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Article

Application of Complex Neutrosophic Graphs in Hospital Infrastructure Design

1
Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, P.O. Box 93499, Riyadh 11673, Saudi Arabia
2
Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and Technology, Chennai 600002, India
3
Department of Mathematics, Faculty of Science, Ajloun National University, P.O. Box 43, Ajloun 26810, Jordan
4
Department of Mathematics, Presidency University, Bangalore 560064, India
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2024, 12(5), 719; https://doi.org/10.3390/math12050719
Submission received: 8 February 2024 / Revised: 21 February 2024 / Accepted: 26 February 2024 / Published: 28 February 2024

Abstract

:
Complex neutrosophic graphs are created by combining complex neutrosophic sets with graph theory ideas. This provides a flexible framework for tackling complex problem-solving circumstances. Various processes, such as union, join, and composition, are thoroughly investigated to improve the administration of complicated neutrosophic graphs. This research also looks into the area of complicated neutrosophic graph homeomorphisms, investigating the transformations and mappings that occur inside these structures. This investigation advances our knowledge of the intrinsic features and linkages found in complicated neutrosophic graphs. Finally, our methodology’s practical usefulness is demonstrated by its use in the design of hospital infrastructure. We illustrate the effectiveness of complicated neutrosophic graphs in addressing difficult design concerns and optimizing infrastructure solutions for practical deployment in this real-world scenario.
MSC:
O5C90; 05C76; 05E30; 05C72

1. Introduction

Many academics have disputed Zadeh’s initial claim that fuzzy sets (FSs) are a special technique to convey uncertainty that occurs in many different fields [1] and for many authors [2,3,4,5,6,7]. An FS is defined by a truth membership function with a range of [ 0 , 1 ] . To expand on the concept of FSs because it is not generally the case that an element in an FS has the falsity degree 1 μ ( x ) , Atanassov created intuitionistic fuzzy sets (IFSs) [8]. Truth and falsity membership functions are used separately to describe an IFS such that the sum of the truth and falsity degrees should not exceed more than one. Fuzzy sets simply offer the degree of membership of an element in a certain set, as opposed to IFSs, which also provide a degree of non-membership that is more or less independent from the other. Different kinds of centroid modifications of IF values were introduced by Liu et al. in their publication [9]. Additionally, Feng et al. [10] developed many additional procedures for generalized IF soft sets. To study the essence, source, and neutralities, as well as a neutrosophic set (NS), Smarandache introduced the concept of neutrosophic as a kind of IFS. An NS is defined by the membership functions for truth, indeterminacy, and falsity, respectively. An NS is used as an effective tool for mathematics to assist with the inconsistent facts that we face every day. Single-valued neutrosophic sets (SVNSs) were introduced by Smarandache [11] and Wang et al. [12] for the use of NSs in science and engineering. To manage uncertain information, an SVNS offers an alternative option. Ye [13] provided a method of decision making that uses the weighted correlation value of SVNSs to rank the options. He also gave an example to illustrate how the suggested decision-making method may be used. The SVN minimal spanning tree and associated clustering algorithm were defined by the same author [5,14,15,16,17,18,19,20], who also put out a multi-criteria decision-making approach for streamlined NSs that makes use of aggregation operators. These theories are useful in a variety of scientific fields, but they have one significant flaw, namely, the inability to represent two-dimensional events. Complex fuzzy sets (CFSs) were proposed by Ramot et al. [21] to overcome this problem. A membership function μ ( x ) that encompasses the complex plane’s unit circle and has a range greater than [ 0 , 1 ] r identifies a CF. Consequently, μ ( x ) is a complex-valued function that gives any element x in the discourse universe a membership grade of the kind v ( x ) e i α x , i = 1 . Thus, the amplitude term v ( x ) , which is in the unit interval [ 0 , 1 ] , and the phase term (periodic term), α ( x ) which is in the interval [ 0 , 2 π ] , are the two components that make up the membership function of the CFS. A CFS model differs from every other model that is accessible in the scientific field due to this phase term. The range of membership degrees for the CFS includes the complex plane with the unit circle as opposed to a fuzzy characteristic function. Ramot et al.’s [22] discussion of the complement, intersection, and union of CFSs included examples for clarity. Yazdanbakhsh and Dick [23] offered a thorough analysis of CFSs. Complex intuitionistic fuzzy sets (CIFSs) were developed by Alkouri and Salleh [24] to build upon the concept of CFSs by introducing a non-membership degree v ( x ) = s ( x ) e i β ( x ) to CFSs that are subject to the restriction r + s 1 . CIFSs are used to manage periodicity and uncertainty knowledge concurrently. Complex-valued truth and falsity membership degrees may be applied to express uncertainty in a variety of physical phenomena, including wave function, impedance in electrical engineering and problem-solving situations. In contrast to the CIFS, which contains two more phase words that are employed in concepts like projections, cylindrical extensions, and distance measurement, the CFS has just one extra aspect term. The concept of complex neutrosophic sets (CNSs), which may handle incorrect information with a periodic nature, was put out by Ali and Smarandache [25]. We can observe that ambiguity, consistency, and falsity in data are recurring phenomena, and the CNS plays a key role in resolving these sorts of problems. A complex-valued truth t ( x ) , complex-valued indeterminate i ( x ) , and complex-valued falsity f ( x ) membership function, each with a range in the complex plane that extends from [0, 1] to the unit disc, are characteristics of a CNS. They showed the use of CNSs in signal processing and offered set-theoretic operations, including complement, union, intersection, complex neutrosophic product, Cartesian product, distance measure, and equalities of CNSs. Fuzzy graphs (FGs), which were initially established by Rosenfeld [26], were required due to the ambiguity in the representation of distinct items and the unpredictable interactions between them. He investigated several fundamental bridges and trees and determined some of their qualities. Bhattacharya [27] made some comments about FGs and demonstrated that the conclusions drawn from (crisp) graph theory are not necessarily applicable to FGs. By Thirunavukarasu et al. [28], FGs were expanded to complex fuzzy graphs (CFGs) to manage hazy and uncertain interactions with a periodic nature. They examined the lower and upper energy constraints of CFGs and provided numerical examples to clarify these ideas. Pairwise relation FGs and CFGs only reveal the truth degrees and uncertainty that occur frequently, respectively [29]. Intuitionistic fuzzy graphs (IFGs) were developed by Parvathi and Karunambigai [30] to take both the truth and falsity degrees of pairwise interactions into account. Complex intuitionistic fuzzy graphs (CIFGs) were developed by Yaqoob et al. [31] to manage the periodic character of falsity degrees in IFGs. They investigated CIFG homomorphisms and supplied a CIFG application to cellular network provider businesses for the testing of their suggested strategy. Complex neutrosophic graphs (CNGs) were introduced by Yaqoob and Akram to generalize the idea of neutrosophic graphs and CIFGs [32]. They talked about certain fundamental CNG operations and described them with the aid of specific illustrations. The existence of uncertain, recurring information in network models serves as the motivating factor behind this research. When dealing with complex indeterminacy and periodicity-related behavior, a complex neutrosophic graph model is crucial. The suggested model generalizes both the complex intuitionistic fuzzy model and the complex fuzzy model. We take into account two voting processes to demonstrate the applicability of our suggested model. Assume that in the first voting process, 0.8 voters choose “yes”, 0.4 choose “no”, and 0.3 choose “undecided”, and that in the second voting process, 0.4 people choose “yes”, 0.3 choose “no”, and 0.4 choose “undecided.” These two surgeries are thought to have taken place on different days. A CFS cannot manage this circumstance since it only presents the true voter participation rate of 0.8 while failing to account for false and uncertain degrees. Similarly to this, a CIFS does not show the 0.4 undecided voters, but it does show the truth 0.6 and falsehood 0.3 degrees of voters. Using a complicated neutrosophic structure, we can currently demonstrate this information if we set the amplitude terms as the membership degrees of the first voting technique and the phase terms as the membership degrees of the second voting procedure: 0.8 e i 0.4 π , 0.3 e i 0.3 π , 0.4 e i 0.4 π . The goal of the proposed study is to cope with the periodic nature of inconsistent information already present in networks by applying an especially generalized idea of complex neutrosophic sets graphs. The suggested study addresses the limitations of earlier research and generalizes the ideas of CNGs, ICNGs, and CCNGs. The suggested model is a more open-ended framework since it takes into account the uses of graphs as well as the hesitant nature of imperfect information. Thus, combining the beneficial effects of CNSs with graph theory is the fundamental goal of this research project (Table 1).

1.1. Motivation

(1)
Traditional techniques fail to cope with the complexity and ambiguity inherent in real-world data.
(2)
A complete mathematical foundation is required for successful management of complicated data structures.
(3)
There is practical relevance in addressing complicated neutrosophic graphs, as it is critical for a variety of problem-solving scenarios across several areas.

1.2. Novelty

(1)
A unique technique is introduced by combining complicated neutrosophic sets with graph theory.
(2)
The unified framework provides a new way to describe and analyze complicated data structures.
(3)
Investigating sophisticated neutrosophic graph homomorphisms elevates the field.

1.3. Goal

(1)
We aim to create an effective approach for dealing with complicated neutrosophic graphs.
(2)
We aim to provide practical solutions to key operations such as union, join, and composition.
(3)
In the field of hospital infrastructure design, we demonstrate the approach’s success through real-world application.
The following are the contents of this essay: Complex neutrosophic graphs, union graphs, sums of graphs, complements of graphs, and compositions of graphs are defined in Section 3. We define the isomorphism of weak and strong complex neutrosophic graphs in Section 4 and go through some of their homomorphism-related characteristics. We provide some specific instances to support the suggested notions. In Section 5, the complex neutrosophic graph complement is defined and discussed. We provide a brief implementation of our suggested paradigm in Section 6. The conclusions and recommendations are covered in Section 7.

2. Preliminaries

Definition 1 ([36]). 
A neutrosophic graph denoted as G = T σ 1 , I σ 2 , F σ 3 , T μ 1 , I μ 2 , F μ 3 is denoted by G * = ( V , E ) , where V and E are the set of vertices and edges. The functions T σ 1 , I σ 2 , and F σ 3 are mappings from V to the closed interval [ 0 , 1 ] , signifying the degrees of true, intermediate, and false membership, respectively. It holds that 0 T σ 1 ( x i ) + I σ 2 ( x i ) + F σ 3 ( x i ) 3 for all x i V . Moreover, in the context of G * , the functions T μ 1 , I μ 2 , and F μ 3 are mappings from V × V to the closed interval [0, 1], representing the degrees of true, intermediate, and false membership, respectively, for each edge ( x i , x j ) E , w ` i , w ` j V × V
T μ 1 ( w ` i , w ` j ) T σ 1 ( w ` i ) T σ 1 ( w ` j ) , I μ 1 ( w ` i , w ` j ) I σ 1 ( w ` i ) I σ 1 ( w ` j ) , F μ 1 ( w ` i , w ` j ) F σ 1 ( w ` i ) F σ 1 ( w ` j ) ,
0 T σ 1 ( w ` i , w ` j ) + I σ 2 ( w ` i , w ` j ) + F σ 3 ( w ` i , w ` j ) 3 .
Definition 2. 
Let J be a universal set. A complex neutrosophic set (CNS) N can be expressed as N = { ( u , T R μ A ( w ` ) e i T R α ( w ` ) , I D μ A ( w ` ) e i I D β ( w ` ) , F S μ A ( w ` ) e i F S γ ( w ` ) | w ` J } , where i = 1 ; T R μ A ( w ` ) , I D μ A ( w ` ) , F S μ A ( w ` ) [ 0 , 1 ] are considered amplitude terms; α ( w ` ) , β ( w ` ) , γ ( w ` ) [ 0 , π ] are known as phase terms; and for every w ` J , T R μ A ( w ` ) + I D μ A ( w ` ) + F S μ A ( w ` ) 3 .
Definition 3. 
A complex neutrosophic graph with an underlying set V is defined to be a pair G = ( A , B ) , where A is a complex neutrosophic set on E V × V such that
T R μ B ( w ` , u ` ) e i α B ( w ` , u ` ) { T R μ A ( w ` ) , T R μ A ( u ` ) } e i { T R α A ( w ` ) , T R α A ( u ` ) } I N μ B ( w ` , u ` ) e i β B ( w ` , u ` ) { I N μ A ( w ` ) , I N μ A ( u ` ) } e i { I N β A ( w ` ) , I N β A ( u ` ) } F S μ B ( w ` , u ` ) e i γ B ( w ` , u ` ) { F S μ A ( w ` ) , F S μ A ( u ` ) } e i { F S γ A ( w ` ) , F S γ A ( u ` ) }
for all w ` , u ` V .
Example 1. 
Let the vertex set V = { a , b , c , d } and edge set E = { a b , a c , c d } on G * = ( V , E ) . Let A be a complex neutrosophic graph subset of V, and let B be a complex neutrosophic graph subset of E V × V , as given: A = ( ( 0.2 e i 0.3 π , 0.3 e i 0.5 π , 0.5 e i 0.6 π ) a , ( 0.9 e i 0.6 π , 0.1 e i 0.4 π , 0.3 e i 0.2 π ) b , ( 0.8 e i 0.4 π , 0.4 e i 1.3 π , 0.5 e i 0.6 π ) c , ( 0.9 e i 1.1 π , 0.3 e i 0.6 π , 0.4 e i 0.4 π ) d ) B = ( ( 0.2 e i 0.3 π , 0.1 e i 0.4 π , 0.5 e i 0.2 π ) a b , ( 0.2 e i 0.3 π , 0.3 e i 0.3 π , 0.5 e i 0.6 π ) a c , ( 0.8 e i 0.4 π , 0.1 e i 0.3 π , 0.5 e i 0.6 π ) b c , ( 0.8 e i 0.4 π , 0.3 e i 0.3 π , 0.5 e i 0.6 π ) c d ) .
1.
It can be seen from simple calculations that the graph in Figure 1 is a complex neutrosophic graph.
2.
O ( G ) = ( 2.8 e i 2.4 π , 1.1 e 2.8 π , 1.7 e 1.4 π )
3.
Each vertex in G has a degree of d e g ( a ) = 0.4 , d e g ( b ) = 1.0 , d e g ( c ) = 1.8 , and d e g ( c ) = 0.8 .
Definition 4. 
Let G = ( A , B ) be a complex neutrosophic graph. The order of a complex neutrosophic graph is defined by
O ( G ) = ( w ` V T R μ A ( w ` ) e w ` V i T R α A ( w ` ) , w ` V I N μ A ( w ` ) e w ` V i I D β A ( w ` ) , w ` V F S μ B ( w ` ) e w ` V i F S γ A ( w ` ) )
and the degree of a vertex w ` in G is defined by
D ( G ) = ( w ` E T R μ B ( w ` u ` ) e w ` E i T R α B ( w ` u ` ) , w ` E I N μ B ( w ` u ` ) e w ` E i I D β B ( w ` u ` ) , w ` E F S μ B ( w ` u ` ) e w ` E i F S γ B ( w ` u ` ) )

3. Operations of Complex Neutrosophic Graph

Definition 5. 
Let G 1 = ( A 1 , B 1 ) and G 2 = ( A 2 , B 2 ) be two complex neutrosophic graphs; the Cartesian product G 1 × G 2 of two complex neutrosophic graphs is defined as a pair G 1 × G 2 = ( A 1 × A 2 , B 1 × B 2 ) , such that
1 . T R μ A 1 × A 2 ( w ` 1 , w ` 2 ) e i α A 1 × A 2 ( w ` 1 , w ` 2 ) = { T R μ A 1 ( w ` 1 ) , T R μ A 2 ( w ` 2 ) } e i { T R α A 1 ( w ` 1 ) , T R α A 2 ( w ` 2 ) } I D μ A 1 × A 2 ( w ` 1 , w ` 2 ) e i β A 1 × A 2 ( w ` 1 , w ` 2 ) = { I D μ A 1 ( w ` 1 ) , I D μ A 2 ( w ` 2 ) } e i { I D β A 1 ( w ` 1 ) , I D β A 2 ( w ` 2 ) } F S μ A 1 × A 2 ( w ` 1 , w ` 2 ) e i γ A 1 × A 2 ( w ` 1 , w ` 2 ) = { F S μ A 1 ( w ` 1 ) , F S μ A 2 ( w ` 2 ) } e i { F S γ A 1 ( w ` 1 ) , F S γ A 2 ( w ` 2 ) }
For all w ` 1 , w ` 2 V .
2 . T R μ B 1 × B 2 ( w ` 1 , w ` 2 ) e i α B 1 × B 2 ( w ` 1 , w ` 2 ) = { T R μ B 1 ( w ` 1 ) , T R μ B 2 ( w ` 2 ) } e i { T R α B 1 ( w ` 1 ) , T R α B 2 ( w ` 2 ) } I D μ B 1 × B 2 ( w ` 1 , w ` 2 ) e i β B 1 × B 2 ( w ` 1 , w ` 2 ) = { I D μ B 1 ( w ` 1 ) , I D μ B 2 ( w ` 2 ) } e i { I D β B 1 ( w ` 1 ) , I D β B 2 ( w ` 2 ) } F S μ B 1 × B 2 ( w ` 1 , w ` 2 ) e i γ B 1 × B 2 ( w ` 1 , w ` 2 ) = { F S μ B 1 ( w ` 1 ) , F S μ B 2 ( w ` 2 ) } e i { F S γ B 1 ( w ` 1 ) , F S γ B 2 ( w ` 2 ) }
For all w ` 1 V 1 and w ` 2 u ` 2 E 2 .
3 . T R μ B 1 × B 2 ( ( w ` 1 , z ) , ( u ` 1 , t ` ) ) e i α B 1 × B 2 ( ( w ` 1 , t ` ) , ( u ` 1 , t ` ) ) = { T R μ B 1 ( ( w ` 1 u ` 1 ) , T R μ A 2 ( t ` ) ) } e i { T R α B 1 ( ( w ` 1 u ` 1 ) , T R α A 2 ( t ` ) ) } I D μ B 1 × B 2 ( ( w ` 1 , t ` ) , ( u ` 1 , t ` ) ) e i β B 1 × B 2 ( ( w ` 1 , t ` ) , ( u ` 1 , t ` ) ) = { I D μ B 1 ( ( w ` 1 u ` 1 ) , I D μ A 2 ( t ` ) ) } e i { I D β B 1 ( ( w ` 1 u ` 1 ) , I D β A 2 ( t ` ) ) } F S μ B 1 × B 2 ( ( w ` 1 , t ` ) , ( u ` 1 , t ` ) ) e i γ B 1 × B 2 ( ( w ` 1 , t ` ) , ( u ` 1 , t ` ) ) = { F S μ B 1 ( ( w ` 1 u ` 1 ) , F S μ B 2 ( ( w ` 1 u ` 1 ) } e i { F S γ B 1 ( ( w ` 1 u ` 1 ) , F S γ A 2 ( t ` ) ) }
For all t ` V 2 and w ` 1 u ` 1 E 1 .
Definition 6. 
Let G 1 × G 2 be two complex neutrosophic graphs. The degree of a vertex in G 1 × G 2 can be defined as follows: for any ( w ` 1 , w ` 2 ) V 1 × V 2 ,
D G 1 × G 2 ( w ` 1 , w ` 2 ) = ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E T R μ B 1 × B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) e i ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E T R α B 1 × B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) , ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E I D μ B 1 × B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) e i ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E I D β B 1 × B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) , ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E F S μ B 1 × B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) e i ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E F S γ B 1 × B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) )
Example 2. 
Consider the two complex neutrosophic graphs G 1 and G 2 , as shown in Figure 2 and Figure 3, and their Cartesian product, as shown in Figure 4.
Then, their corresponding Cartesian product G 1 × G 2 is shown in Figure 4.
Theorem 1. 
The Cartesian product of two complex neutrosophic graphs is a complex neutrosophic graph.
Proof. 
The conditions for A 1 × A 2 are obvious; therefore, we verify only the conditions for B 1 × B 2 . Let w ` V 1 and w ` 2 u ` 2 E 2 . Then,
T R μ B 1 × B 2 ( ( w ` , w ` 2 ) , ( w ` , u ` 2 ) ) e i α B 1 × B 2 ( ( w ` , w ` 2 ) , ( w ` , u ` 2 ) )
= { T R μ A 1 ( w ` ) , T R μ B 2 ( w ` 2 u ` 2 ) ) } e i { T R α A 1 ( w ` ) , T R α B 2 ( w ` 2 u ` 2 ) } { T R μ A 1 ( w ` ) , { T R μ B 2 ( w ` 2 , u ` 2 ) } } e i { T R μ A 1 ( w ` ) , { T R μ A 2 ( w ` 2 ) , T R μ A 2 ( u ` 2 ) } } = { { T R μ A 1 ( w ` ) , T R μ A 2 ( w ` 2 ) } , { T R μ A 1 ( w ` ) , T R μ A 2 ( u ` 2 ) } } , e i { { T R μ A 1 ( w ` ) , T R μ A 2 ( w ` 2 ) } , { T R μ A 1 ( w ` ) , T R μ A 2 ( u ` 2 ) } } = { T R μ A 1 × A 2 ( ( w ` , w ` 2 ) , T R μ A 1 × A 2 ( ( w ` , u ` 2 ) } e i { T R μ A 1 × A 2 ( ( w ` , w ` 2 ) , T R μ A 1 × A 2 ( ( w ` , u ` 2 ) }
I D μ B 1 × B 2 ( ( w ` , w ` 2 ) , ( w ` , u ` 2 ) ) e i β B 1 × B 2 ( ( w ` , w ` 2 ) , ( w ` , u ` 2 ) )
= { I D μ A 1 ( w ` ) , I D μ B 2 ( w ` 2 u ` 2 ) ) } e i { I D β A 1 ( w ` ) , I D β B 2 ( w ` 2 u ` 2 ) } { I D μ A 1 ( w ` ) , { I D μ B 2 ( w ` 2 , u ` 2 ) } } e i { I D β A 1 ( w ` ) , { I D β A 2 ( w ` 2 ) , I D β A 2 ( u ` 2 ) } } = { { I D μ A 1 ( w ` ) , I D μ A 2 ( w ` 2 ) } , { I D μ A 1 ( w ` ) , I D μ A 2 ( u ` 2 ) } } , e i { { I D β A 1 ( w ` ) , I D β A 2 ( w ` 2 ) } , { I D β A 1 ( w ` ) , I D β A 2 ( u ` 2 ) } } = { I D μ A 1 × A 2 ( ( w ` , w ` 2 ) , I D μ A 1 × A 2 ( ( w ` , u ` 2 ) } e i { I D β A 1 × A 2 ( ( w ` , w ` 2 ) , I D β A 1 × A 2 ( ( w ` , u ` 2 ) }
F S μ B 1 × B 2 ( ( w ` , w ` 2 ) , ( w ` , u ` 2 ) ) e i γ B 1 × B 2 ( ( w ` , w ` 2 ) , ( w ` , u ` 2 ) )
= { F S μ A 1 ( w ` ) , F S μ B 2 ( w ` 2 u ` 2 ) ) } e i { A S γ A 1 ( w ` ) , F S γ B 2 ( w ` 2 u ` 2 ) } { F S μ A 1 ( w ` ) , { F S μ B 2 ( w ` 2 , u ` 2 ) } } e i { F S γ A 1 ( w ` ) , { T R γ A 2 ( w ` 2 ) , T R γ A 2 ( u ` 2 ) } } = { { F S μ A 1 ( w ` ) , F S μ A 2 ( w ` 2 ) } , { F S μ A 1 ( w ` ) , F S μ A 2 ( u ` 2 ) } } , e i { { F S γ A 1 ( w ` ) , F S γ A 2 ( w ` 2 ) } , { F S γ A 1 ( w ` ) , F S γ A 2 ( u ` 2 ) } } = { F S γ A 1 × A 2 ( ( w ` , w ` 2 ) , F S γ A 1 × A 2 ( ( w ` , u ` 2 ) } e i { F S γ A 1 × A 2 ( ( w ` , w ` 2 ) , F S γ A 1 × A 2 ( ( w ` , u ` 2 ) }
Similarly, we can prove it for t ` V 2 and w ` 1 u ` 1 E 1 .
Definition 7. 
Let G 1 = ( A 1 , B 1 ) and G 2 = ( A 2 , B 2 ) be two complex neutrosophic graphs; then, the composition G 1 G 2 of two complex neutrosophic graphs is defined as a pair G 1 G 2 = ( A 1 A 2 , B 1 B 2 ), such that
1 . T R μ A 1 A 2 ( ( w ` 1 , w ` 2 ) ) e i T R α A 1 A 2 ( ( w ` 1 , w ` 2 ) ) = { T R μ A 1 ( w ` 1 ) , T R μ A 1 ( w ` 2 ) } e i { T R α A 1 ( w ` 1 ) , T R α A 2 ( w ` 2 ) } I D μ A 1 A 2 ( ( w ` 1 , w ` 2 ) ) e i I D _ A 1 A 2 ( ( w ` 1 , w ` 2 ) ) = { I D μ A 1 ( w ` 1 ) , I D μ A 1 ( w ` 2 ) } e i { I D β A 1 ( w ` 1 ) , I D β A 1 ( w ` 2 ) } F S μ A 1 A 2 ( ( w ` 1 , w ` 2 ) ) e i F S γ A 1 A 2 ( ( w ` 1 , w ` 2 ) ) = { F S μ A 1 ( w ` 1 ) , F S μ A 1 ( w ` 2 ) } e i { F S γ A 1 ( w ` 1 ) , F S γ A 2 ( w ` 2 ) }
For all w ` 1 , w ` 2 V .
2 . T R μ B 1 B 2 ( ( w ` 1 , w ` 2 ) ) e i T R α B 1 B 2 ( ( w ` 1 , w ` 2 ) ) = { T R μ B 1 ( w ` 1 ) , T R μ B 1 ( w ` 2 ) } e i { T R α B 1 ( w ` 1 ) , T R α B 2 ( w ` 2 ) } I D μ B 1 B 2 ( ( w ` 1 , w ` 2 ) ) e i I D β B 1 B 2 ( ( w ` 1 , w ` 2 ) ) = { I D μ B 1 ( w ` 1 ) , I D μ B 1 ( w ` 2 ) } e i { I D β B 1 ( w ` 1 ) , I D β B 1 ( w ` 2 ) } F S μ B 1 B 2 ( ( w ` 1 , w ` 2 ) ) e i F S γ B 1 B 2 ( ( w ` 1 , w ` 2 ) ) = { F S μ B 1 ( w ` 1 ) , F S μ B 1 ( w ` 2 ) } e i { F S γ B 1 ( w ` 1 ) , F S γ B 2 ( w ` 2 ) }
For all w ` 1 V 1 , and w ` 2 u ` 2 E 2 .
3 . T R μ B 1 B 2 ( ( w ` 1 , t ` ) ( u ` 1 , t ` ) ) e i T R α B 1 B 2 ( ( w ` 1 , t ` ) ( u ` 1 , t ` ) ) = { T R μ B 1 ( w ` 1 u ` 1 ) , T R μ A 2 ( t ` ) } e i { T R α B 1 ( w ` 1 u ` 1 ) , T R α A 2 ( t ` ) } I D μ B 1 B 2 ( ( w ` 1 , t ` ) ( u ` 1 , t ` ) ) e i I D _ B 1 B 2 ( ( w ` 1 , t ` ) ( u ` 1 , t ` ) ) = { I D μ B 1 ( w ` 1 u ` 1 ) , I D μ A 2 ( t ` ) } e i { I D β B 1 ( w ` 1 u ` 1 ) , I D β A 2 ( t ` ) } F S μ B 1 B 2 ( ( w ` 1 , t ` ) ( u ` 1 , t ` ) ) e i F S γ B 1 B 2 ( ( w ` 1 , t ` ) ( u ` 1 , t ` ) ) = { F S μ B 1 ( w ` 1 u ` 1 ) , F S μ A 2 ( t ` ) } e i { F S γ B 1 ( w ` 1 u ` 1 ) , F S γ A 2 ( t ` ) }
For all t ` V 2 , and w ` 1 u ` 1 E 1 .
4 . T R μ B 1 B 2 ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) ) e i T R α B 1 B 2 ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) ) = { T R μ A 2 ( w ` 2 ) , T R μ A 2 ( u ` 2 ) , T R μ B 1 ( w ` 1 u ` 1 ) } e i { T R α A 2 ( w ` 2 ) , T R α A 2 u ` 2 ) , T R α B 1 ( w ` 1 u ` 1 ) } I D μ B 1 B 2 ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) ) e i I D β B 1 B 2 ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) ) = { I D μ A 2 ( w ` 2 ) , I D μ A 2 ( u ` 2 ) , I D μ B 1 ( w ` 1 u ` 1 ) } e i { I D β A 2 ( w ` 2 ) , I D β A 2 u ` 2 ) , I D β B 1 ( w ` 1 u ` 1 ) } F S μ B 1 B 2 ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) ) e i F S γ B 1 B 2 ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) ) = { F S μ A 2 ( w ` 2 ) , F S μ A 2 ( u ` 2 ) , F S μ B 1 ( w ` 1 u ` 1 ) } e i { F S γ A 2 ( w ` 2 ) , F S γ A 2 u ` 2 ) , F S γ B 1 ( w ` 1 u ` 1 ) }
For all w ` 2 , u ` 2 V 2 , w ` 2 u ` 2 , and w ` 1 u ` 1 E 1 .
Definition 8. 
Let G 1 = ( A 1 , B 1 ) and G 2 = ( A 2 , B 2 ) be two complex neutrosophic graphs; then, the G 1 G 2 be two complex neutrosophic graphs. The degree of a vertex in G 1 G 2 can be defined as follows: for any ( w ` 1 , w ` 2 ) V 1 V 2 ,
D G 1 G 2 ( w ` 1 , w ` 2 ) = ( ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E T R μ B 1 B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) e i ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E T R α B 1 B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) , ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E I D μ B 1 B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) e i ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E I D β B 1 B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) , ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E F S μ B 1 B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) e i ( w ` 1 , w ` 2 ) ( u ` 1 , u ` 2 ) E F S γ B 1 B 2 ( ( w ` 1 , w ` 2 ) , ( u ` 1 , u ` 2 ) ) ) .
Example 3. 
Consider the two complex neutrosophic graphs, as shown in Figure 5, and their composition, as shown in Figure 6.
Then, their composition G 1 G 2 is shown in Figure 6.
Definition 9. 
Let   G 1 = ( A 1 , B 1 )   and   G 2 = ( A 2 , B 2 )   be two complex neutrosophic graphs; then, the union G 1 G 2 = A 1 A 2 , B 1 B 2 of two complex neutrosophic graphs is defined as follows:
1.
T R μ A 1 A 2 ( w ` ) e i T R α A 1 A 2 ( w ` ) = T R μ A 1 ( w ` ) e i T R α A 1 ( w ` )
I D μ A 1 A 2 ( w ` ) e i I D β A 1 A 2 ( w ` ) = I D μ A 1 ( w ` ) e i I D β A 1 ( w ` )
F S μ A 1 A 2 ( w ` ) e i F S γ A 1 A 2 ( w ` ) = F S μ A 1 ( w ` ) e i F S γ A 1 ( w ` ) , for w ` V 1 and w ` V 2 .
2.
T R μ A 1 A 2 ( w ` ) e i T R α A 1 A 2 ( w ` ) = T R μ A 2 ( w ` ) e i T R α A 2 ( w ` )
I D μ A 1 A 2 ( w ` ) e i I D β A 1 A 2 ( w ` ) = I D μ A 2 ( w ` ) e i I D β A 2 ( w ` )
F S μ A 1 A 2 ( w ` ) e i F S γ A 1 A 2 ( w ` ) = F S μ A 2 ( w ` ) e i F S γ A 2 ( w ` ) , for w ` V 2 and w ` V 1 .
3.
T R μ A 1 A 2 ( w ` ) e i T R α A 1 A 2 ( w ` ) = { T R μ A 2 ( w ` ) , T R μ A 2 ( w ` ) } e i { T R α A 2 ( w ` ) , T R α A 2 ( w ` ) }
I D μ A 1 A 2 ( w ` ) e i I D β A 1 A 2 ( w ` ) = { I D μ A 2 ( w ` ) , I D μ A 2 ( w ` ) } e i { I D β A 2 ( w ` ) , I D β A 2 ( w ` ) }
F S μ A 1 A 2 ( w ` ) e i F S γ A 1 A 2 ( w ` ) = { F D μ A 2 ( w ` ) , F S μ A 2 ( w ` ) } e i { F S γ A 2 ( w ` ) , F S γ A 2 ( w ` ) } ,
for w ` V 1 V 2 .
4.
T R μ B 1 B 2 ( w ` u ` ) e i T R α B 1 B 2 ( w ` u ` ) = T R μ B 1 ( w ` u ` ) e i T R α B 1 ( w ` u ` )
I D μ B 1 B 2 ( w ` u ` ) e i I D β B 1 B 2 ( w ` u ` ) = I D μ B 1 ( w ` u ` ) e i I D β B 1 ( w ` u ` )
F S μ B 1 B 2 ( w ` u ` ) e i F S γ B 1 B 2 ( w ` u ` ) = F S γ B 1 ( w ` u ` ) e i F S γ B 1 ( w ` u ` ) , for w ` u ` E 1 and w ` u ` E 2 .
5.
T R μ B 1 B 2 ( w ` u ` ) e i T R α B 1 B 2 ( w ` u ` ) = T R μ B 2 ( w ` u ` ) e i T R α B 2 ( w ` u ` )
I D μ B 1 B 2 ( w ` u ` ) e i I D β B 1 B 2 ( w ` u ` ) = I D μ B 2 ( w ` u ` ) e i I D β B 2 ( w ` u ` )
F S μ B 1 B 2 ( w ` u ` ) e i F S γ B 1 B 2 ( w ` u ` ) = F S μ B 2 ( w ` u ` ) e i F S γ B 2 ( w ` u ` ) , for w ` u ` E 2 and w ` u ` E 1 .
6.
T R μ B 1 B 2 ( w ` u ` ) e i T R α B 1 B 2 ( w ` u ` ) = { T R μ B 2 ( w ` u ` ) , T R μ B 2 ( w ` u ` ) } e i { T R α B 2 ( w ` u ` ) , T R α B 2 ( w ` u ` ) }
I D μ B 1 B 2 ( w ` u ` ) e i I D β B 1 B 2 ( w ` u ` ) = { I D μ B 2 ( w ` u ` ) , I D μ B 2 ( w ` u ` ) } e i { I D β B 2 ( w ` u ` ) , I D β B 2 ( w ` u ` ) }
F S μ B 1 B 2 ( w ` u ` ) e i F S γ B 1 B 2 ( w ` u ` ) = { I D μ B 2 ( w ` u ` ) , F S μ B 2 ( w ` u ` ) } e i { F S γ B 2 ( w ` u ` ) , F S γ B 2 ( w ` u ` ) } ,
for w ` u ` E 1 E 2 .
Example 4. 
Consider the two complex neutrosophic graphs, as shown in Figure 7, and their union, as shown in Figure 8.
Definition 10. 
Let G 1 = ( A 1 , B 1 ) and G 2 = ( A 2 , B 2 ) be two complex neutrosophic graphs; then, the join G 1 + G 2 = ( A 1 + A 2 , B 1 + B 2 ) of the two complex neutrosophic graphs, where V 1 V 2 = , is defined as follows:
1.
T R μ A 1 | A 2 ( w ` ) e i T R α A 1 + A 2 ( w ` ) = T R μ A 1 A 2 ( w ` ) e i T R α A 1 A 2 ( w ` )
I D μ A 1 | A 2 ( w ` ) e i I D β A 1 + A 2 ( w ` ) = I D μ A 1 A 2 ( w ` ) e i I D β A 1 A 2 ( w ` )
F S μ A 1 + A 2 ( w ` ) e i F S γ A 1 + A 2 ( w ` ) = F S μ A 1 A 2 ( w ` ) e i F S γ A 1 A 2 ( w ` ) , if w ` V 1 V 2 .
2.
T R μ B 1 | B 2 ( w ` u ` ) e i T R α B 1 + B 2 ( w ` ) = T R μ A 1 A 2 ( w ` u ` ) e i T R α A 1 A 2 ( w ` )
I D μ B 1 | B 2 ( w ` u ` ) e i I D β B 1 + B 2 ( w ` ) = I D μ A 1 A 2 ( w ` u ` ) e i I D β A 1 A 2 ( w ` )
F S μ B 1 + B 2 ( w ` u ` ) e i F S γ B 1 + B 2 ( w ` ) = F S μ A 1 A 2 ( w ` u ` ) e i F S γ A 1 A 2 ( w ` ) , for w ` u ` E 1 E 2 .
3.
T R μ B 1 + B 2 ( w ` u ` ) e i T R α B 1 + B 2 ( w ` u ` ) = { T R μ A 1 ( w ` ) , T R μ A 1 ( u ` ) } e i { T R α A 1 ( w ` ) , T R α A 1 ( u ` ) }
I D μ B 1 + B 2 ( w ` u ` ) e i I D β B 1 + B 2 ( w ` u ` ) = { I D μ A 1 ( w ` ) , I D μ A 1 ( u ` ) } e i { I D β A 1 ( w ` ) , I D β A 1 ( u ` ) }
F S μ B 1 + B 2 ( w ` u ` ) e i F S γ B 1 + B 2 ( w ` u ` ) = { F S μ A 1 ( w ` ) , F S μ A 1 ( u ` ) } e i { F S γ A 1 ( w ` ) , F S γ A 1 ( u ` ) } .
Here, w ` u ` E , where E is the set of all edges joining the vertices of V 1 and V 2 .
Example 5. 
Consider the two complex neutrosophic graphs, as shown in Figure 9, and their join, as shown in Figure 10.
Proposition 1. 
The join of two complex neutrosophic graphs is a neutrosophic graph.
Theorem 2. 
Let G 1 = ( A 1 , B 1 ) and G 2 = ( A 2 , B 2 ) be complex neutrosophic graphs of the graphs G 1 * and G 2 * and let V 1 V 2 = . Then, union G 1 G 2 = ( A 1 A 2 , B 1 B 2 ) is a complex neutrosophic graph of G * if and only if G 1 and G 2 are complex neutrosophic graphs of the graphs G 1 * and G 2 * , respectively.
Proof. 
Suppose that G 1 G 2 is a complex neutrosophic graph. Let w ` u ` E 1 and w ` , u ` V 1 V 2 . Thus,
T R μ B 1 ( w ` u ` ) e i T R α B 1 ( w ` ) = T R μ B 1 B 2 ( w ` u ` ) e i T R α B 1 B 2 ( w ` ) = { T R μ A 1 A 2 ( w ` ) , T R μ A 1 A 2 ( u ` ) } e i { T R α A 1 A 2 ( w ` ) , T R α A 1 A 2 ( u ` ) } = { T R μ A 1 ( w ` ) , T R μ A 1 ( u ` ) } e i { T R α A 1 ( w ` ) , T R α A 1 ( u ` ) } I D μ B 1 ( w ` u ` ) e i I D β B 1 ( w ` ) = I D μ B 1 B 2 ( w ` u ` ) e i I D β B 1 B 2 ( w ` ) = { I D μ A 1 A 2 ( w ` ) , I D μ A 1 A 2 ( u ` ) } e i { I D β A 1 A 2 ( w ` ) , I D α A 1 A 2 ( u ` ) } = { I D μ A 1 ( w ` ) , I D μ A 1 ( u ` ) } e i { I D β A 1 ( w ` ) , I D β A 1 ( u ` ) } F S μ B 1 ( w ` u ` ) e i F S γ B 1 ( w ` ) = F S μ B 1 B 2 ( w ` u ` ) e i F S γ B 1 B 2 ( w ` ) = { F S μ A 1 A 2 ( w ` ) , F S μ A 1 A 2 ( u ` ) } e i { F S γ A 1 A 2 ( w ` ) , F S γ A 1 A 2 ( u ` ) } = { F S γ A 1 ( w ` ) , F S γ A 1 ( u ` ) } e i { F S γ A 1 ( w ` ) , F S γ A 1 ( u ` ) }
This shows that G 1 = ( A 1 , B 1 ) is a complex neutrosophic graph. Similarly, we can show that G 2 = ( A 2 , B 2 ) is a complex neutrosophic graph. The converse part is obvious. □
Theorem 3. 
Let G 1 = ( A 1 , B 1 ) and G 2 = ( A 2 , B 2 ) be complex neutrosophic graphs of the graphs G 1 * and G 2 * , and let V 1 V 2 = . Then, join G 1 + G 2 = ( A 1 + A 2 , B 1 + B 2 ) is a complex neutrosophic graph of G * if and only if G 1 and G 2 are complex neutrosophic graphs of the graphs G 1 * and G 2 * , respectively.

4. Isomorphism of Complex Neutrosophic Graphs

Definition 11. 
Let G 1 = ( A 1 , B 1 ) and G 2 = ( A 2 , B 2 ) be two complex neutrosophic graphs. A homomorphism g : G 1 G 2 is a mapping g : V 1 V 2 such that:
1 . T R μ A 1 ( w ` 1 ) e i T R α A 1 ( w ` 1 ) T R μ A 2 ( φ ( w ` 1 ) ) e i T R α A 2 ( φ ( w ` 1 ) ) I D μ A 1 ( w ` 1 ) e i I D β A 1 ( w ` 1 ) I D μ A 2 ( φ ( w ` 1 ) ) e i I D β A 2 ( φ ( w ` 1 ) ) f o r a l l w ` 1 u ` 1 V 1 F S μ A 1 ( w ` 1 ) e i F S γ A 1 ( w ` 1 ) F S μ A 2 ( φ ( w ` 1 ) ) e i F S γ A 2 ( φ ( w ` 1 ) )
2 . T R μ B 1 ( w ` 1 u ` 1 ) e i T R α B 1 ( w ` 1 u ` 1 ) T R μ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) e i T R α B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) I D μ B 1 ( w ` 1 u ` 1 ) e i I D β B 1 ( w ` 1 u ` 1 ) I D μ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) e i I D β B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) f o r a l l w ` 1 u ` 1 E 1 F S μ B 1 ( w ` 1 u ` 1 ) e i I D γ B 1 ( w ` 1 u ` 1 ) F S μ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) e i F S γ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) )
A bijective homomorphism with the property
3 . T R μ A 1 ( w ` 1 ) e i T R α A 1 ( w ` 1 ) = T R μ A 2 ( φ ( w ` 1 ) ) e i T R α A 2 ( φ ( w ` 1 ) ) I D μ A 1 ( w ` 1 ) e i I D β A 1 ( w ` 1 ) = I D μ A 2 ( φ ( w ` 1 ) ) e i I D β A 2 ( φ ( w ` 1 ) ) f o r a l l w ` 1 V 1 F S μ A 1 ( w ` 1 ) e i F S γ A 1 ( w ` 1 ) = F S μ A 2 ( φ ( w ` 1 ) ) e i F S γ A 2 ( φ ( w ` 1 ) )
is called a week isomorphism. A bijective homomorphism with the property
4 . T R μ B 1 ( w ` 1 u ` 1 ) e i T R α B 1 ( w ` 1 u ` 1 ) = T R μ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) e i T R α B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) I D μ B 1 ( w ` 1 u ` 1 ) e i I D β B 1 ( w ` 1 u ` 1 ) = I D μ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) e i I D β B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) f o r a l l w ` 1 u ` 1 E 1 F S μ B 1 ( w ` 1 u ` 1 ) e i I D γ B 1 ( w ` 1 u ` 1 ) = F S μ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) ) e i F S γ B 2 ( φ ( w ` 1 ) φ ( u ` 1 ) )
is called a strong co-isomorphism. A bijective mapping φ : G 1 G 2 satisfying 3 and 4 is called an isomorphism.
Theorem 4. 
An isomorphism between complex neutrosophic graphs is an equivalence relation.
Proof. 
The reflexivity and symmetry are obvious. To prove the transitivity, we let ψ : V 1 V 2 and φ : V 2 V 3 be the isomorphism of G 1 onto G 2 and G 2 onto G 3 , respectively. Then, φ ψ : V 1 V 3 is a bijective map from V 1 to V 3 , where ( φ ψ ) ( w ` 1 ) = φ ( ψ ( w ` 1 ) ) for all w ` 1 V 1 . Since a map ψ : V 1 V 2 defined by ψ ( w ` 1 ) = w ` 2 for w ` 1 V 1 is an isomorphism, here,
T R μ A 1 ( w ` 1 ) e i T R α A 1 ( w ` 1 ) = T R μ A 2 ( ψ ( w ` 1 ) ) e i T R α A 2 ( ψ ( w ` 1 ) ) (1) = T R μ A 2 ( w ` 2 ) e i T R α A 2 ( w ` 2 ) , f o r a l l w ` 1 V 1 I D μ A 1 ( w ` 1 ) e i I D β A 1 ( w ` 1 ) = I D μ A 2 ( ψ ( w ` 1 ) ) e i I D β A 2 ( ψ ( w ` 1 ) ) (2) = I D μ A 2 ( w ` 2 ) e i I D β A 2 ( w ` 2 ) , f o r a l l w ` 1 V 1 F S μ A 1 ( w ` 1 ) e i F S γ A 1 ( w ` 1 ) = F S μ A 2 ( φ ( w ` 1 ) ) e i F S γ A 2 ( φ ( w ` 1 ) ) (3) = F S μ A 2 ( w ` 2 ) e i F S γ A 2 ( w ` 2 ) , f o r a l l w ` 1 V 1 T R μ B 1 ( w ` 1 u ` 1 ) e i T R α B 1 ( w ` 1 u ` 1 ) = T R μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i T R α B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) (4) = T R μ B 2 ( w ` 2 u ` 2 ) e i T R α B 2 ( w ` 2 u ` 2 ) , f o r a l l w ` 1 u ` 1 E 1 I D μ B 1 ( w ` 1 u ` 1 ) e i I D β B 1 ( w ` 1 u ` 1 ) = I D μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i I D β B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) (5) = I D μ B 2 ( w ` 2 u ` 2 ) e i I D β B 2 ( w ` 2 u ` 2 ) , f o r a l l w ` 1 u ` 1 E 1 F S μ B 1 ( w ` 1 u ` 1 ) e i F S γ B 1 ( w ` 1 u ` 1 ) = F S μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i F S γ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) (6) = F S μ B 2 ( w ` 2 u ` 2 ) e i F S γ B 2 ( w ` 2 u ` 2 ) , f o r a l l w ` 1 u ` 1 E 1 .
Since a map φ : V 2 V 3 defined by φ ( w ` 2 ) = w ` 3 for w ` 2 V 2 is an isomorphism,
T R μ A 2 ( w ` 2 ) e i T R α A 2 ( w ` 2 ) = T R μ A 3 ( ψ ( w ` 2 ) ) e i T R α A 3 ( ψ ( w ` 2 ) ) (7) = T R μ A 3 ( w ` 3 ) e i T R α A 3 ( w ` 3 ) , f o r a l l w ` 2 V 1 I D μ A 2 ( w ` 2 ) e i I D β A 2 ( w ` 2 ) = I D μ A 3 ( ψ ( w ` 2 ) ) e i I D β A 3 ( ψ ( w ` 2 ) ) (8) = I D μ A 3 ( w ` 3 ) e i I D β A 3 ( w ` 3 ) , f o r a l l w ` 2 V 1 F S μ A 2 ( w ` 2 ) e i F S γ A 2 ( w ` 2 ) = F S μ A 3 ( φ ( w ` 2 ) ) e i F S γ A 3 ( φ ( w ` 2 ) ) (9) = F S μ A 3 ( w ` 3 ) e i F S γ A 3 ( w ` 3 ) , f o r a l l w ` 2 V 1 T R μ B 2 ( w ` 2 u ` 2 ) e i T R α B 2 ( w ` 2 u ` 2 ) = T R μ B 3 ( ψ ( w ` 2 ) ψ ( u ` 2 ) ) e i T R α B 3 ( ψ ( w ` 2 ) ψ ( u ` 2 ) ) (10) = T R μ B 3 ( w ` 3 u ` 3 ) e i T R α B 3 ( w ` 3 u ` 3 ) , f o r a l l w ` 2 u ` 2 E 1 I D μ B 2 ( w ` 2 u ` 2 ) e i I D β B 2 ( w ` 2 u ` 2 ) = I D μ B 3 ( ψ ( w ` 2 ) ψ ( u ` 2 ) ) e i I D β B 3 ( ψ ( w ` 2 ) ψ ( u ` 2 ) ) (11) = I D μ B 3 ( w ` 3 u ` 3 ) e i I D β B 3 ( w ` 3 u ` 3 ) , f o r a l l w ` 2 u ` 2 E 1 F S μ B 2 ( w ` 2 u ` 2 ) e i F S γ B 2 ( w ` 2 u ` 2 ) = F S μ B 3 ( φ ( w ` 2 φ ( u ` 2 ) ) e i F S γ B 3 ( φ ( w ` 2 ) φ ( u ` 2 ) ) (12) = F S μ B 3 ( w ` 3 u ` 3 ) e i F S γ B 3 ( w ` 3 u ` 3 ) , f o r a l l w ` 2 u ` 2 E 1 .
From (1), (7), and ψ ( w ` 1 ) = w ` 2 , w ` 1 V 1 , we have
T R μ A 1 ( w ` 1 ) e i T R α A 1 ( w ` 1 ) = T R μ A 2 ( ψ ( w ` 1 ) ) e i T R α A 2 ( ψ ( w ` 1 ) ) = T R μ A 2 ( w ` 2 ) e i T R α A 2 ( w ` 2 ) = T R μ A 2 ( φ ( w ` 2 ) ) e i T R α A 3 ( φ ( w ` 2 ) ) = T R μ A 2 ( φ ( ψ ( w ` 1 ) ) ) e i T R α A 3 ( φ ( ψ ( w ` 2 ) ) ) .
From (2), (8), and ψ ( w ` 1 ) = w ` 2 , w ` 1 V 1 , we have
I D μ A 1 ( w ` 1 ) e i I D β A 1 ( w ` 1 ) = I D μ A 2 ( ψ ( w ` 1 ) ) e i I D β A 2 ( ψ ( w ` 1 ) ) = I D μ A 2 ( w ` 2 ) e i I D β A 2 ( w ` 2 ) = I D μ A 2 ( φ ( w ` 2 ) ) e i I D β A 3 ( φ ( w ` 2 ) ) = I D μ A 2 ( φ ( ψ ( w ` 1 ) ) ) e i I D β A 3 ( φ ( ψ ( w ` 2 ) ) ) .
From (3), (9), and ψ ( w ` 1 ) = w ` 2 , w ` 1 V 1 , we have
F S μ A 1 ( w ` 1 ) e i F S γ A 1 ( w ` 1 ) = F S μ A 2 ( ψ ( w ` 1 ) ) e i F S γ A 2 ( ψ ( w ` 1 ) ) = F S μ A 2 ( w ` 2 ) e i F S γ A 2 ( w ` 2 ) = F S μ A 2 ( φ ( w ` 2 ) ) e i F S γ A 3 ( φ ( w ` 2 ) ) = F S μ A 2 ( φ ( ψ ( w ` 1 ) ) ) e i F S γ A 3 ( φ ( ψ ( w ` 2 ) ) ) .
From (4) and (10), we have
T R μ B 1 ( w ` 1 u ` 1 ) e i T R α B 1 ( w ` 1 u ` 1 ) = T R μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i T R α B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) = T R μ B 2 ( w ` 2 u ` 2 ) e i T R α B 2 ( w ` 2 u ` 2 ) = T R μ B 3 ( φ ( w ` 2 ) φ ( u ` 2 ) ) e i T R α B 3 ( φ ( w ` 2 ) φ ( u ` 2 ) ) = T R μ B 3 ( φ ( ψ ( w ` 2 ) ) φ ( ψ ( u ` 2 ) ) ) e i T R α B 3 ( φ ( ψ ( w ` 2 ) ) φ ( ψ ( u ` 2 ) ) ) .
From (5) and (11), we have
I D μ B 1 ( w ` 1 u ` 1 ) e i I D β B 1 ( w ` 1 u ` 1 ) = I D μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i I D β B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) = I D μ B 2 ( w ` 2 u ` 2 ) e i I D β B 2 ( w ` 2 u ` 2 ) = I D μ B 3 ( φ ( w ` 2 ) φ ( u ` 2 ) ) e i I D β B 3 ( φ ( w ` 2 ) φ ( u ` 2 ) ) = I D μ B 3 ( φ ( ψ ( w ` 2 ) ) φ ( ψ ( u ` 2 ) ) ) e i I D β B 3 ( φ ( ψ ( w ` 2 ) ) φ ( ψ ( u ` 2 ) ) ) .
From (6) and (12), we have
F S μ B 1 ( w ` 1 u ` 1 ) e i F S γ B 1 ( w ` 1 u ` 1 ) = F S μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i F S γ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) = F S μ B 2 ( w ` 2 u ` 2 ) e i F S γ B 2 ( w ` 2 u ` 2 ) = F S μ B 3 ( φ ( w ` 2 ) φ ( u ` 2 ) ) e i F S γ B 3 ( φ ( w ` 2 ) φ ( u ` 2 ) ) = F S μ B 3 ( φ ( ψ ( w ` 2 ) ) φ ( ψ ( u ` 2 ) ) ) e i F S γ B 3 ( φ ( ψ ( w ` 2 ) ) φ ( ψ ( u ` 2 ) ) ) .
For all w ` 1 u ` 2 E 1 . Here, φ ψ is an isomorphism between G 1 and G 3 . This completes the proof. □
Theorem 5. 
A weak isomorphism between complex neutrosophic graphs is a partial ordering relation.
Proof. 
The reflexivity and transitivity are obvious. To prove the anti-symmetry, we let ψ : V 1 V 2 be a strong isomorphism of G 1 onto G 2 . Then, ψ is a bijective map defined by ψ ( w ` 1 ) = w ` 2 for all w ` 2 V 1 , satisfying
T R μ A 2 ( w ` 1 ) e i T R α A 2 ( w ` 1 ) = T R μ A 2 ( ψ ( w ` 1 ) ) e i T R α A 2 ( ψ ( w ` 1 ) ) , f o r a l l w ` 1 V 1 , I D μ A 2 ( w ` 1 ) e i T R β A 2 ( w ` 1 ) = I D μ A 2 ( ψ ( w ` 1 ) ) e i I D β A 2 ( ψ ( w ` 1 ) ) , f o r a l l w ` 1 V 1 , F S μ A 2 ( w ` 1 ) e i F S γ A 2 ( w ` 1 ) = F S μ A 2 ( ψ ( w ` 1 ) ) e i F S α A 2 ( ψ ( w ` 1 ) ) , f o r a l l w ` 1 V 1
T R μ B 1 ( w ` 1 u ` 1 ) e i T R α B 1 ( w ` 1 u ` 1 ) T R μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i T R α B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) , f o r a l l w ` 1 u ` 1 E 1
I D μ B 1 ( w ` 1 u ` 1 ) e i I D β B 1 ( w ` 1 u ` 1 ) I D μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i I D β B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) , f o r a l l w ` 1 u ` 1 E 1
F S μ B 1 ( w ` 1 u ` 1 ) e i F S γ B 1 ( w ` 1 u ` 1 ) I D μ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) e i I D γ B 2 ( ψ ( w ` 1 ) ψ ( u ` 1 ) ) , f o r a l l w ` 1 u ` 1 E 1 .
Let φ : V 2 V 1 be a strong isomorphism of G 2 and G 1 . Then, g is a bijective map defined by φ ( w ` 2 ) = w ` 1 for all w ` 2 V 2 , satisfying
T R μ A 2 ( w ` 2 ) e i T R α A 2 ( w ` 2 ) = T R μ A 1 ( φ ( w ` 2 ) ) e i T R α A 1 ( φ ( w ` 2 ) ) , f o r a l l w ` 2 V 2 I D μ A 2 ( w ` 2 ) e i I D β A 2 ( w ` 2 ) = I D μ A 1 ( φ ( w ` 2 ) ) e i I D β A 1 ( φ ( w ` 2 ) ) , f o r a l l w ` 2 V 2 F S μ A 2 ( w ` 2 ) e i F S γ A 2 ( w ` 2 ) = F S μ A 1 ( φ ( w ` 2 ) ) e i F S γ A 1 ( φ ( w ` 2 ) ) , f o r a l l w ` 2 V 2
T R μ B 2 ( w ` 2 u ` 2 ) e i T R α B 2 ( w ` 2 u ` 2 ) T R μ B 2 ( φ ( w ` 2 ) φ ( u ` 2 ) ) e i T R α B 2 ( φ ( w ` 2 ) φ ( u ` 2 ) ) , f o r a l l w ` 2 u ` 2 E 1
I D μ B 2 ( w ` 2 u ` 2 ) e i I D β B 2 ( w ` 2 u ` 2 ) I D μ B 2 ( φ ( w ` 2 ) φ ( u ` 2 ) ) e i I D β B 2 ( φ ( w ` 2 ) φ ( u ` 2 ) ) , f o r a l l w ` 2 u ` 2 E 1
F S μ B 2 ( w ` 2 u ` 2 ) e i F S γ B 2 ( w ` 2 u ` 2 ) I D μ B 2 ( φ ( w ` 2 ) φ ( u ` 2 ) ) e i I D γ B 2 ( φ ( w ` 2 ) φ ( u ` 2 ) ) , f o r a l l w ` 2 u ` 2 E 1 .
Inequalities (13)–(18) hold on the finite sets V 1 and V 2 only when G 1 and G 2 have the same number of edges and the corresponding edges have the same weight. Hence, G 1 and G 2 are identical. Therefore, φ ψ is a strong isomorphism between G 1 and G 3 . This completes the proof. □

5. Complement of Complex Neutrosophic Graphs

Definition 12. 
The complement of a week complex neutrosophic graph G = ( A , B ) of G * = ( V , E ) is a weak complex neutrosophic graph G ¯ = ( A ¯ , B ¯ ) on G * ¯ , as defined by
1. V ¯ = V
2 . T R μ A ¯ ( w ` ) e i T R α A ¯ ( w ` ) = T R μ A ( w ` ) e i T R α A ( w ` ) I D μ A ¯ ( w ` ) e i I D β A ¯ ( w ` ) = I D μ A ( w ` ) e i I D β A ( w ` ) f o r a l l w ` V F S μ A ¯ ( w ` ) e i F S γ A ¯ ( w ` ) = F S μ A ( w ` ) e i F S γ A ( w ` )
3 . T R μ B ¯ ( w ` u ` ) e i T R α B ¯ ( w ` u ` ) = 0 i f T R μ B ( w ` u ` ) e i T R α B ( w ` u ` ) 0 { T R μ A ( w ` ) , T R μ A ( u ` ) } e i { T R α A ( w ` ) , T R α A ( u ` ) } i f T R μ B ( w ` u ` ) e i T R α B ( w ` u ` ) = 0 I D μ B ¯ ( w ` u ` ) e i T R β B ¯ ( w ` u ` ) = 0 i f I D μ B ( w ` u ` ) e i I D β B ( w ` u ` ) 0 { I D μ A ( w ` ) , I D μ A ( u ` ) } , e i { I D β B ( w ` ) , I D β B ( u ` ) } i f I D μ B ( w ` u ` ) e i I D β B ( w ` u ` ) = 0 F S μ B ¯ ( w ` u ` ) e i F S γ B ¯ ( w ` u ` ) = 0 i f F S μ B ( w ` u ` ) e i F S γ B ( w ` u ` ) 0 { F S μ A ( w ` ) , F S μ A ( u ` ) } , e i { F S γ B ( w ` ) , F S γ B ( u ` ) } i f F S μ B ( w ` u ` ) e i F S