Conormal Product for Intutionistic Anti-Fuzzy Graphs

K. Kalaiarasi; L. Mahalakshmi; Nasreen Kausar; A. B. M. Saiful Islam

doi:10.5391/IJFIS.2023.23.1.79

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 79-90

Published online March 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.1.79

Conormal Product for Intutionistic Anti-Fuzzy Graphs

K. Kalaiarasi¹, L. Mahalakshmi¹, Nasreen Kausar², and A. B. M. Saiful Islam³

¹Department of Mathematics, Cauvery College for Women (Autonomous), Affiliated to Bharathidasan University, Tamil Nadu, India
²Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Istanbul, Turkey
³Department of Civil & Construction Engineering, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia

Correspondence to :
K. Kalaiarasi (kalaishruthi1201@gmail.com)

Received: October 22, 2021; Revised: November 29, 2022; Accepted: January 4, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Go to

Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
Figures
References
Biographies

This study introduces and analyzes the conormal product of intuitionistic anti-fuzzy graphs (IAFGs) and analyzes certain fundamental theorems and applications. Further, new notions on complete and regular IAFGs were introduced, and the conormal product operation was applied to these IAFGs. We showed that the conormal product of two IAFGs could be used and analyzed important results showing that the conormal product of complete, regular, and strong IAFGs is an IAFG.

Keywords: Conormal product of IFGs, Conormal product of IAFGs, Conormal product of complete IAFGs, Conormal product of regular IAFGs, Conormal product of strong IAFGs

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
Figures
References
Biographies

The concept of fuzzy sets was first introduced by Zadeh [1] in 1965 and has since gained popularity. It has been used to solve many real-world decision-making problems with uncertainty. Additionally, fuzzy set theory is investigated in various domains, including mathematics, computer science, and signal processing. There have been many attempts at generalizing fuzzy sets, including interval-valued fuzzy sets (IVFSs) bipolar fuzzy sets (BPFs), intuitionistic fuzzy sets (IFSs), and picture fuzzy sets (PFSs). Graphs are a simple method of communicating data, as are connections between different substances. The substances are demonstrated by vertices and their relations by edges. Kaufmann and Zadeh [2] was the first to introduce the concept of fuzzy graphs. In 1975, Rosenfield [3] fuzzy graph concepts that broadened the scope of fuzzy sets to include graph theory. Later, Bhattachaya [4] analyzed several ideas in fuzzy graphs. Mordeson and his colleagues [5,6] analyzed certain operations on fuzzy graphs and hypergraphs. Parvathi and his colleagues [7,8] introduced intuitionistic fuzzy graphs (IFGs). Chaira and Ray [9] explained the application of intuitionistic fuzzy set theory. Nagoorgani and Radha [10] proposed certain properties and the conjunction of two fuzzy graphs. Xu and Hu [11] suggested some ideas in the projection models of IFGs. Mordeson and Nar [6] put forward basic ideas for fuzzy graphs and hypergraphs. Radha and Arumugam [12, 13] summarized the properties of strong product fuzzy graphs. The product of fuzzy graphs was established by Dogra [14], and Sahoo and Pal [15] presented product concepts based on IFGs in the same year. Rashmanlou et al. [16] explained some properties of IFGs. Products on IFGs were introduced in the same year by Sahoo et al.[18]. Sahoo and Pal [15]. The authors of [15, 17] also developed intuitionistic fuzzy competition graphs in 2016. Karunambigai et al. [18] explained the concepts of IFGs. In the same year, Mohideen et al. [19] described the properties of regular IFGs. Peter [20] established the conormal product of two fuzzy graphs. Muthuraj and Sasireka [21, 22] provided notions on anti-fuzzy graphs. Complex IFGs were introduced by Yaqoob et al. [23]. Pal et al. [24] established important notions of fuzzy graph theory. Rashmanlou and his colleagues [16, 25–28] explained various IVFSs in 2014. In 2015, IFGs with categorical properties were developed by Rashmanlou et al. [16]. Several studies [29–33] discussed the principles of fuzzy graphs and different products in fuzzy graphs. X. Hong et al. introduced some applications of fuzzy graphs. From 2012 to 2023, the authors of [18, 34–38] explained the concepts of intuitionistic and domination fuzzy graphs. Later, they discovered the novel conormal product, utilized exclusively for IFGs. Subsequently, the conormal product was utilized for intuitionistic anti-fuzzy graphs (IAFGs).

This manuscript is organized as follows: Section 2 provides basic and important definitions. We introduce and study conormal products on IAFGs in Section 3 on fascinating IAFGs, including complete, regular, and strong IAFGs. Using a conormal product yielded interesting outcomes. The conormal product of complete IAFGs yields a complete IAFG, the conormal product of regular IAFGs yields a regular IAFG, and the conormal product of pseudo-strong IAFGs yields a strong IAFG. In Section 4, we discovered an application of the conormal product on IAFGs. Future conormal product work on IAFGs is discussed in Section 5.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Definition 2.1 [2]

A fuzzy graph is an ordered triple G_F (V_F , σ_F , μ_F ), where V_F is a set of vertices {u_F_₁, u_F_₂, ..., u_F_{_n}} and σ_F is a fuzzy subset of V_F such that σ_F : V_F → [0, 1] and is denoted by σ_F = {(u_F_₁, σ_F (u_F_₁)), (u_F_₂, σ(u_F_₂)), ..., (u_F_{_n}, σ(u_Fn))}, and μ_F is a fuzzy relation on σ_F such that μ_F (u_F , v_F ) ≤ σ_F (u_F ) ∧ σ_F (v_F ).

Definition 2.2 [11]

A fuzzy graph G is regular if all its vertices have the same degree. In a fuzzy graph, if the degree of each vertex is k_F , that is, d_F (v_F) = ∑μ_F (u_F , v_F) = k_F , the graph is called a k_F -regular fuzzy graph.

Definition 2.3 [6]

An IFG has the form G_F : (V_F,E_F ), where

(i) V_F = {v_F_₁, v_F_₂, ..., v_F_{_n}} such that μ_F_₁: V_F → [0, 1] and V_F_₁: v_F → [0, 1] denote the degree of membership (MS) and non-membership (NMS) of the element v_F_{_i}∈ V_F_{_i}, respectively, and 0 ≤ μ_F_{_i}(v_F_{_i}) + V_F_{_i}(v_F_{_i}) ≤ 1 for every v_F_{_i}∈ V_F_{_i}(i = 1, 2, ..., n);
(ii) E_F ⊆ V_F × V_F , where μ_F_₂: V_F × V_F → [0, 1] and V_F_₂: V_F × V_F → [0, 1] are such that

μF2(vFi,vFj)≤{μF1(vFi)∧μF1(vFj)},VF2(vFi,vFj)≤{vF1(vFI)∨vF1(vFj)},

and 0 ≤ μ_F_₂(v_F_{_i}, v_F_{_j}) + V_F_₂(v_F_{_i}, v_F_{_j}) ≤ 1 for every (v_F_{_i}, v_F_{_j}) ∈ E_F , (i, j = 1, 2, ..., n).

Definition 2.4 [10]

Let G_F_₁: (μ_F_₁, μ_F_₂) and GF2:(μF1′,μF2′) be two fuzzy graphs with underlying vertex sets V_F_₁and V_F_₂and edge sets E_F_₁and E_F_₂, respectively. Then, the conormal product of IFGs G_F_₁and G_F₂ is a pair of functions GF1*GF2:(μF1*μF1′,μF2*μF2′) with the underlying vertex set V_F_₁× V_F_₂= {(u_F_₁, v_F_₁); u_F_₁∈ V_F_₁, v_F_₁∈ V_F_₂} and underlying the edge in E_F_₂× E_F_₂= {(u_F_₁, v_F_₁)(u_F_₂, v_F_₂) : u_F_₁= u_F_₂, v_F_₁v_F_₂∈ E_F_₂(or)u_F_₁u_F_₂∈ E_F_₁, V_F_₁= V_F_₂}. The vertex’s MS and NMS values (u_F , v_F ) in G_F_₁* G_F_₂are given by

(σF1*σF1′)(uF1,vF1)=σF1(uF1)∧σF1′(vF1),(σF2*σF2′)(uF1,vF1)=σF2(uF1)∧σF2′(vF1).

The extension of edge values is given by

(i) μF1*μF1′((uF1,vF1)(uF2,vF2))={μF1(uF1uF2)∧σF1′(vF1)if uF1uF2∈EF1 and vF1=vF2,μF1′(vF1vF2)∧σF1(uF1)if vF1vF2∈EF2 and uF1=uF2,μF1(uF1uF2)∧σF1′(vF1)∧σF1′(vF2)if uF1uF2∈EF1 and vF1vF2∉EF2,μF1′(vF1vF2)∧σF1(uF1)∧σF1(uF2)if vF1vF2∈EF2 and uF1uF2∉EF1,μF1(uF1uF2)∧μF1′(vF1vF2)if uF1uF2∈EF1 and vF1vF2∈EF2.}
(ii) μF2*μF2′((uF1,vF1)(uF2,vF2))={μF2(uF1uF2)∨σF2′(vF2)if uF1uF2∈EF1 and vF1=vF2,μF2′(vF1vF2)∨σF2(uF2)if vF1vF2∈EF2 and uF1=uF2,μF2(uF1uF2)∨σF2′(vF1)∨σF2′(vF2)if uF1uF2∈EF1 and vF1vF2∉EF2,μF2′(vF1vF2)∨σF2(uF1)∨σF2(uF2)if vF1vF2∈EF2 and uF1uF2∉EF1,μF2(uF1uF2)∨μF2′(vF1vF2)if uF1uF2∈EF1 and vF1vF2∈EF2.}

3. Conormal Product of IAFGs

Go to

Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
Figures
References
Biographies

The preceding paper discussed certain concepts of IAFGs. This section introduces conormal products and illustrates how to use them in IAFGs. We use the conormal product on two IFGs.

Definition 3.1

Let G_AF_₁: (μ_AF_₁, μ_AF_₂) and GAF2:(μAF1′,μAF2′) be anti-fuzzy graphs with underlying vertex sets V_AF_₁and V_AF_₂and edge sets E_AF_₁and E_AF_₂, respectively. Then, the conormal product of IAFGs G_AF_₁and G_AF_₂is a pair of functions GAF1*GAF2:(μAF1*μAF1′,μAF2*μAF2′) with the underlying vertex set V_AF_₁× V_AF_₂= {(u_AF_₁, v_AF_₁); u_AF_₁∈ V_AF_₁, v_AF_₁∈ V_AF_₂} and underlying the edge in E_AF_₁× E_AF_₂= {(u_AF_₁, v_AF_₁)(u_AF_₂, v_AF_₂): u_AF_₁= u_AF_₂, v_AF_₁v_AF_₂∈ E_AF_₂(or)u_AF_₁u_AF_₂∈ E_AF_₁, v_AF_₁= v_AF_₂}.

The vertex’s MS and NMS values (u_AF , v_AF ) in G_AF_₁* G_AF_₂are given by

(i) (σAF1*σAF1′)(uAF1,vAF1)=σAF1(uAF1)∧σAF1′(vAF1),(σAF2*σAF2′)(uAF1,vAF1)=σAF2(uAF1)∨σAF2′(vAF1),
(ii) (σAF1*σAF1′)(uAF2,vAF2)=σAF1(uAF2)∧σAF1′(vAF2),(σAF2*σAF2′)(uAF2,vAF2)=σAF2(uAF2)∨σAF2′(vAF2).

The extension of edge values is given by

(i) μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))={μAF1(uAF1uAF2)∨σAF1′(vAF1)if uAF1uAF2∈EAF1 and vAF1=vAF2,μAF1′(vAF1vAF2)∨σAF1(uAF1)if vAF1vAF2∈EAF2 and uAF1=uAF2,μAF1(uAF1uAF2)∨σAF1′(vAF1)∨σAF1′(vAF2)if uAF1uAF2∈EAF1 and vAF1vAF2∉EAF2,μAF1′(vAF1vAF2)∨σAF1(uAF1)∨σAF1(uAF2)if vAF1vAF2∈EAF2 and uAF1uAF2∉EAF1,μAF1(uAF1uAF2)∨μAF1′(vAF1vAF2)if uAF1uAF2∈EAF1 and vAF1vAF2∈EAF2.}
(ii) μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))={μAF2(uAF1uAF2)∨σAF2′(vAF2)if uAF1uAF2∈EAF1 and vAF1=vAF2,μAF2′(vAF1vAF2)∧σAF2(uAF2)if vAF1vAF2∈EAF2 and uAF1=uAF2,μAF2(uAF1uAF2)∧σAF2′(vAF1)∧σAF2′(vAF2)if uAF1uAF2∈EAF1 and vAF1vAF2∉EAF2,μAF2′(vAF1vAF2)∧σAF2(uAF1)∧σAF2(uAF2)if vAF1vAF2∈EAF2 and uAF1uAF2∉EAF1,μAF2(uAF1uAF2)∧μAF2′(vAF1vAF2)if uAF1uAF2∈EAF1 and vAF1vAF2∈EAF2.}

Example 3.1

See Figure 1.

Definition 3.2

LetG_AF_₁: (μ_AF_₁, μ_AF_₂) and G_AF_₂: (μ’_AF_₁, μ’_AF_₂) be complete and conormal products of two IAFGs; then, the fuzzy graph is called a conormal product of complete IAFGs and is denoted by G_CAF_₁* G_CAF_₂.

Definition 3.3

Let G_AF_₁: (μ_AF_₁, μ_AF_₂) and G_AF_₂: (μ’_AF_₁, μ’_AF_₂) be strong, conormal products of two IAFGs; then, the fuzzy graph is called a conormal product of strong IAFGs and denoted by G_SAF_₁* G_SAF_₂.

Example 3.5

See Figure 5.

Definition 3.4

Let G_AF_₁* G_AF_₂be the conormal product of two IAFGs; then, the graph is said to be a conormal product

Example 3.6

See Figure 6.

Theorem 3.1

A conormal product of two IAFGs is an IAFGs.

Proof

Let G_AF_₁: (μ_AF_₁, μ_AF_₂) and GAF2:(μAF1′,μAF2′) represent IAFGs with the underlying crisp graphs GAF1*:(VAF1,EAF1) and GAF2*:(VAF2,EAF2), respectively, and G_AF_₁* G_AF_₂be their conormal product; we must prove that G_AF_₁* G_AF_₂is an IAFG.

From definition, it follows that

Case (i)

If uAF1uAF2∈EAF1 and vAF1=vAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1uAF2)∨σAF1′(vAF1)≤[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF2′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF2′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1,uAF2)∧σAF2′(vAF1)≤[σAF2(uAF1)∧σAF2(uAF2)]∧ [σAF2′(vAF1)∧σAF2′(vAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (ii)

If vAF1vAF2∈EAF2 and uAF1=uAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1′(vAF1,vAF2)∨σAF1(uAF1)≤[σAF1′(vAF1)∨σAF2′(v2)]∨ [σAF1(uAF1)∨σAF1(uAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2′(vAF1,vAF2)∧σAF2′(uAF2)≤[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (iii)

If uAF1uAF2∈EAF1 and vAF1vAF2∉EAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1uAF2)∨σAF1′(vAF1)∨σAF1′(vAF2)≤[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1uAF2)∧σAF2′(vAF1)∧σAF2′(vAF2)≤[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (iv)

If vAF1vAF2∈EAF2 and uAF1uAF2∉EAF1,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(vAF1vAF2)∨σAF1′(uAF1)∨σAAF1′(uAF2)≤[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μA2′(vAF1vAF2)∧σA2(uAF1)∧σAF2(uAF2)≤[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(u2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (v)

If uAF1uAF2∈EAF1 and vAF1vAF2∈EAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1,uAF2)∨σAF1′(vAF1vAF2)≤[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1uAF2)∨μAF2′(vAF1vAF2)≤[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2),

then

μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).

This proves that an IAFG is the conormal product of two IAFGs.

Theorem 3.2

If G_AF_₁: (μ_AF_₁, μ_AF_₂) and GAF2:(μAF1′,μAF2′) are complete IAFGs, thenG_AF_₁*G_AF_₂is also a complete IAFG.

Proof

W.K.T conormal product of two IAFGs is an IAFG.

To prove

The conormal product of both IAFGs is complete.

An IAFG is said to be complete if μ_AF_₁(v_AF_{_i}, v_AF_{_j}) = σ_AF_₁(v_AF_{_i})∨σ_AF_₁(v_AF_{_j}) and μ_AF_₂(v_AF_{_i}, v_AF_{_j}) = σ_AF_₂(v_AF_{_i}) ∧σ_AF_₂(v_AF_{_j}) for all v_AF_{_i}, v_AF_{_j}∈ V_AF .

Case (i)

If uAF1uAF2∈EAF1 and vAF1=vAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1uAF2)∨σAF1′(vAF1)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF2′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF2′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1,uAF2)∧σAF2′(vAF1),=[σAF2(uAF1)∧σAF2(uAF2)]∧ [σAF2′(vAF1)∧σAF2′(vAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (ii)

If vAF1vAF2∈EAF2 and uAF1=uAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1′(vAF1,vAF2)∨σAF1(uAF1)=[σAF1′(vAF1)∨σAF2′(v2)]∨ [σAF1(uAF1)∨σAF1(uAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2′(vAF1,vAF2)∧σAF2′(uAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (iii)

If uAF1uAF2∈EAF1 and vAF1vAF2∉EAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1uAF2)∨σAF1′(vAF1)∨σAF1′(vAF2)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1uAF2)∧σAF2′(vAF1)∧σAF2′(vAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (iv)

If vAF1vAF2∈EAF2 and uAF1uAF2∉EAF1,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(vAF1vAF2)∨σAF1′(uAF1)∨σAAF1′(uAF2)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μA2′(vAF1vAF2)∧σA2(uAF1)∧σAF2(uAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(u2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (v)

If uAF1uAF2∈EAF1 and vAF1vAF2∈EAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1,uAF2)∨σAF1′(vAF1vAF2)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1uAF2)∨μAF2′(vAF1vAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).

Hence, G_AF_₁* G_AF_₂is a complete IAFG.

Theorem 3.3

If G_AF_₁: (μ_AF_₁, μ_AF_₂) and GAF2:(μAF1′,μAF2′) are two strong IAFGs, then G_AF_₁*G_AF_₂is also a strong IAFG.

Proof

W.K.T conormal product of two IAFGs is an IAFG.

To prove

Conormal product of both IAFGs is strong.

An IAFG is said to be strong if μ_AF_₁(v_AF_{_i}, v_AF_{_j}) = σ_AF_₁(v_AF_{_i}) ∨ σ_AF_₁(v_AF_{_j}) and μ_AF_₂(v_AF_{_i}, v_AF_{_j}) = σ_AF_₂(v_AF_{_i}) ∧ σ_AF_₂(v_AF_{_j}) for all v_AF_{_i}, v_AF_{_j}∈ V_AF .

Case (i)

If uAF1uAF2∈EAF1 and vAF1=vAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1uAF2)∨σAF1′(vAF1)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF2′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF2′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1,uAF2)∧σAF2′(vAF1)=[σAF2(uAF1)∧σAF2(uAF2)]∧ [σAF2′(vAF1)∧σAF2′(vAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (ii)

If vAF1vAF2∈EAF2 and uAF1=uAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1′(vAF1,vAF2)∨σAF1(uAF1)=[σAF1′(vAF1)∨σAF2′(v2)]∨ [σAF1(uAF1)∨σAF1(uAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2′(vAF1,vAF2)∧σAF2′(uAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (iii)

If uAF1uAF2∈EAF1 and vAF1vAF2∉EAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1uAF2)∨σAF1′(vAF1)∨σAF1′(vAF2)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1uAF2)∧σAF2′(vAF1)∧σAF2′(vAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (iv)

If vAF1vAF2∈EAF2 and uAF1uAF2∉EAF1,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(vAF1vAF2)∨σAF1′(uAF1)∨σAAF1′(uAF2)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μA2′(vAF1vAF2)∧σA2(uAF1)∧σAF2(uAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(u2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).
Case (v)

If uAF1uAF2∈EAF1 and vAF1vAF2∈EAF2,μAF1*μAF1′((uAF1,vAF1)(uAF2,vAF2))=μAF1(uAF1,uAF2)∨σAF1′(vAF1vAF2)=[σAF1(uAF1)∨σAF1(uAF2)]∨ [σAF1′(vAF1)∨σAF1′(vAF2)]=[σAF1(uAF1)∨σAF1′(vAF1)]∨ [σAF1(uAF2)∨σAF1′(vAF2)]=(σAF1*σAF1′)(uAF1vAF1)∨ (σAF1*σAF1′)(uAF2vAF2),μAF2*μAF2′((uAF1,vAF1)(uAF2,vAF2))=μAF2(uAF1uAF2)∨μAF2′(vAF1vAF2)=[σAF2′(vAF1)∧σAF2′(vAF2)]∧ [σAF2(uAF1)∧σAF2(uAF2)]=[σAF2(uAF1)∧σAF2′(vAF1)]∧ [σAF2(uAF2)∧σAF2′(vAF2)]=(σAF2*σAF2′)(uAF1vAF1)∧ (σAF2*σAF2′)(uAF2vAF2).

Hence, G_AF_₁* G_AF_₂is also a strong IAFG.

Theorem 3.4

If G_AF_₁: (μ_AF_₁, μ_AF_₂) and of regular IAFGs if each vertex has the same degree. GAF2:(μAF1′,μAF2′) are two regular IAFGs, then G_AF_₁* G_AF_₂is also a regular IAFG.

Proof

Let G_AF_₁= (V_AF , E_AF , σ_AF , μ_AF ) and GAF2=(VAF′,EAF′,σAF′,μAF′) be the conormal product of IAFGs.

To prove

The conormal product of IAFGs is also an IAFG.

Let G_AF_₁= (V_AF , E_AF , σ_AF , μ_AF ) and GAF2=(VAF′,EAF′,σAF′,μAF′) be IAFGs and G_AF_₁* G_AF_₂be their conormal product; then, d_FG_{_AF_₁}_*G_{_AF_₂}(u_AF_{_i}, v_AF_{_j}) = ℑ for all i, j = 1, 2, ..., n. A regular fuzzy graph of degree ℑ, or a ℑ- regular fuzzy graph, is one in which each vertex has the same degree. An IAFG is a regular IAFG in this sense.

Example 3.7

See Figure 7.

In Fig. 7,

dAF(uAF1,vAF1)=1.7,dAF(uAF1,vAF2)=1.7,dAF(uAF2,vAF1)=1.7,dAF(uAF2,vAF2)=1.7.

From the above we get

dAF(uAF1,vAF1)=dAF(uAF1,vAF2)=dAF(uAF2,vAF1)=dAF(uAF2,vAF2).

Hence, G_AF_₁* G_AF_₂is a regular IAFG.

Remark 3.1

Although the conormal product of IAFGs is regular, regular fuzzy graphs need not be the conormal product of IAFGs.

Example 3.8

See Figure 8.

The above example is a regular and pseudo-regular fuzzy graph. However, it is not a conormal product of IAFGs.

4. Application

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
Figures
References
Biographies

In this example, we demonstrated how to use the conormal product of two IAFGs. Consider the following example, where G_AF_₁and G_AF_₂represent two IFAGs and two vertices representing distinct grocery goods. The MS and NMS values of the vertices denote the proportion of a product’s quality and cost, respectively.

In G_AF_₁, the MS and NMS values of the vertices reflect the percentage of a product’s quality and cost, respectively. The MS and NMS values of the edges show the maximum relative product quality and minimum product cost, respectively. In addition, for G_AF_₂, the MS and NMS values of the vertices represent the percentage of a product’s quality and cost, respectively. The MS and NMS values of the edges represent the maximum relative product quality and minimum product cost, respectively. The conormal product G_AF_₁*G_AF_₂of G_AF_₁and G_AF_₂shows the percentage change in product quality for the highest MS value and the percentage change in product cost for the lowest NMS value.

Consequently, the most effective combinations with the highest percentage of quality improvement and the lowest percentage of cost increase were discovered across all supermarkets.

5. Conclusion

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
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Biographies

In this manuscript, we present the conormal product of IAFGs. We define the conormal product of IAFGs, and use a few examples to demonstrate its properties. Each of the given definitions and hypothetical methodologies is appropriate for regular, complete, and strong IAFGs.

In future work, different kinds of products of different types of IFGs, like m-polar, interval-valued, vague, and hesitancy anti-fuzzy graphs, will be discussed. The above anti-fuzzy graphs are useful for practical applications. We are committed to managing other maintainable improvement objectives for a better world.

Conflict of Interest

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
Figures
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
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Figures
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Fig. 1.

Conormal product of two IAFGs.

Fig. 2.

Conormal product of complete IAFGs.

Fig. 3.

Conormal product of complete IAFGs.

Fig. 4.

Conormal product of complete IAFGs.

Fig. 5.

Conormal product of strong IAFGs.

Fig. 6.

Conormal product of regular IAFGs.

Fig. 7.

Conormal product of regular IAFG.

Fig. 8.

Regular IAFG.

References

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
Figures
References
Biographies

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Rashmanlou, H, Samanta, S, Pal, M, and Borzooei, RA (2015). Intuitionistic fuzzy graphs with categorical properties. Fuzzy Information and Engineering. 7, 317-334. https://doi.org/10.1016/j.fiae.2015.09.005
Sahoo, S, and Pal, M (2016). Intuitionistic fuzzy competition graphs. Journal of Applied Mathematics and Computing. 52, 37-57. https://doi.org/10.1007/s12190-015-0928-0
Karunambigai, MG, Akram, M, and Buvaneswari, R (2016). Strong and superstrong vertices in intuitionistic fuzzy graphs. Journal of Intelligent & Fuzzy Systems. 30, 671-678. https://doi.org/10.3233/IFS-151786
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Biographies

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Abstract
1. Introduction
2. Preliminaries
3. Conormal Product of IAFGs
4. Application
5. Conclusion
Conflict of Interest
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Biographies

K. Kalaiarasi received a Ph.D. degree in Mathematics from Manonmanium University, Thirunelveli. She is currently an assistant professor of Mathematics in faculty of PG and Research Department of Mathematics, Cauvery College for Women, Trichy. Her research interests include fuzzy inventory models, operations research, data science, and machine learning.

E-mail: kalaishruthi1201@gmail.com

L. Mahalakshmi received a Ph.D degree in Mathematics from Cauvery College for Women Autonomous and affiliated to Bharathidasan University, Tiruchirappalli. She is currently an assistant professor of Mathematics in Faculty of PG and Research Department of Mathematics, Cauvery College for Women, Trichy. Her research interests are associative and commutative (non-associative and non-commutative fuzzy graphs and its applications on fuzzy graphs).

E-mail: mahaparthi19@gmail.com

Nasreen Kausar received a Ph.D. degree in Mathematics from Quaid-i-Azam university Islamabad, Pakistan. She is currently an assistant professor of Mathematics in Faculty of Arts and Science, Yildiz Technical University, Istanbul, Tur-key. Her research interests are associative and commutative (non-associative and non-commutative algebraic structure and its applications on fuzzy structures.

E-mail: nkausar@yildiz.edu.tr

A. B. M. Saiful Islam received a Ph.D. in Structural Engineering with an excellence award from the Department of Civil Engineering, University of Malaya (UM), Malaysia. He did his Post-Doc as a High Impact Research Bright Sparks (HIR BSP) and earlier he obtained a Master of Science in Structural Engineering and Bachelor of Science in Civil Engineering from Bangladesh University of Engineering and Technology (BUET). His research interests include nonlinear dynamics, offshore structures, structural retrofitting, base isolation and lightweight concrete. Dr. Islam authored/coauthored 100+ referred publications. Currently, he is an associate professor in the Department of Civil & Construction Engineering, College of Engineering, Imam Abdulrahman Bin Faisal University (IAU), Saudi Arabia. His involvement includes as Editor-in-Chief/Editor of 4 journals and reviewer of 60+ journals.

E-mail: abm.saiful@gmail.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 79-90

Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.79

Conormal Product for Intutionistic Anti-Fuzzy Graphs

K. Kalaiarasi¹, L. Mahalakshmi¹, Nasreen Kausar², and A. B. M. Saiful Islam³

Correspondence to:K. Kalaiarasi (kalaishruthi1201@gmail.com)

Received: October 22, 2021; Revised: November 29, 2022; Accepted: January 4, 2023

Abstract

Keywords: Conormal product of IFGs, Conormal product of IAFGs, Conormal product of complete IAFGs, Conormal product of regular IAFGs, Conormal product of strong IAFGs

1. Introduction

2. Preliminaries

Definition 2.1 [2]

Definition 2.2 [11]

Definition 2.3 [6]

An IFG has the form G_F : (V_F,E_F ), where

(i) V_F = {v_F_₁, v_F_₂, ..., v_F_{_n}} such that μ_F_₁: V_F → [0, 1] and V_F_₁: v_F → [0, 1] denote the degree of membership (MS) and non-membership (NMS) of the element v_F_{_i}∈ V_F_{_i}, respectively, and 0 ≤ μ_F_{_i}(v_F_{_i}) + V_F_{_i}(v_F_{_i}) ≤ 1 for every v_F_{_i}∈ V_F_{_i}(i = 1, 2, ..., n);
(ii) E_F ⊆ V_F × V_F , where μ_F_₂: V_F × V_F → [0, 1] and V_F_₂: V_F × V_F → [0, 1] are such that

$\begin{array}{l} μ_{F_{2}} (v_{F_{i}}, v_{F_{j}}) \leq {μ_{F_{1}} (v_{F_{i}}) \land μ_{F_{1}} (v_{F_{j}})}, \\ V_{F_{2}} (v_{F_{i}}, v_{F_{j}}) \leq {v_{F_{1}} (v_{F_{I}}) \lor v_{F_{1}} (v_{F_{j}})}, \end{array}$

and 0 ≤ μ_F_₂(v_F_{_i}, v_F_{_j}) + V_F_₂(v_F_{_i}, v_F_{_j}) ≤ 1 for every (v_F_{_i}, v_F_{_j}) ∈ E_F , (i, j = 1, 2, ..., n).

Definition 2.4 [10]

Let G_F_₁: (μ_F_₁, μ_F_₂) and $G_{F_{2}} : (μ_{F_{1}}^{'}, μ_{F_{2}}^{'})$ be two fuzzy graphs with underlying vertex sets V_F_₁and V_F_₂and edge sets E_F_₁and E_F_₂, respectively. Then, the conormal product of IFGs G_F_₁and G_F₂ is a pair of functions $G_{F_{1}} * G_{F_{2}} : (μ_{F_{1}} * μ_{F_{1}}^{'}, μ_{F_{2}} * μ_{F_{2}}^{'})$ with the underlying vertex set V_F_₁× V_F_₂= {(u_F_₁, v_F_₁); u_F_₁∈ V_F_₁, v_F_₁∈ V_F_₂} and underlying the edge in E_F_₂× E_F_₂= {(u_F_₁, v_F_₁)(u_F_₂, v_F_₂) : u_F_₁= u_F_₂, v_F_₁v_F_₂∈ E_F_₂(or)u_F_₁u_F_₂∈ E_F_₁, V_F_₁= V_F_₂}. The vertex’s MS and NMS values (u_F , v_F ) in G_F_₁* G_F_₂are given by

\begin{array}{l} (σ_{F_{1}} * σ_{F_{1}}^{'}) (u_{F_{1}}, v_{F_{1}}) = σ_{F_{1}} (u_{F_{1}}) \land σ_{F_{1}}^{'} (v_{F_{1}}), \\ (σ_{F_{2}} * σ_{F_{2}}^{'}) (u_{F_{1}}, v_{F_{1}}) = σ_{F_{2}} (u_{F_{1}}) \land σ_{F_{2}}^{'} (v_{F_{1}}) . \end{array}

The extension of edge values is given by

(i) $\begin{array}{l} μ_{F_{1}} * μ_{F_{1}}^{'} ((u_{F_{1}}, v_{F_{1}}) (u_{F_{2}}, v_{F_{2}})) \\ = {\begin{cases} μ_{F_{1}} (u_{F_{1}} u_{F_{2}}) \land σ_{F_{1}}^{'} (v_{F_{1}}) \\ if u_{F_{1}} u_{F_{2}} \in E_{F_{1}} and v_{F_{1}} = v_{F_{2}}, \\ μ_{F_{1}}^{'} (v_{F_{1}} v_{F_{2}}) \land σ_{F_{1}} (u_{F_{1}}) \\ if v_{F_{1}} v_{F_{2}} \in E_{F_{2}} and u_{F_{1}} = u_{F_{2}}, \\ μ_{F_{1}} (u_{F_{1}} u_{F_{2}}) \land σ_{F_{1}}^{'} (v_{F_{1}}) \land σ_{F_{1}}^{'} (v_{F_{2}}) \\ if u_{F_{1}} u_{F_{2}} \in E_{F_{1}} and v_{F_{1}} v_{F_{2}} \notin E_{F_{2}}, \\ μ_{F_{1}}^{'} (v_{F_{1}} v_{F_{2}}) \land σ_{F_{1}} (u_{F_{1}}) \land σ_{F_{1}} (u_{F_{2}}) \\ if v_{F_{1}} v_{F_{2}} \in E_{F_{2}} and u_{F_{1}} u_{F_{2}} \notin E_{F_{1}}, \\ μ_{F_{1}} (u_{F_{1}} u_{F_{2}}) \land μ_{F_{1}}^{'} (v_{F_{1}} v_{F_{2}}) \\ if u_{F_{1}} u_{F_{2}} \in E_{F_{1}} and v_{F_{1}} v_{F_{2}} \in E_{F_{2}} . \end{cases}} \end{array}$
(ii) $\begin{array}{l} μ_{F_{2}} * μ_{F_{2}}^{'} ((u_{F_{1}}, v_{F_{1}}) (u_{F_{2}}, v_{F_{2}})) \\ = {\begin{cases} μ_{F_{2}} (u_{F_{1}} u_{F_{2}}) \lor σ_{F_{2}}^{'} (v_{F_{2}}) \\ if u_{F_{1}} u_{F_{2}} \in E_{F_{1}} and v_{F_{1}} = v_{F_{2}}, \\ μ_{F_{2}}^{'} (v_{F_{1}} v_{F_{2}}) \lor σ_{F_{2}} (u_{F_{2}}) \\ if v_{F_{1}} v_{F_{2}} \in E_{F_{2}} and u_{F_{1}} = u_{F_{2}}, \\ μ_{F_{2}} (u_{F_{1}} u_{F_{2}}) \lor σ_{F_{2}}^{'} (v_{F_{1}}) \lor σ_{F_{2}}^{'} (v_{F_{2}}) \\ if u_{F_{1}} u_{F_{2}} \in E_{F_{1}} and v_{F_{1}} v_{F_{2}} \notin E_{F_{2}}, \\ μ_{F_{2}}^{'} (v_{F_{1}} v_{F_{2}}) \lor σ_{F_{2}} (u_{F_{1}}) \lor σ_{F_{2}} (u_{F_{2}}) \\ if v_{F_{1}} v_{F_{2}} \in E_{F_{2}} and u_{F_{1}} u_{F_{2}} \notin E_{F_{1}}, \\ μ_{F_{2}} (u_{F_{1}} u_{F_{2}}) \lor μ_{F_{2}}^{'} (v_{F_{1}} v_{F_{2}}) \\ if u_{F_{1}} u_{F_{2}} \in E_{F_{1}} and v_{F_{1}} v_{F_{2}} \in E_{F_{2}} . \end{cases}} \end{array}$

3. Conormal Product of IAFGs

The preceding paper discussed certain concepts of IAFGs. This section introduces conormal products and illustrates how to use them in IAFGs. We use the conormal product on two IFGs.

Definition 3.1

Let G_AF_₁: (μ_AF_₁, μ_AF_₂) and $G_{A F_{2}} : (μ_{A F_{1}}^{'}, μ_{A F_{2}}^{'})$ be anti-fuzzy graphs with underlying vertex sets V_AF_₁and V_AF_₂and edge sets E_AF_₁and E_AF_₂, respectively. Then, the conormal product of IAFGs G_AF_₁and G_AF_₂is a pair of functions $G_{A F_{1}} * G_{A F_{2}} : (μ_{A F_{1}} * μ_{A F_{1}}^{'}, μ_{A F_{2}} * μ_{A F_{2}}^{'})$ with the underlying vertex set V_AF_₁× V_AF_₂= {(u_AF_₁, v_AF_₁); u_AF_₁∈ V_AF_₁, v_AF_₁∈ V_AF_₂} and underlying the edge in E_AF_₁× E_AF_₂= {(u_AF_₁, v_AF_₁)(u_AF_₂, v_AF_₂): u_AF_₁= u_AF_₂, v_AF_₁v_AF_₂∈ E_AF_₂(or)u_AF_₁u_AF_₂∈ E_AF_₁, v_AF_₁= v_AF_₂}.

The vertex’s MS and NMS values (u_AF , v_AF ) in G_AF_₁* G_AF_₂are given by

(i) $\begin{array}{l} (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}}, v_{A F_{1}}) = σ_{A F_{1}} (u_{A F_{1}}) \land σ_{A F_{1}}^{'} (v_{A F_{1}}), \\ (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}}, v_{A F_{1}}) = σ_{A F_{2}} (u_{A F_{1}}) \lor σ_{A F_{2}}^{'} (v_{A F_{1}}), \end{array}$
(ii) $\begin{array}{l} (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}}, v_{A F_{2}}) = σ_{A F_{1}} (u_{A F_{2}}) \land σ_{A F_{1}}^{'} (v_{A F_{2}}), \\ (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}}, v_{A F_{2}}) = σ_{A F_{2}} (u_{A F_{2}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}}) . \end{array}$

The extension of edge values is given by

(i) $\begin{array}{l} μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = {\begin{cases} μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \\ if u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} = v_{A F_{2}}, \\ μ_{A F_{1}}^{'} (v_{A F_{1}} v_{A F_{2}}) \lor σ_{A F_{1}} (u_{A F_{1}}) \\ if v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} = u_{A F_{2}}, \\ μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}}) \\ if u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F_{2}} \notin E_{A F_{2}}, \\ μ_{A F_{1}}^{'} (v_{A F_{1}} v_{A F_{2}}) \lor σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}}) \\ if v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} u_{A F_{2}} \notin E_{A F_{1}}, \\ μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor μ_{A F_{1}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ if u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} . \end{cases}} \end{array}$
(ii) $\begin{array}{l} μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = {\begin{cases} μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}}) \\ if u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} = v_{A F_{2}}, \\ μ_{A F_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \land σ_{A F_{2}} (u_{A F_{2}}) \\ if v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} = u_{A F_{2}}, \\ μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}}) \\ if u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F_{2}} \notin E_{A F_{2}}, \\ μ_{A F_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \land σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}}) \\ if v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} u_{A F_{2}} \notin E_{A F_{1}}, \\ μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \land μ_{A F_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ if u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} . \end{cases}} \end{array}$

Example 3.1

See Figure 1.

Definition 3.2

Example 3.2

See Figure 2.

Example 3.3

See Figure 3.

Example 3.4

See Figure 4.

Definition 3.3

Example 3.5

See Figure 5.

Definition 3.4

Let G_AF_₁* G_AF_₂be the conormal product of two IAFGs; then, the graph is said to be a conormal product

Example 3.6

See Figure 6.

Theorem 3.1

A conormal product of two IAFGs is an IAFGs.

Proof

Let G_AF_₁: (μ_AF_₁, μ_AF_₂) and $G_{A F_{2}} : (μ_{A F_{1}}^{'}, μ_{A F_{2}}^{'})$ represent IAFGs with the underlying crisp graphs $G_{A F_{1}}^{*} : (V_{A F_{1}}, E_{A F_{1}})$ and $G_{A F_{2}}^{*} : (V_{A F_{2}}, E_{A F_{2}})$ , respectively, and G_AF_₁* G_AF_₂be their conormal product; we must prove that G_AF_₁* G_AF_₂is an IAFG.

From definition, it follows that

Case (i)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} = v_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \\ \leq [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}}, u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}}) \\ \leq [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ \land [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (ii)

$\begin{array}{l} If v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} = u_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}}^{'} (v_{A F_{1}}, v_{A F_{2}}) \lor σ_{A F_{1}} (u_{A F_{1}}) \\ \leq [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{2}}^{'} (v_{2})] \\ \lor [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}}^{'} (v_{A F_{1}}, v_{A F_{2}}) \land σ_{A F_{2}}^{'} (u_{A F_{2}}) \\ \leq [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (iii)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F 2} \notin E_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}}) \\ \leq [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}}) \\ \leq [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (iv)

$\begin{array}{l} If v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} u_{A F 2} \notin E_{A F_{1}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (v_{A F_{1}} v_{A F_{2}}) \lor σ_{A F_{1}}^{'} (u_{A F_{1}}) \lor σ_{A_{A F_{1}}}^{'} (u_{A F_{2}}) \\ \leq [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \land σ_{A_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}}) \\ \leq [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{2}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (v)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F 2} \in E_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}}, u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ \leq [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \lor μ_{A F_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ \leq [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \end{array}$

then

$\begin{array}{l} μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$

This proves that an IAFG is the conormal product of two IAFGs.

Theorem 3.2

If G_AF_₁: (μ_AF_₁, μ_AF_₂) and $G_{A F_{2}} : (μ_{A F_{1}}^{'}, μ_{A F_{2}}^{'})$ are complete IAFGs, thenG_AF_₁*G_AF_₂is also a complete IAFG.

Proof

W.K.T conormal product of two IAFGs is an IAFG.

To prove

The conormal product of both IAFGs is complete.

Case (i)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} = v_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}}, u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}}), \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ \land [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (ii)

$\begin{array}{l} If v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} = u_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}}^{'} (v_{A F_{1}}, v_{A F_{2}}) \lor σ_{A F_{1}} (u_{A F_{1}}) \\ = [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{2}}^{'} (v_{2})] \\ \lor [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}}^{'} (v_{A F_{1}}, v_{A F_{2}}) \land σ_{A F_{2}}^{'} (u_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (iii)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F 2} \notin E_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (iv)

$\begin{array}{l} If v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} u_{A F 2} \notin E_{A F_{1}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (v_{A F_{1}} v_{A F_{2}}) \lor σ_{A F_{1}}^{'} (u_{A F_{1}}) \lor σ_{A_{A F_{1}}}^{'} (u_{A F_{2}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \land σ_{A_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{2}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (v)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F 2} \in E_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}}, u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \lor μ_{A F_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$

Hence, G_AF_₁* G_AF_₂is a complete IAFG.

Theorem 3.3

If G_AF_₁: (μ_AF_₁, μ_AF_₂) and $G_{A F_{2}} : (μ_{A F_{1}}^{'}, μ_{A F_{2}}^{'})$ are two strong IAFGs, then G_AF_₁*G_AF_₂is also a strong IAFG.

Proof

W.K.T conormal product of two IAFGs is an IAFG.

To prove

Conormal product of both IAFGs is strong.

Case (i)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} = v_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}}, u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}}) \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ \land [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (ii)

$\begin{array}{l} If v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} = u_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}}^{'} (v_{A F_{1}}, v_{A F_{2}}) \lor σ_{A F_{1}} (u_{A F_{1}}) \\ = [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{2}}^{'} (v_{2})] \\ \lor [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}}^{'} (v_{A F_{1}}, v_{A F_{2}}) \land σ_{A F_{2}}^{'} (u_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (iii)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F 2} \notin E_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}} u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (iv)

$\begin{array}{l} If v_{A F_{1}} v_{A F_{2}} \in E_{A F_{2}} and u_{A F_{1}} u_{A F 2} \notin E_{A F_{1}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (v_{A F_{1}} v_{A F_{2}}) \lor σ_{A F_{1}}^{'} (u_{A F_{1}}) \lor σ_{A_{A F_{1}}}^{'} (u_{A F_{2}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \land σ_{A_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{2}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$
Case (v)

$\begin{array}{l} If u_{A F_{1}} u_{A F_{2}} \in E_{A F_{1}} and v_{A F_{1}} v_{A F 2} \in E_{A F_{2}}, \\ μ_{A F_{1}} * μ_{A F_{1}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{1}} (u_{A F_{1}}, u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}} (u_{A F_{2}})] \\ \lor [σ_{A F_{1}}^{'} (v_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = [σ_{A F_{1}} (u_{A F_{1}}) \lor σ_{A F_{1}}^{'} (v_{A F_{1}})] \\ \lor [σ_{A F_{1}} (u_{A F_{2}}) \lor σ_{A F_{1}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \lor (σ_{A F_{1}} * σ_{A F_{1}}^{'}) (u_{A F_{2}} v_{A F_{2}}), \\ μ_{A F_{2}} * μ_{A F_{2}}^{'} ((u_{A F_{1}}, v_{A F_{1}}) (u_{A F_{2}}, v_{A F_{2}})) \\ = μ_{A F_{2}} (u_{A F_{1}} u_{A F_{2}}) \lor μ_{A F_{2}}^{'} (v_{A F_{1}} v_{A F_{2}}) \\ = [σ_{A F_{2}}^{'} (v_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ \land [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}} (u_{A F_{2}})] \\ = [σ_{A F_{2}} (u_{A F_{1}}) \land σ_{A F_{2}}^{'} (v_{A F_{1}})] \\ \land [σ_{A F_{2}} (u_{A F_{2}}) \land σ_{A F_{2}}^{'} (v_{A F_{2}})] \\ = (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{1}} v_{A F_{1}}) \\ \land (σ_{A F_{2}} * σ_{A F_{2}}^{'}) (u_{A F_{2}} v_{A F_{2}}) . \end{array}$

Hence, G_AF_₁* G_AF_₂is also a strong IAFG.

Theorem 3.4

If G_AF_₁: (μ_AF_₁, μ_AF_₂) and of regular IAFGs if each vertex has the same degree. $G_{A F_{2}} : (μ_{A F_{1}}^{'}, μ_{A F_{2}}^{'})$ are two regular IAFGs, then G_AF_₁* G_AF_₂is also a regular IAFG.

Proof

Let G_AF_₁= (V_AF , E_AF , σ_AF , μ_AF ) and $G_{A F_{2}} = (V_{A F}^{'}, E_{A F}^{'}, σ_{A F}^{'}, μ_{A F}^{'})$ be the conormal product of IAFGs.

To prove

The conormal product of IAFGs is also an IAFG.

Let G_AF_₁= (V_AF , E_AF , σ_AF , μ_AF ) and $G_{A F_{2}} = (V_{A F}^{'}, E_{A F}^{'}, σ_{A F}^{'}, μ_{A F}^{'})$ be IAFGs and G_AF_₁* G_AF_₂be their conormal product; then, d_FG_{_AF_₁}_*G_{_AF_₂}(u_AF_{_i}, v_AF_{_j}) = ℑ for all i, j = 1, 2, ..., n. A regular fuzzy graph of degree ℑ, or a ℑ- regular fuzzy graph, is one in which each vertex has the same degree. An IAFG is a regular IAFG in this sense.

Example 3.7

See Figure 7.

In Fig. 7,

\begin{array}{l} d_{A F} (u_{A F_{1}}, v_{A F_{1}}) = 1.7, d_{A F} (u_{A F_{1}}, v_{A F_{2}}) \\ = 1.7, d_{A F} (u_{A F_{2}}, v_{A F_{1}}) \\ = 1.7, d_{A F} (u_{A F_{2}}, v_{A F_{2}}) \\ = 1.7. \end{array}

From the above we get

\begin{array}{l} d_{A F} (u_{A F_{1}}, v_{A F_{1}}) = d_{A F} (u_{A F_{1}}, v_{A F_{2}}) \\ = d_{A F} (u_{A F_{2}}, v_{A F_{1}}) \\ = d_{A F} (u_{A F_{2}}, v_{A F_{2}}) . \end{array}

Hence, G_AF_₁* G_AF_₂is a regular IAFG.

Remark 3.1

Although the conormal product of IAFGs is regular, regular fuzzy graphs need not be the conormal product of IAFGs.

Example 3.8

See Figure 8.

The above example is a regular and pseudo-regular fuzzy graph. However, it is not a conormal product of IAFGs.

4. Application

Consequently, the most effective combinations with the highest percentage of quality improvement and the lowest percentage of cost increase were discovered across all supermarkets.

Fig 1.

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