A Study of Complex Dombi Fuzzy Graph With Application in Decision Making Problems

A complex fuzzy set (CFS) is a generalization of a fuzzy set (FS) in which a limit of degrees occurs on the complex plane with unit disc. The averaging operators are a key part of turning all the data into one value. Dombi operators have exceptional flexibility with operating factors, and they are particularly efficient in decision-making problems. In this paper, we establish a complex dombi fuzzy graph (CDFG). We implement dombi operators on CFSs to extend graph nomenclature. We define the complement of CDFG with an example. The idea of self-complementary in CDFG is discussed. The concepts of homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism of two CDFGs are discussed. We define regular and entirely regular graphs with sufficient elaboration and examine their key properties. Furthermore, significant characteristics are used to explain the edge regularity of CDFG. Lastly, we establish an application of CDFG in decision making problems.

certain application, one must evaluate their features, model 29 applicability, simplicity, software and hardware implementa-30 tion, etc. As the study of these operators has grown, more 31 alternatives for selecting T-operators have emerged. 32 Graphs have many applications in the field of operational 33 research and computer science. A graph is a visual repre-34 sentation of links between several items that is useful for 35 elaborating on information. However, haziness turns a graph 36 into a fuzzy graph. Fuzzy graphs are intended to portray as 37 a matter of degree structures of connections (in the form of 38 edges) between tangible objects (nodes). Fuzzy graphs have 39 a wide variety of applications, including decision-making, 40 database theory, cluster analysis, and network optimization. 41 Kaufman [11] firstly proposed the concept of fuzzy graph. 42 After that, Rosenfeld [14] studied fuzzy relations on fuzzy 43 sets and used max and min operations to construct the 44 structure of fuzzy graphs, resulting in analogues of numer-45 ous graph-theoretical ideas. Bhattacharya [4] made some 46 comments on fuzzy graphs. Reference [16]  By putting λ = 1 in dombi's t-norm, we obtain one more 118 T-operator that is T (x, y) = xy x+y−xy .

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Definition 8: A CDFG on a universe ψ is an ordered pair 148 τ = (C, F), where C = (g, ς C e iϑ C ) : ψ → {z : z ∈ C, |Z | ≤ 149 1} is a CFS subset in ψ and F = (gh, ς F e iϑ F ) : ψ × ψ → 150 {z : z ∈ C, |Z | ≤ 1} is a complex fuzzy relation (CFR) on C 151 such that for amplitude term and for phase term We call C and F the complex fuzzy node set and complex fuzzy 157 edge set, respectively.
By calculations, one can see that τ = (C, F) is a CDFG.

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The degree of a node g ∈ ψ for a phase term is expressed .

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• The total degree of node g ∈ ψ for amplitude term is The total degree of a node g ∈ ψ for a phase term is 182 Example 2: From above example, we have 186 • The degree of nodes in τ are as follows:

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• The total degree of nodes in τ are as follows: (1.14) , Definition 10: Let τ = (C, F) be a CDFG on a graph 197 τ * = (ψ, A). The complement of τ for amplitude term is 198 determined by:

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Similarly the complement of τ for phase term is determined 202 by: Further, the complement of a CDFG τ is denoted byτ = 206 (C,F).
for all gh ∈ A. ),

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Proof: Suppose that τ is a self complementary CDFG, 237 then there occurs an isomorphism Z : ψ → ψ such that By using definition of complement, we have Similarly, the phase term can be proved.

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Proof: Consider τ is CDFG that satisfies for all g,h∈ψ, then the identity mapping I : ψ → ψ 264 is an isomorphism from τ toτ that satisfies the following 265 conditions: Since the membership value of an edge set gh is given by ), for all g, h ∈ ψ. 269 We have

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Similarly the phase term condition of isomorphism Proof: Suppose that 281 τ and G are two isomorphic CDFGs. Then by definition of 282 isomorphism, there occur a bijective mapping Z : ψ → ψ 283 that satisfies

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By using definition of complement, the membership of an edge 288 gh is Similarly for the phase term, . 298 We conclude that the complement of τ is isomorphic to the 299 complement of G . Similarly, we can prove its converse part.
For the membership value of an edge, we have Similarly for the phase term . 327 Hence we conclude that complement of τ is weak isomorphic 328 to the complement of G .

Definition 16: A CDFG is said to be complete if
for all g,h∈ ψ.

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Definition 17: A CDFG is said to be strong if for all gh∈A.

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Definition 18: Let τ = (C, F) be a strong CDFG on a 338 graph τ * = (ψ, A). The complement of τ for amplitude term 339 is represented by: Similarly the complement of τ for phase term is represented 343 by: Further, the complement of a strong CDFG τ is expressed 347 byτ = (C,F).
Example 3: Let τ = (C, F) be a CDFG on τ * = (ψ, A) 356 as shown in Figure 2, where ψ = {g, h, r, s} and A = 357 {gh, qr, ps, rs}. The set of nodes C and set of edges F of τ 358 are defined on ψ and A, respectively.
totally regular if its each node has same total degree, i.e, for all g ∈ ψ.   Proof: 1. Suppose that τ is isomorphic to G and τ is 387 R 1 e iR * 1 -regular CDFG, therefore degree of each node of τ is 388 given by Thus, G is a R 1 e iR * 1 -regular CDFG. 406 2. Suppose that τ is isomorphic to G and τ is T 1 e iT * 1 -totally 407 regular CDFG, therefore the total degree of each node of G 408 is given by = T 1 .

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Also for phase term,

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Since τ ∼ = G , we must have Suppose that τ = (C, F) is R 1 e iR * 1 -regular CDFG, then The total degree of a node is given by Hence, τ is a (R 1 + c 1 )e i(R * 1 +c * 1 ) -totally regular CDFG. 452 Conversely, suppose that τ = (C, F) is T 1 e iT * 1 -totally 453 regular CDFG, then Theorem 3: Suppose that τ = (C, F) is a CDFG on a 467 graph τ * = (ψ, A). If τ is both R 1 e iR * 1 -regular and T 1 e iT * 1 -468 totally regular CDFG, then ς C e iϑ C is a constant function. 469 Proof: Suppose that τ is R 1 e iR * 1 -regular and T 1 e iT * 1 -totally 470 regular CDFG. Then, then the degree of a node is given by The total degree of a node is given by It follows that ).

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• The total degree of an edge gh ∈ A is represented by ).

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Definition 22: A CDFG τ = (C, F) is known as an edge 525 regular, if the degree of its each edge is equal, i.e, Definition 23: A CDFG τ = (C, F) is known as totally 535 edge regular, if the total degree of its each edge is equal, i.e, 536 for all gh ∈ A. τ is known as K 1 e iK * 1 -totally edge regular 543 CDFG.
Since the degree of an edge gh ∈ A is given by

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Theorem 5: Suppose a CDFG τ is L 1 e iL * 1 -edge regular 575 and K 1 e iK * 1 -totally edge regular, then ς F e iϑ F is a constant 576 function. Proof: Suppose that τ is L 1 e iL * 1 -edge regular 577 CDFG, then the degree of its every arc is

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Also τ is K 1 e iK * 1 -totally edge regular CDFG, then the degree 583 of its each edge is Further, it follows that Hence, ς F e iϑ F is a constant function.  Proof: Suppose that τ is a CDFG. Assume that ς F e iϑ F 604 is a constant function, therefore, ς F (gh)e iϑ F (gh) = c 1 e ic * 1 for 605 all gh ∈ A, where c 1 e ic * 1 is a constant.

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So τ is both regular-CDFG and totally edge regular-CDFG.

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Conversely, Let τ be L 1 e iL * 1 -edge regular and K 1 e iK * 1 -627 totally edge regular CDFG. Furthermore, the total degree of 628 an edge is given by Hence, ς F e iϑ F is a constant function.

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In this part, we present an algorithm and resolve a problem of   .     Telecommunication plays an important role in the develop-672 mental level of any country. Developed countries have strong 673 telecommunication system. A variety of factors like social 674 interaction, employee, economic growth, job creation, busi-675 ness productivity are all dependent on telecommunication 676 system. There are many ways of telecommunication includ-677 ing smart phones features, skype, whatsapp, imo, snapchat, 678 facebook etc. It plays an important role in globalization. The 679 telecommunication through these softwares is dependent on 680 high speed internet. A private internet company decided to 681 build their office in a city for the convenience of its service to 682 the public. They decided three place in a city P k (k=1,2,3). 683 The company make a pairwise comparison in these three 684 places to build an internet office. Following are some param-685 eters that are to be observed.

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• a desirable location to open an office.

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The specialists of company give their preference infor-692 mation in the form of CFPR Q = (d kq ) 3×3 as shown in 693 Table 1, where d kq = ς kq e iϑ kq is a complex fuzzy element 694 (CFE) preferred by the expert. Consider 0.8e i2π(0.7) , For the 695 value 0.8, the amplitude term shows that eighty percent of 696 the specialist says P 1 is best choice to establish an office 697 VOLUME 10, 2022 over place P 2 . Now the phase term 0.7 represents that seventy 698 percent of the specialists conclude that P 1 location will create 699 more time profit for the company over location P 2 . The CFPR 700 Q = (d kq ) 3×3 is given in Table 1.

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The directed network of CFPR Q represented in 702   Table 1 and is shown as in Figure 6. We get the ranking order of the four terminals P k from the 716 score functions as follows: The ranking leads to the conclusion that P 3 is best place to 719 establish an internet office.  In the future work, we will describe some operation on 738 CDFG. Energy of CDFG may be discuss.

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The data used to support the findings of the study are included 741 with in the article.

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All authors are here with confirm that there are no competing 744 interests between them.