2)(n1,n2))={min{TM1(m1),TM1(n1),TN2(m2n2)} m1n1E1 and m2n2E2,min{TM2(m2),TM2(n2),TN1(m1n1)} m1n1E1 and m2n2E2,       (FN1FN2)((m1,m2)(n1,n2))={max{FM1(m1),FM1(n1),FN2(m2n2)} m1n1E1 and m2n2E2,max{FM2(m2),FM2(n2),FN1(m1n1)} m1n1E1 and m2n2E2.

Example 3.23. Consider the vague graphs G1 and G2 as in Figures 13, 14. The symmetric difference of G1 and G2, i.e., G1G2, is shown in Figure 15.

FIGURE 13
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Figure 13. G1.

FIGURE 14
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Figure 14. G2.

FIGURE 15
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Figure 15. G1G2.

For the vertex (a, f), we find the membership and non-membership values as follows:

(TM1TM2)((a,f))=min{TM1(a),TM2(f)}                                             =min{0.2,0.3}=0.2,(FM1FM2)((a,f))=max{FM1(a),FM2(f)}                                             =max{0.4,0.3}=0.4

for aV1 and fV2.

For the edge (a, d)(a, e), the membership and non-membership values are given by

(TN1TN2)((a,d)(a,e))=min{TM1(a),TN2(de)}                                                      =min{0.2,0.1}=0.1,(FN1FN2)((a,d)(a,e))=max{FM1(a),FN2(de)}                                                      =max{0.4,0.6}=0.6

for aV1 and deE2.

For the edge (a, d)(b, d) we have

(TN1TN2)((a,d)(b,d))=min{TN1(ab),TM2(d)}                                                      =min{0.2,0.2}=0.2,(FN1FN2)((a,d)(b,d))=max{FN1(ab),FM2(d)}                                                       =max{0.7,0.4}=0.7

for abE1 and dV2.

For the edge (a, c)(b, f), the membership and non-membership values are

(TN1TN2)((a,c)(b,f))=min{TM2(c),TM2(f),TN1(ab)}                                                      =min{0.4,0.3,0.2}=0.2,(FN1FN2)((a,c)(b,f))=max{FM2(c),FM2(f),FN1(ab)}                                                      =max{0.5,0.3,0.7}=0.7

for abE1 and cfE2.

In the same way, we can find the membership and non-membership values for all remaining vertices and edges.

Proposition 3.24. The symmetric difference of two vague graphs G1 and G2 is a vague graph.

Proof: Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs on crisp graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, and let ((m1, m2)(n1, n2)) ∈ E1 × E2. Then by Definition 3.22 we have the following cases:

(i) If m1 = n1 = m, then

(TN1TN2)((m,m2)(m,n2))=min{TM1(m),TN2(m2n2)} min{TM1(m),min{TM2(m2),TM2(n2)}} =min{min{TM1(m),TM2(m2)},min{TM1(m),TM2(n2)}} =min{(TM1TM2)(m,m2),(TM1TM2)(m,n2)},(FN1FN2)((m,m2)(m,n2))=max{FM1(m),FN2(m2n2)} max{FM1(m),max{FM2(m2),FM2(n2)}} =max{max{FM1(m),FM2(m2)},max{FM1(m),FM2(n2)}} =max{(FM1FM2)(m,m2),(FM1FM2)(m,n2)}.

(ii) If m2 = n2 = z, then

(TN1TN2)((m1,z)(n1,z))=min{TN1(m1n1),TM2(z)} min{min{TN1(m1n1),TM2(z)} =min{min{TM1(m1),TM2(z)},min{TM1(n1),TM2(z)}} =min{(TM1TM2)(m1,z),(TM1TM2)(n1,z)},(FN1FN2)((m1,z)(n1,z))=max{FN1(m1n1),FM2(z)}  max{max{FN1(m1n1),FM2(z)} =max{max{FM1(m1),FM2(z)},max{FM1(n1),FM2(z)}} =max{(FM1FM2)(m1,z),(FM1FM2)(n1,z)}.

(iii) If m1n1E1 and m2n2E2, then

(TN1TN2)((m1,m2)(n1,n2))=min{TM1(m1),TM1(n1),TN2(m2n2)} min{TM1(m1),TM1(n1),min{TM2(m2)TM2(n2)}} =min{min{TM1(m1),TM2(m2)},{TM1(m1),TM2(n2)}} =min{(TM1TM2)(m1,m2),(TM1TM2)(n1,n2)},(FN1FN2)((m1,m2)(n1,n2))=max{FM1(m1),FM1(n1),FN2(m2n2)} max{FM1(m1),FM1(n1),max{FM2(m2)FM2(n2)}} =max{max{FM1(m1),FM2(m2)},{FM1(m1),FM2(n2)}} =max{(FM1FM2)(m1,m2),(FM1FM2)(n1,n2)}.

(iv) If m1n1E1 and m2n2E2, then

(TN1TN2)((m1,m2)(n1,n2))=min{TM2(m2),TM2(n2),TN1(m1n1)} min{TM2(m2),TM2(n2),min{TM1(m1)TM1(n1)}} =min{min{TM1(m1),TM2(m2)},{TM1(n1),TM2(n2)} =min{(TM1TM2)(m1,m2),(TM1TM2)(n1,n2)},(FN1FN2)((m1,m2)(n1,n2))=max{FM2(m2),FM2(n2),FN1(m1n1)} max{FM2(m2),FM2(n2),max{FM1(m1)FM1(n1)}} =max{max{FM2(m2),FM1(m1)},{FM2(m2),FM1(n1)} =max{(FM1FM2)(m1,m2),(FM1FM2)(n1,n2)}.

Hence G1G2 is a vague graph.

Remark 3.5. The symmetric difference of two connected vague graphs G1 = (M1, N1) and G2 = (M2, N2) is connected, because we include the case where (m1, m2) ∈ E1 and (n1, n2) ∈ E2 in the definition of the symmetric difference of two vague graphs.

Definition 3.25. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. For any vertex (m1, m2) ∈ V1 × V2 we define

(dT)G1G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1TN2)((m1,m2)(n1,n2))                                              =m1=n1,m2n2E2min{TM1(m1),TN2(m2n2)}                                                    +m1n1E1,m2=n2min{TN1(m1n1),TM2(m2)}                                              +m1n1E1 and m2n2E2min{TM1(m1),TM1(n1),TN2(m2n2)}                                              +m1n1E1 and m2n2E2min{TN1(m1n1),TM2(m2),TM2(n2)}

and

(dF)G1G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1FN2)((m1,m2)(n1,n2))                                              =m1=n1,m2n2E2max{FM1(m1),FN2(m2n2)}                                                    +m1n1E1,m2=n2max{FN1(m1n1),FM2(m2)}                                              +m1n1E1 and m2n2E2max{FM1(m1),FM1(n1),FN2(m2n2)}                                              +m1n1E1 and m2n2E2max{FN1(m1n1),FM2(m2),FM2(n2)}.

Theorem 3.26. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. If TM1TN2, FM1FN2, TM2TN1, and FM2FN1, then for every (m1, m2) ∈ V1 × V2 we have (d)G1G2(m1, m2) = q(d)G1(m1) + s(d)G2(m2), where s = |V1| − (d)G1(m1) and q = |V2| − (d)G2(m2).

Proof: Using Definition 3.25,

(dT)G1G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1TN2)((m1,m2)(n1,n2))                                              =m1=n1,m2n2E2min{TM1(m1),TN2(m2n2)}                                                  +m1n1E1,m2=n2min{TN1(m1n1),TM2(m2)}                                              +​​​​​​​m1n1E1 and m2n2E2​​​​​​min{TM1(m1),TM1(n1),TN2(m2n2)}                                              +​​​​​​​m1n1E1 and m2n2E2​​​​​​min{TN1(m1n1),TM2(m2),TM2(n2)}                                              =m2n2E2TN2(m2n2)+m1n1E1TN1(m1n1)                                              +​​​​​​​​​m1n1E1 and m2n2E2​​​​​​​​​​TN2(m2n2)}+​​​​​​​​​​​m1n1E1 and m2n2E2​​​​​​​​​​​​TN1(m1n1)                                             =q(dT)G1(m1)+s(dT)G2(m2),
(dF)G1G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1FN2)((m1,m2)(n1,n2))                                              =m1=n1,m2n2E2max{FM1(m1),FN2(m2n2)}                                              +m1n1E1,m2=n2max{FN1(m1n1),FM2(m2)}                                              +m1n1E1 and m2n2E2max{FM1(m1),FM1(n1),FN2(m2n2)}                                              +m1n1E1 and m2n2E2max{FN1(m1n1),FM2(m2),FM2(n2)}                                              =m2n2E2FN2(m2n2)+m1n1E1FN1(m1n1)                                              +m1n1E1 and m2n2E2FN2(m2n2)}+m1n1E1 and m2n2E2FN1(m1n1)                                              =q(dF)G1(m1)+s(dF)G2(m2),

and hence the result is proved.

Example 3.27. Consider the two vague graphs G1 and G2 in Figure 9 and their symmetric difference in Figure 16. In Figure 9, TM1TN2, FM1FN2, TM2TN1, and FM2FN1. Then, the total degree of a vertex in the symmetric difference is calculated by the following formula:

(dT)G1G2(m1,m2)=q(dT)G1(m1)+s(dT)G2(m2),(dF)G1G2(m1,m2)=q(dF)G1(m1)+s(dF)G2(m2).
FIGURE 16
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Figure 16. G1G2 of the vague graphs in Figure 9.

Using the formula we find that

(dT)G1G2(a,c)=1·(0.2)+1·(0.2)=0.4,(dF)G1G2(a,c)=1·(0.4)+1·(0.5)=0.9,(dT)G1G2(a,d)=1·(0.2)+1·(0.2)=0.4,(dF)G1G2(a,d)=1·(0.4)+1·(0.5)=0.9.

Hence, (d)G1G2(a, c) = (0.4, 0.9) and (d)G1G2(a, d) = (0.4, 0.9).

In the same way, we can show that (d)G1G2(b, c) = (d)G1G2(b, d) = (0.4, 0.9). Direct calculations give

(dT)G1G2(a,c)=0.2+0.2=0.4,(dF)G1G2(a,c)=0.4+0.5=0.9,(dT)G1G2(a,d)=0.2+0.2=0.4,(dF)G1G2(a,d)=0.4+0.5=0.9,(dT)G1G2(b,c)=0.2+0.2=0.4,(dF)G1G2(b,c)=0.4+0.5=0.9,(dT)G1G2(b,d)=0.2+0.2=0.4,(dF)G1G2(b,d)=0.4+0.5=0.9.

It is obvious from the above that the degrees of vertices calculated using the formula and by the direct method are the same.

Definition 3.28. Let G1 = (M1, N1) and G2 = (M2, N2) be two vague graphs. For any vertex (m1, m2) ∈ V1 × V2 we define

(tdT)G1G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(TN1TN2)((m1,m2)(n1,n2))+(TM1TM2(m1,m2)=m1=n1,m2n2E2min{TM1(m1),TN2(m2n2)}     +m1n1E1,m2=n2min{TN1(m1n1),TM2(m2)}     +m1n1E1 and m2n2E2min{TM1(m1),TM1(n1),TN2(m2n2)}     +m1n1E1 and m2n2E2min{TN1(m1n1),TM2(m2),TM2(n2)}     +min{TM1(m1),TM2(m2)}

and

(tdF)G1G2(m1,m2)=(m1,m2)(n1,n2)E1×E2(FN1FN2)((m1,m2)(n1,n2))+(FM1FM2(m1,m2)     =m1=n1,m2n2E