Hybrid Obfuscation of Encryption

*Asma'a Al-Hakimi and Abu Bakar Md Sultan*

### **Abstract**

Obfuscation is an encryption method. It allows the programmer to reform the code for protection. Obfuscation has promising chance to change the way of coding, where the programmer has the ability to program with any language, not necessarily English. Obfuscation, Unicode, and mathematical equations have the possibility to change strings and identifiers and to hide secret algorithms and business rules. Special dictionary is used for the string obfuscation to hide the logic of the program. The hybrid obfuscation technique will be implemented into a tool that automatically converts the code. It can be can be used for games and mobile applications for protection. With obfuscation, the application still has the ability to perform sufficiently and provide the desired output without any delays in performing timing. After obfuscating the source file, reverser still has the ability to break the object file but will not be able to read or understand and when obfuscation technique is complicated, reversing leads to error where original code disappears. In this chapter, hybrid obfuscation will be presented with examples, and obfuscation table is presented as well for future use.

**Keywords:** obfuscation, encryption, anti-reverse engineering, reverse engineering, hacking prevention

### **1. Introduction**

Obfuscation is considered as anti-reverse engineering to prevent hacking and code theft. It mainly works in the source file to change the form of the code to confuse the reverser or the hacker and also to prevent the compiler from reading the hacked code. The obfuscation technique converts all the code into unreadable text, but it functions like the original code and produces the same output. There are many forms of obfuscation, such as string encryption, hiding, changing identifier names, junk code obfuscation, packing, byte code obfuscation, string encryption, stealth obfuscation, chaotic encryption, and junk code obfuscation. In this chapter, a new method of obfuscation is introduced to produce a different kind of chaotic code that is almost impossible to read and understand but still produces the desired output [1]. For this case, Java code will be used to implement and test the code. **Figure 1** presents most common categories of obfuscation.

The decision of using any category of any obfuscation or merging them together depends on the level of complication the programmer or author wish to make the code and also depends on the part that wanted to be obfuscated, such as a business rule or an algorithm that is important to the code or the business of the company that is developing the code. Following section describes the categories of the obfuscation.

**Figure 1.** *Obfuscation categories.*

### **1.1 Lexical obfuscation**

This technique is used to transform or alter the compiler information. Other information will be removed from the byte code such as comments and identifiers. Programmers use this technique alone without merging it with any other technique that does not guarantee protection [2].

### **1.2 Stealthy obfuscation**

This is an obfuscator that contains several obfuscation techniques to obfuscate the code when it is read. Stealthy obfuscation provides a sort of false sense of the actual program structure. This technique works with the assembly file. After applying this technique, two files are created, one of them is the assembly file and the second file is the obfuscated file. In this technique, the source file is not encrypted. However, there is a possibility that the reverser will be confused when reading the code [3].

### **1.3 Key hiding obfuscation**

This technique is used to protect intellectual property, and it is based on key hiding. This method should not be used alone. It must be combined with another technique to provide more protection. A symmetric mechanism is used to combine with key hiding. Key hiding focuses on executable software. The software protection key is then encrypted with a threshold key to make it difficult for the reverser to find it and break the code. This technique focuses on executable software and leaves the source code and class file as they are [4].

### **1.4 Junk obfuscation**

This technique converts identifiers into an unreadable but performs and produces output that can be read by the compiler. This technique is particularly useful when combined with another technique to increase code security level. Junk obfuscation misleads and does not allow the reverser to read the identifier or understand what is the purpose for it but can only see the output [5].

### **1.5 Control obfuscation**

This technique hides the actual flow of the code and creates a fake one. It controls flow by using a structured exception handling mechanism in Windows. It disguises the control flow by adding exception statements. When the exception occurs, the exception handler is called and the flow of execution is changed in the exception handler. This technique is provided by Windows operating system for exception handling. It focuses on basic blocks that can be obfuscated by further splitting them into few parts. This technique does not modify the class file. It changes the source file, which is a good point to protect the code. However, newer reverse engineering tools can change the flow of the software and even create new flows [6].

### **1.6 String obfuscation**

This technique uses many approaches such as encryption, mathematical equations, or chaotic obfuscation. It depends on the programmer to decide how complicated the obfuscated strings should be. String obfuscation is very effective in protecting the code from theft. When the string is obfuscated, only the compiler can read and output it, while it becomes unreadable to humans [7].

### **1.7 Chaotic obfuscation**

Here, a mathematical modeling is used for string encoding. It is up to the programmer to determine the form of the equation for encrypting the string. The programmer has the option to encrypt all strings or some of them. Chaos theory involves stems generated from mathematical equations that produce random numbers and chaos that are not readable by the user; however, the chaos sequences are readable by the compiler at runtime. The chaotic equations are deterministic by nature, which means that they go into saturation after several iterations at a single value. **Figure 2** presents sample of string after applying chaotic obfuscation [8].

### **1.8 Cipher algorithm**

This technique uses session keys instead of permeant. Session keys are symmetric keys that are regenerated for each encryption. The keys in Cipher are automatically generated in the algorithm itself to prevent the inverter from guessing the permanent key. The user using Cipher can purchase a permanent key from the developer; however, the key can be compromised by determined reversers. String encryption in Cipher follows certain steps: the first step is to choose the secret key, which can be an x-value. The second step is to assign the equation used for encryption that will to cause the series of chaos. The encryption is a secret function that only the developer

#### **Figure 2.** *String chaotic obfuscation.*

knows. The third step is the iteration of x-value to produce the ciphertext. **Figure 3** presents sample of cipher obfuscation [9].

### *1.8.1 Cipher block chaining*

This technique divides the data or code into blocks of bits and chains. The encrypted data are blocked together to avoid eavesdroppers from inserting their own blocks of bits among the blocks of encrypted code. A mathematical equation is used for the for the Cipher block changing, and the equation is given as follows:

$$\mathbf{C\_1 = \mathbf{ek}(m\_1 \mathbf{XORIV})} \tag{1}$$

*Hybrid Obfuscation of Encryption DOI: http://dx.doi.org/10.5772/intechopen.109662*

$$\mathbf{C}\_{1} = \mathbf{e} \mathbf{k}(m\_{1} \mathbf{X} \mathbf{O} \mathbf{R} \mathbf{C}\_{i-1}) \quad for \text{ i} > 1 \tag{2}$$

The technique involves a specific (N) value passed between the plaintext to ensure that the ciphertext blocks look different. The N value is the second layer of encryption, while the first layer of encryption is done by the secret key. Each generated text is encrypted with the same secret key. If an error occurs in one of the blocks, it will also occur in all other blocks that follow the affected block [10].

### **1.9 Symmetric cipher**

The symmetric Cipher is well known and common for string encryption and decryption. It can encrypt large data. This technique uses one key of encryption and decryption. The reverser or the end user must find the meaning of exchanging the key securely. Without the key of the encryption algorithm, the reverser will not be able to reveal or translate or decrypt the encrypted string. Below figure illustrates the sample of code before and after applying Cipher algorithm. **Figure 4** presents sample of symmetric obfuscation after applying on J**ava code [11]**.

### **2. Discussion of current obfuscation techniques**

Obfuscation techniques based on the identifiers renaming have been recently presented. Such techniques can be classified as a form of layout obfuscation, since they reduce the information available to a human reader which examines the target program, or of preventive obfuscation since they aim to prevent the decompilation from producing original code with full meaning or to produce an incorrect Java source code. Such techniques try to hide the structure and the behavior information embedded in the identifiers of a Java program by replacing them with meaningless or confounding identifiers to make more difficult the task of the reverse engineer. It is worth to notice that the information associated with an identifier is completely lost after the renaming [12]. By replacing the identifiers of a Java bytecode with new ones that are illegal with respect to the Java language specification, such techniques try to make the decompilation process impossible or make the decompiler return unusable source code. After applying any obfuscation technique, it is very important to test the program, especially if there are many loops that execute many times such as games or algorithmically intensive method. The constant test is to ensure that all obfuscated

**Figure 4.** *Symmetric cipher obfuscation.*

parts are well working with no error recorded. During test for several obfuscation techniques, there were several limitations that can be a vulnerable entrance for the any strong decompiler [13].

Control flow obfuscation is only able to defeat decompilers when the method contains basic blocks of code. This technique is not fully deterministic, whereby it is only applicable to methods if the developer sees the performance degradation during testing. If the control flow obfuscation was implemented on highly complicated code that contains extensive loops, it will not be useful as it will be difficult to trace the errors during implementation, and it does not work sufficiently during reversing. However, it is useful with small applications. Control flow obfuscation does not have the ability to be nested in the source file, as it will be difficult to trace the loops during execution. Error management will not be as possible as it should be [14].

The obfuscation techniques offered by various developers have several gaps. The obfuscation techniques are able to protect the code to some extent; however, the code contains some debugging information. There is no obfuscator tool that can be completely declared as the best obfuscation technique. If the secret of an obfuscator is known, reverse engineers can easily accomplish their tasks by constructing deobfuscators. These de-obscuscator tools have not yet been published, but in the future there may be the possibility of developing de-obscuscators [15].

The bytecode contains unknown characters and symbols from the source code. Reverse engineers have cracked the secrets of the byte code using reverse engineering tools. Therefore, it is possible to copy the original code after reversal, improve it, and resell it on the market to gain an advantage over the original author who developed the code in the first place. Some software development companies hire a hacker or a reverser to crack their code and find out the weaknesses and vulnerabilities of the software so that the company can fix it before it is actually hacked. All software programs contain a security key or registry file that ensures the protection of the software. Reverse engineers convert this file into source code when they remove the registration file from the software and use the exposed code for their own illegal development purpose [16].

Most of the obfuscation techniques are applied in the source file. These obfuscation techniques are applied individually in the source file. Most obfuscation techniques focus on renaming the identifiers and hiding the meaning of the code. Most reverse engineering tools are capable of analyzing the obfuscated code. According to the discussion in the papers highlighted in this research, they do not include mathematical equations to convert or encrypt the strings in the source file, and they do not include garbage conversion to change the layout of the code [17].

Obfuscation techniques are applied in the source code as a single technique. For example, the developer uses only variable names or hides only the names of classes. None of the papers discussed the use of a hybrid obfuscation technique, and none discussed a hybrid obfuscation technique with a mathematical equation for protection. For the obfuscation technique to be strong, it must be merged or joined. If the developer uses more than one obfuscation technique, there is a good chance that the code is protected from the reversal tools. The developer selects the obfuscation techniques that work together based on the layout and complexity of the original code. From the work examined in this study, the use of combined or hybrid obfuscation techniques guarantees strong protection against prohibited reverse engineering [18].

**Table 1** presents limitations of most common obfuscation techniques.

### *Hybrid Obfuscation of Encryption DOI: http://dx.doi.org/10.5772/intechopen.109662*


#### **Table 1.**

*Current obfuscation limitation.*

### **3. Implementing hybrid obfuscation of encryption**

In this section, we introduce a new hybrid obfuscation technique based on identifier renaming and string encryption. The technique relies on hybrid identifier renaming in the program's source file to cause extreme confusion for both reversal tools and humans when they examine the source file without permission. Regardless of the obfuscation strategy used, it was possible to contrast the obfuscation by renaming the identifiers and string encoding in two phases to first overcome the preemptive obfuscation and then add type information to the identifiers in the source code to contrast the layout obfuscation.

The first phase is renaming, and the hybrid obfuscation technique consists of two sections. The first section is obfuscating the identifiers to junk code to hide the meaning and increase complexity and confuse the decompiler during reversing. The second section is replacing the system keywords with Unicode.

The second phase is string encryption, where a set of random mathematical equations are injected into the strings to encrypt them. A transformation framework has been implemented to represent the steps of the hybrid obfuscation technique. The proposed technique can be used for many languages such as Arabic, English, Chinese, and so on. Using this technique creates the possibility of programming in different languages instead of English, which increases the protection of the code.

Following sections discuss the hybrid obfuscation encryption in detail:

### **3.1 Unicode approach**

In the Java language, each character or symbol is represented using Unicode, which creates a possibility of changing the form of the code while reading. This technique is used in the source file. If this file is stolen, there will be no way to read it. The thief has to translate any Unicode to understand the meaning and figure out the code. The compiler is able to read Unicode and produce output. Combining Unicode with other encoding techniques in the source file makes it stronger. **Table 2** presents examples of Unicode [26].



#### **Table 2.**

*Uniocode characters.*

In this approach, a Unicode transformation was used to rename the system keywords. The purpose of this renaming is to make the code in the source file more complicated. In this case, when reading the source file, the attacker will not be able to recognize the actual meaning of the code. This approach is very beneficial because in case of stealing the source file, the reader is not able to recognize the actual meaning of the code. He has to translate the whole code to understand the purpose of the code. However, even if the Unicode is easy to translate, the keywords of the system do not have much meaning, because the classes and variables in the functions and methods are.

### **3.2 String encryption approach**

In this approach, a mathematical equation with a character field and loops were used to encode the strings in the source file. The encoding of the strings causes confusion while decompiling. The reversing tool is not able to translate the symbols generated by the mathematical equation; moreover, the compiler cannot translate the symbols that were converted to bytecode during compilation. The purpose of string encoding is to create a chaos stream in the source file and in the reverse file after decompiling [27]. The advantage of string encoding is that the mathematical formula used to create the chaos stream that can be used **N** times in the source code, and multiple (**X**) sets of mathematical equations can be used in the same source file. The more the chaos streams are created in the source file, the more the confusion is created during decompiling. The mathematical equations used in the source file were derived from the concept that Java programming language provides a function that can be used to convert the mathematical equation characters into different symbols. Normally, the equation contains a fixed value to ensure accurate output [28]. For the proposed technique, the value for the equation is two which will assigned to (**P**). There is other two values in the equation that are the values of (**Y**) and (**Z**). The values of (**Y**) and (**Z**) have to be carefully declared and assigned to produce the accurate output.

If the value of **Y** is **17** then the value of **Z** is **2**.

If the value of **Y** is **19** then the value of **Z** is **4**.

If the value of **Y** is **16** then the value of **Z** is **1**.

According to the above conditions, if the value of (**Y**) increases by one value, then the value of (**Z**) has to increase by one as well. The assigned value of (**P**) is 2, it can be changed as well to increment by one, and then the value of (**Y**) has to decrease by three values in order to get the calculation right for accurate output. The final result of calculating the three values have to be always 17; therefore, the value of (**P**) is fixed

but it can decrease by one value, to increase the value of (**Y**) by one value as well. To prevent errors, the value of (**P**) was fixed at 2. The values of (**Y**) and (**Z**) can be increased and decreased accurately to allow using more mathematical equations in the source file. The final equation is:

$$\text{Char} = \frac{V}{2+Y+Z}.\tag{3}$$

#### **3.3 Mathematical equation to encrypt strings**

The equation that was used to encrypt the strings in the source code is associated with beneficial attributes, and **(Y)** indicates the ideal (best) value of the considered attribute among the values of the attribute for different alternatives, and the fixed and best value for the equation is **2**; this value will not be changed. In the case of beneficial attributes for instance, those of which higher values are desirable for the given application, **(Y)** indicates the higher value of the attribute, and the highest value which will be used for the equation is **17** [29].

Lower values are desired for the given application*, and* **(Z)** indicates the lower value of the attribute**. (Z)** indicates the lowest value of the considered attribute among the values of the attribute for different alternatives, and the lowest value which will be used is **2**. In the case of beneficial attributes**, (Z)** indicates the lower value of the attribute. In the case of non-beneficial attributes, **(Y)** indicates the higher value of the attribute [30]. Following equation presents the string encryption transformation:

$$\text{Char} = \frac{V}{2+Y+Z} \tag{4}$$

#### **3.4 Identifiers renaming to junk obfuscation**

The main purpose of junk renaming is to create complicated code that is difficult to read and understand and make sense out of it. Junk renaming is used to confuse the reversing tool which leads to incorrect analysis and thus produces incorrect codes. Junk conversion provides the ability to create a variety of languages during the development of the software to protect it. The class file contains the junk code after compiling the source file. After using junk conversion, the converted code in the class file is converted back to junk code, which increases protection. Applying this feature means compromising some of the software quality factors that are readable code and manageable size. These features are compromised to increase the security of the code.

### **4. Hybrid obfuscation of encryption**

Java development is based on object orientation, while the compiler executes the application based on components, unlike structured programs developed with the C programming language. Therefore, code obfuscation will not be a problem when compiling to machine language or bytecode. To use this hybrid obfuscation technique, certain steps must be followed. The first step is to use Object Junk Renaming Obfuscation [31–35]. This conversion must be done first to avoid confusion and errors when the obfuscation process is running. The second step is to encrypt strings. This

technique must be performed second to have smooth conversion without errors. The last step is the Unicode renaming technique for obfuscation. Performing the hybrid obfuscation technique increases the security level of the code where reversing is nearly impossible. **Table 3** presents a sample of code after merging three approaches of obfuscation and after reversing.

There is possibility to change names to junk and can be used for any purposes such as emails, login, and so on. With this logic, the hybrid obfuscation encryption can be used to write encrypted letters and create a whole system using only junk code. **Table 4** presents names before and after obfuscation. Every time the obfuscated name is copied or used, it changes automatically.

The string encryption makes the obfuscation technique more effective in terms of securing the code, as it contains so many symbols that help to confuse the decompiler while parsing and analysis. **Figure 5** presents the framework of the proposed hybrid obfuscation encryption (**Figure 6**).


**Table 3.** *Obfuscated code before and after reversing.*

#### **Figure 5.**

*Reversing hybrid obfuscation.*


#### **Table 4.**

*Names before and after obfuscating.*

**Figure 6.** *Hybrid obfuscation encryption framework.*

### **5. Empirical evaluation of the hybrid obfuscation**

Four reversing tools were used to test the effectiveness of the technique and to determine how much can the reversing tool uncover and read from the obfuscated code. Four reversing tools were used for this experiment; the tools are CAVAJ, JAD, DJ, and JD. The parameters are distributed among the reversing tools based on their behavior toward the obfuscated code. For instance, JD only tested the identifiers names because it has the ability to reveal the entire code; therefore, there was no need to test the rest of parameters. **Figure 7** presents experiment design.

### **5.1 Testing with CAVAJ**

CAVAJ as reversing tool for Java class file *is* used to determine the ability of it to read the code after obfuscating. **Figure 8** presents the results of CAVAJ testing.

### **5.2 Testing with Java decompiler (JD)**

DJ Reversing tool is used to determine the ability to reverse Java class file that contains hybrid obfuscated technique. The test will determine if the tool is able to read the obfuscated code, and how much can the tool read and discover. **Figure 9** presents the output after reversing.

**Figure 7.** *Experiment design.*

## **Figure 8.**

**Figure 9.** *Reversing result of JD.*

### **5.3 Testing with JAD**

After installing JAD, prompt command is used to find the Java class file, then the file is opened in command, and the file name.jad is typed to reverse the file. **Figure 10** presents the result of reversing.

First and second classes test for output correctness and reversed code error*:*

The tool was not able the code after obfuscation with hybrid technique, and it has presented errors while reading and just revealed the Unicode without the ability to read the identifiers.

First and second classes test for methods and classes and identifiers:

### *Hybrid Obfuscation of Encryption DOI: http://dx.doi.org/10.5772/intechopen.109662*


#### **Figure 10.**

*Reversing result with JAD.*

#### **Figure 11.**

*JAD reversing result for methods and identifiers.*

Based on **Figure 11**, the tool was not able to get a meaning of the encrypted strings and identifiers; in fact, it has changed the names further which can be considered for the another level of protection. This way the reverser will not be able to read the code or get a meaning of it, and also the name of the Java file was encrypted to mislead the reverser if the source file is stolen. **Figure 12** presents the form of the Java file name after encryption.

### **5.4 Testing to Decompiler java (DJ)**

DJ reversing tool Java is a tool that reverses the class file. This tool is used to determine the ability to reverse Java class file that contains hybrid obfuscated

**Figure 12.** *File name after encryption.*

### **Figure 13.**

*Reversing result with DJ.*

technique. The test will determine if the tool is able to read the obfuscated code, and how much can the tool reveal. **Figure 13** presents the reversing result of reversing the class file of output correctness.

First class test/output correctness*.*

The tool was not able to read the first-class test to reveal the code. Therefore, there is no code to test its correctness. This is promising results of having hybrid obfuscation technique. An error message is appeared to define syntax error.

Second class test/identifiers.

According to **Figure 14**, the tool was not able to read the code after obfuscation. This results the proof that using hybrid obfuscation is more beneficial than just applying one technique.

### **6. Conclusion**

The hybrid obfuscation technique was effective to protect the code. The reversing tools were not able to read and translate the encrypted strings. Renaming to junk in the obfuscation technique was effective as the reversing tool has converted the junk to a series of random numbers and symbols. The reversing tool was able to read the

### *Hybrid Obfuscation of Encryption DOI: http://dx.doi.org/10.5772/intechopen.109662*


#### **Figure 14.** *Identifiers test.*

system keywords only. Furthermore, the reversing tool has added methods and preprocessors while parsing the file. The reversing tool was not able to analyze the obfuscated code to get appropriate output. This means that the hybrid obfuscation technique is effective to protect the source file from prohibited reverse engineering. Third objective of this research was successfully met; according to the experimentation, a series of junk and chaos was created after reversing the obfuscated code.

The extreme chaos was generated due to the merge of string encryption and renaming approaches in one source file which has led to confusion while reversing as the reversing tool was not able to translate or read or analyze the code. To summarize the results of the experiments that were conducted before and after obfuscation, we calculate the lines of code (LOC) of original file before and after reversing, calculate the total errors appeared during running the reversed file before and after obfuscation, and then find the difference to determine the strength.

Based on the results of the reversing tools, they were not able to discover fully functioning code; in all cases, the reversing tools have generated a series of chaos and random numbers and symbols while attempting to translate the obfuscated code. The code that was generated from the reversing tools did not provide an output, and there was always an error while trying to compile the obfuscated code after reversing.

**Table 5**. The summary of errors occurred for the four tested cases.



**Table 5.** *Error summary.*

### **7. Future work**

The number and type of obfuscators we used for our research were fairly small. Future work could explore a wider variety of noncommercial and research obfuscators to provide a broader picture of protection possibilities. Due to time constraints, we were also not able to take advantage of all commercial obfuscators that we had access to. In the future, more commercial obfuscators and reversing tools can be used for the sake of this research. The proposed hybrid obfuscation technique can be further used for games and mobile applications to protect financially from being illegally reversed.

The technique can be developed with C/C++ programming language instead of Java, as Java is closer to the hardware level and communicate with it easily due to the pointer feature it has. Having the technique implemented with C/C++ is an advantage which makes the tool stronger for more defensive.

The technique can be as an added tool in the programming environment such as NetBeans or eclipse where programmer can customize which part of the code to be encrypted and which approach to use. Programmer has full freedom to mix and match encryption approaches in the code to increase security. Having such encryption tool prevents errors while encryption and saves time.

The proposed technique's concept can be used in any programming language to what fits its requirements and mechanisms and also opens an opportunity to have an option to insert different verbal languages, such as Arabic, Chinese, or any other language, for the sake of encryption to increase the level of security.

*Hybrid Obfuscation of Encryption DOI: http://dx.doi.org/10.5772/intechopen.109662*

### **Author details**

Asma'a Al-Hakimi<sup>1</sup> \* and Abu Bakar Md Sultan<sup>2</sup>

1 Faculty of Information Sciences and Engineering, Management and Science University, University Drive, Off Persiaran Olahraga, Shah Alam, Selangor Darul Ehsan, Malaysia

2 Department of Software Engineering and Information System, Faculty of Computer Science and Information Technology, Universiti Putra Malaysia (UPM), Serdang, Malaysia

\*Address all correspondence to: selfemoon@gmail.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### **Chapter 5**

## Extended Intuitionistic Fuzzy Line Graphs: Theory and Properties

*Venkata Naga Srinivasa Rao Repalle, Keneni Abera Tola and Maamo Abebe Ashebo*

### **Abstract**

The introduction of fuzzy set theory was given by Zadeh. The introduction of fuzzy graph theory was given by Kauffman. Later the structure of fuzzy graph was developed Rosenfeld. The traditional fuzzy set cannot be used to completely describe all the evidence in problems where someone wants to know in how much degree of non-membership. Such a problem got the solution by Atanassov who introduced intuitionistic fuzzy set which described by a membership, a non-membership and a hesitation functions. An intuitionistic fuzzy set is used to solve problems involving uncertainty and imprecision that can't be handled by a traditional fuzzy set. This chapter introduced the interval-valued intuitionistic fuzzy line graphs (IVIFLG) and explored the results related to IVIFLG. As a result, many theorems and propositions related to IVIFLG are developed and supported by proof. Moreover, some remarkable isomorphic properties, strong IVIFLG, and complete IVIFLG have been investigated, and the proposed concepts are illustrated with the examples.

**Keywords:** fuzzy set, interval-valued intuitionistic fuzy graph, interval-valued intuitionistic fuzzy line graph, isomorphism, isomorphic properties

### **1. Introduction**

Since Euler was presented with the impression of the Königsberg bridge problem, graph theory has received recognition in a variety of academic fields, including natural science, social science, engineering, and medical science. In the field of graph theory, some operations such as the Wiener index of graphs, line graphs, total graphs, cluster and corona operations of graphs, edge join of graphs, and semi-total line have been useful. In addition, some properties of boiling point, heat of evaporation, surface tension, vapor pressure, total electron energy of polymers, partition coefficients, ultrasonic sound velocity, and internal energy can be analyzed in chemical graph theory. These operations are not only useful in classical graphs but also in fuzzy graphs and generalizations of fuzzy graphs. Because real-world problems are frequently fraught with uncertainty and imprecision, Zadeh proposed fuzzy sets and membership degrees [1]. Accordingly, Kaufman presented the concept of fuzzy relations based on Zedeh's work in [2]. Rosenfeld [3] assembled both Zedeh's and Kaufman's work and then introduced fuzzy graphs.

Later on, Atanassov observed that fuzzy sets (FS) did not handle many problems with uncertainty and imprecision [4]. Based on these observations, he combined the membership degree with the falsehood degree and presented intuitionistic fuzzy sets (IFS) with relations and IFG, which is a generalization of FS [4–6]. It has many applications in fuzzy control, and defuzzification is the most computationally intensive part of fuzzy control. Mordeson investigated the concept of fuzzy line graphs (FLG) for the first time and explored both sufficient and necessary conditions for FLG to be a bijective homomorphism to its FG. He developed some theorems and propositions [7]. Firouzian et.al [8] introduced the notion of degree of an edge in fuzzy line graphs and congraphs.

Akram and Dudek discussed interval valued fuzzy graph (IVFG) and its properties in [9]. Later, different classes of IVIFGs such as regular, irregular, highly irregular, strongly irregular and neighbourly irregular IVIFGs were discussed [10]. Then, Akram drived IVFLG from IVFG [11]. Interval-valued intuitionistic ð Þ� *S*, *T* fuzzy graphs were introduced by Rashmanlou and Borzooei [12]. Afterward, the idea of intuitionistic fuzzy line graph (IFLG) studied by Akram and Davvaz [13]. Furthermore, IFLG and its properties are investigated in [14].

Based on the defined concepts, we gave the definition of IVIFLG in this chapter. Our works are novel in the following ways: (1) IVIFLG is presented and illustrated with an example, (2) numerous theorems and propositions are developed and proved; (3) further, interval-valued intuitionistic weak line isomorphism and interval-valued intuitionistic weak vertex homomorphism are proposed. Readers should refer [5, 7, 11] for notations that are not declared in this chapter.

### **2. Discussion**

This section contains some basic definitions used to introduce IVIFLG. Throughout this chapter we considered only simple graph.

**Definition 1.1.** The graph *G* ¼ ð Þ *V*, *E* is an intuitionistic fuzzy graph (IFG) if the following conditions are satisfied [15]


**Definition 1.2.** The line graph L(G) of graph G is defined as any node in *L G*ð Þ that corresponds to an edge in *G*, and pair of nodes in *L G*ð Þ are adjacent if and only if their correspondence edges *ei*,*ej* ∈ *G* share a common node *v*∈ *G*.

**Definition 1.3.** For the given graph *G* ¼ ð Þ *V*, *E* with *n*�vertices and *Si* ¼

*vi*,*ei*<sup>1</sup> , ⋯,*eip* n o such that 1<sup>≤</sup> *<sup>i</sup>*≤*n*, 1≤*j*<sup>≤</sup> *pi* and *eij* <sup>∈</sup> *<sup>E</sup>* has *vi* as a vertex. Then ð Þ *<sup>S</sup>*, *<sup>T</sup>* is called intersection graph where *S* ¼ f g *Si* is the vertex set of (S, T) and *T* ¼ *SiSj*j*Si*, *Sj* <sup>∈</sup>*S*; *Si*∩*Sj* 6¼ <sup>∅</sup>, for *<sup>i</sup>* 6¼ *<sup>j</sup>* � � is an edge set of (S, T).

**Definition 1.4.** The line(edge) graph *L G*ð Þ¼ ð Þ *H*, *J* is where *H* ¼ f g*<sup>e</sup>* <sup>∪</sup> *ue* f g , *ve* : *<sup>e</sup>*∈*E*, *ue* <sup>f</sup> , *ve* <sup>∈</sup>*V*,*<sup>e</sup>* <sup>¼</sup> *ueve* and *<sup>J</sup>* <sup>¼</sup> *SeSf* : *<sup>e</sup>*, *<sup>f</sup>* <sup>∈</sup> *<sup>E</sup>*,*<sup>e</sup>* 6¼ *<sup>f</sup>*, *Se*∩*Sf* 6¼ <sup>∅</sup> � � with *Se* ¼ f g*e* ∪ *ue*, *ve* f g ,*e*∈*E* [11].

*Extended Intuitionistic Fuzzy Line Graphs: Theory and Properties DOI: http://dx.doi.org/10.5772/intechopen.110182*

**Definition 1.5.** Let *G* ¼ ð Þ *A*1, *B*<sup>1</sup> is an IFG with *A*<sup>1</sup> ¼ *σA*<sup>1</sup> , *γA*<sup>1</sup> and *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>B*<sup>1</sup> , *<sup>γ</sup>B*<sup>1</sup> be IFS on V and E respectively. Then ð Þ¼ *S*, *T* ð Þ *A*2, *B*<sup>2</sup> is an intuitionistic fuzzy intersection graph of *G* whose membership and nonmembership functions are defined as [14]

$$\begin{aligned} \text{i. } \sigma\_{A\_2}(\mathbb{S}\_i) &= \sigma\_{A\_1}(v\_i), & \chi\_{A\_2}(\mathbb{S}\_i) &= \chi\_{A\_1}(v\_i), \forall \mathbb{S}\_i, \mathbb{S}\_j \in \mathbb{S} \\\\ \text{ii. } \sigma\_{B\_2}\left(\mathbb{S}\_i \,\mathbb{S}\_j\right) &= \sigma\_{B\_1}\left(v\_i v\_j\right), & \chi\_{B\_2}\left(\mathbb{S}\_i \mathbb{S}\_j\right) &= \chi\_{B\_1}\left(v\_i v\_j\right) \forall \mathbb{S}\_i \mathbb{S}\_j \in T. \end{aligned}$$

where *A*<sup>2</sup> ¼ *σA*<sup>2</sup> , *γA*<sup>2</sup> , *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *<sup>σ</sup>B*<sup>2</sup> , *<sup>γ</sup>B*<sup>2</sup> on S and T respectively. So, IFG of the intersection graph ð Þ *S*, *T* is isomorphic to G(means, ð Þffi *S*, *T G*).

**Definition 1.6.** Consider *L G*<sup>∗</sup> ð Þ¼ ð Þ *<sup>H</sup>*, *<sup>J</sup>* be line graph of *<sup>G</sup>*<sup>∗</sup> <sup>¼</sup> ð Þ *<sup>V</sup>*, *<sup>E</sup>* . Let *<sup>G</sup>* <sup>¼</sup> ð Þ *<sup>A</sup>*1, *<sup>B</sup>*<sup>1</sup> be IFG of *<sup>G</sup>*<sup>∗</sup> with *<sup>A</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup><sup>A</sup>*<sup>1</sup> , *<sup>γ</sup><sup>A</sup>*<sup>1</sup> and *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup><sup>B</sup>*<sup>1</sup> , *<sup>γ</sup><sup>B</sup>*<sup>1</sup> be IFS on X and *E* receptively. Then we define the intuitionistic fuzzy line graph *L G*ð Þ¼ ð Þ *A*2, *B*<sup>2</sup> of G as

$$\begin{aligned} \text{i. } \sigma\_{A\_2}(\mathcal{S}\_\epsilon) &= \sigma\_{B\_1}(e) = \sigma\_{B\_1}(u\_\epsilon v\_\epsilon), \\ \gamma\_{A\_2}(\mathcal{S}\_\epsilon) &= \gamma\_{B\_1}(e) = \gamma\_{B\_1}(u\_\epsilon v\_\epsilon), \text{ for all } \mathcal{S}\_\epsilon, \mathcal{S}\_\epsilon \in H \\\\ \text{ii. } \sigma\_{B\_2}(\mathcal{S}\_\epsilon \mathcal{S}\_f) &= \sigma\_{B\_1}(e) \land \sigma\_{B\_1}(f) \\ \gamma\_{B\_2}(\mathcal{S}\_\epsilon \mathcal{S}\_f) &= \gamma\_{B\_1}(e) \lor \gamma\_{B\_1}(f), \forall \mathcal{S}\_\epsilon \mathcal{S}\_f \in f. \end{aligned}$$

where *A*<sup>2</sup> ¼ *σ<sup>A</sup>*<sup>2</sup> , *γ<sup>A</sup>*<sup>2</sup> and *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *<sup>σ</sup><sup>B</sup>*<sup>2</sup> , *<sup>γ</sup><sup>B</sup>*<sup>2</sup> are IFS on H and J respectively. The *L G*ð Þ¼ ð Þ *A*2, *B*<sup>2</sup> of IFG G is always IFG.

**Definition 1.7.** Let *G*<sup>1</sup> ¼ ð Þ *A*1, *B*<sup>1</sup> and *G*<sup>2</sup> ¼ ð Þ *A*2, *B*<sup>2</sup> be two IFGs. The homomorphism of *ψ* : *G*<sup>1</sup> ! *G*<sup>2</sup> is mapping *ψ* : *V*<sup>1</sup> ! *V*<sup>2</sup> such that [14].

$$\begin{aligned} \text{i. } \sigma\_{A\_1}(v\_i) \le \sigma\_{A\_2}(\psi(v\_i)), \quad \gamma\_{A\_1}(v\_i) \le \gamma\_{A\_2}(\psi(v\_i)) \\\\ \text{ii. } \sigma\_{B\_1}(v\_i, v\_j) \le \sigma\_{B\_2}(\psi(v\_i)\psi\left(v\_j\right)), \\ \quad \gamma\_{B\_1}(v\_i, v\_j) \le \gamma\_{B\_2}(\psi(v\_i)\psi\left(v\_j\right)) \text{ } \forall v\_i \in V\_1, v\_i v\_j \in E\_1. \end{aligned}$$

**Definition 1.8.** The interval valued FS *A* is characterized by [9].

$$A = \left\{ v\_i, \left[ \sigma\_A^-(v\_i), \sigma\_A^+(v\_i) \right] : v\_i \in X \right\}.$$

Here, *σ*� *<sup>A</sup>*ð Þ *vi* and *σ*<sup>þ</sup> *<sup>A</sup>*ð Þ *vi* are lower and upper interval of fuzzy subsets *A* of X respectively, such that *σ*� *<sup>A</sup>*ð Þ *vi* ≤*σ*<sup>þ</sup> *<sup>A</sup>*ð Þ *vi* ∀*vi* ∈*V*.

For simplicity, we used IVFS for interval valued fuzzy set.

**Definition 1.9.** Let *A* ¼ *σ*� *<sup>A</sup>*ð Þ*v* , *σ*<sup>þ</sup> *<sup>A</sup>*ð Þ*<sup>v</sup>* : *<sup>v</sup>*<sup>∈</sup> *<sup>X</sup>* be IVFS. Then, the graph *<sup>G</sup>*<sup>∗</sup> <sup>¼</sup> <sup>ð</sup>*V*, *<sup>E</sup>*) is called IVFG if the following conditions are satisfied;

$$
\sigma\_B^- \left( v\_i v\_j \right) \le \left( \sigma\_A^- \left( v\_i \right) \land \sigma\_A^- \left( v\_j \right) \right)
$$

$$
\sigma\_B^+ \left( v\_i v\_j \right) \le \sigma\_A^+ \left( v\_i \right) \land \sigma\_A^+ \left( v\_j \right)
$$

∀*vi*, *vj* ∈*V*, ∀*vivj* ∈*E* and where *A* ¼ *σ*� *<sup>A</sup>*, *σ*<sup>þ</sup> *A* , *<sup>B</sup>* <sup>¼</sup> *<sup>σ</sup>*� *<sup>B</sup>* , *σ*<sup>þ</sup> *B* is IVFS on V and E respectively.

**Definition 1.10.** Let *G* ¼ ð Þ *A*1, *B*<sup>1</sup> be simple IVFG. Then we define IVF intersection graph ð Þ¼ *S*, *T* ð Þ *A*2, *B*<sup>2</sup> as follows:

1.*A*<sup>2</sup> and *B*<sup>2</sup> are IFS of S and T respectively,

$$\begin{aligned} \text{2.} \sigma\_{A\_2}^-(\mathbb{S}\_i) &= \sigma\_{A\_1}^-(v\_i) \text{ and } \sigma\_{A\_2}^+(\mathbb{S}\_i) = \sigma\_{A\_1}^+(v\_i), \forall \mathbb{S}\_i, \mathbb{S}\_j \in \mathbb{S} \text{ and} \\\\ \text{3.} \sigma\_{B\_2}^-(\mathbb{S}\_i \mathbb{S}\_j) &= \sigma\_{B\_1}^-(v\_i v\_j), \sigma\_{B\_2}^+(\mathbb{S}\_i \mathbb{S}\_j) = \sigma\_{B\_1}^+(v\_i v\_j), \quad \forall \mathbb{S}\_i \mathbb{S}\_j \in T. \end{aligned}$$

Remark: The given IVFG G and its intersection graph (S, T) are always isomorphic to each other.

**Definition 1.11.** An interval valued fuzzy line graph (IVFLG) *L G*ð Þ¼ ð Þ *A*2, *B*<sup>2</sup> of IVFG *G* ¼ ð Þ *A*1, *B*<sup>1</sup> is defined as follows [11]:

• *<sup>A</sup>*<sup>2</sup> and *<sup>B</sup>*<sup>2</sup> are IVFS of H and J respectively, where *L G*<sup>∗</sup> ð Þ¼ ð Þ *<sup>H</sup>*, *<sup>J</sup>*

$$
\sigma\_{A\_2}^-(\mathbb{S}\_i) = \sigma\_{B\_1}^-(e) = \sigma\_{B\_1}^-(u\_\epsilon v\_\epsilon), \ \sigma\_{A\_2}^+(\mathbb{S}\_i) = \sigma\_{B\_1}^+(e) = \sigma\_{B\_1}^+(u\_\epsilon v\_\epsilon),
$$

$$\bullet \ \sigma\_{B\_2}^- \left( \mathbb{S}\_t \mathbb{S}\_f \right) = \sigma\_{B\_1}^-(e) \wedge \sigma\_{B\_1}^-(f), \ \sigma\_{B\_2}^+ \left( \mathbb{S}\_t \mathbb{S}\_f \right) = \sigma\_{B\_1}^+(e) \wedge \sigma\_{B\_1}^+(f) \text{ for all } \mathbb{S}\_t, \mathbb{S}\_f \in H, \mathbb{S}\_t \mathbb{S}\_f \in I.$$

**Definition 1.12.** A graph *G* ¼ ð Þ *A*, *B* with underlying fuzzy set V is IVIFG if



where 0 ≤*σ*<sup>þ</sup> *<sup>B</sup> vivj* <sup>þ</sup> *<sup>γ</sup><sup>B</sup>* <sup>þ</sup> *vivj* ≤ 1 and ∀*vivj* ∈*E*.

In the next section, we begin the main findings of this chapter by introducing and demonstrating examples of IVIFLG.

**Definition 1.13.** Consider *L G*ð Þ¼ ð Þ *H*, *J* is IVIFLG of IVIFG *G* ¼ ð Þ *A*1, *B*<sup>1</sup> and denoted by *L G*ð Þ¼ ð Þ *A*2, *B*<sup>2</sup> whose membership and non membership function is defined as

i. *A*<sup>2</sup> and *B*<sup>2</sup> are IVIFS of H and J respectively, such that

$$\begin{aligned} \sigma\_{A\_2}^-(\mathcal{S}\_\epsilon) &= \sigma\_{B\_1}^-(e) = \sigma\_{B\_1}^-(\boldsymbol{u}\_\epsilon \boldsymbol{v}\_\epsilon) \\ \sigma\_{A\_2}^+(\mathcal{S}\_\epsilon) &= \sigma\_{B\_1}^+(e) = \sigma\_{B\_1}^+(\boldsymbol{u}\_\epsilon \boldsymbol{v}\_\epsilon) \\ \gamma\_{A\_2}^-(\mathcal{S}\_\epsilon) &= \gamma\_{B\_1}^-(e) = \gamma\_{B\_1}^-(\boldsymbol{u}\_\epsilon \boldsymbol{v}\_\epsilon) \\ \gamma\_{A\_2}^+(\mathcal{S}\_\epsilon) &= \gamma\_{B\_1}^+(e) = \gamma\_{B\_1}^+(\boldsymbol{u}\_\epsilon \boldsymbol{v}\_\epsilon) \ \forall \mathcal{S}\_\epsilon \in H. \end{aligned}$$

*Extended Intuitionistic Fuzzy Line Graphs: Theory and Properties DOI: http://dx.doi.org/10.5772/intechopen.110182*

ii. The edge set of L(G) is

$$\begin{aligned} \sigma\_{B\_2}^- \left( \mathbb{S}\_{\mathfrak{C}} \mathbb{S}\_f \right) &= \sigma\_{B\_1}^-(e) \wedge \sigma\_{B\_1}^-(f), & \sigma\_{B\_2}^+ \left( \mathbb{S}\_{\mathfrak{C}} \mathbb{S}\_f \right) &= \sigma\_{B\_1}^+(e) \wedge \sigma\_{B\_1}^+(f) \\ \chi\_{B\_2}^- \left( \mathbb{S}\_{\mathfrak{C}} \mathbb{S}\_f \right) &= \sigma\_{B\_1}^-(e) \vee \chi\_{B\_1}^-(f), & \chi\_{B\_2}^+ \left( \mathbb{S}\_{\mathfrak{C}} \mathbb{S}\_f \right) &= \chi\_{B\_1}^+(e) \vee \chi\_{B\_1}^+(f) \text{ for all } , \mathbb{S}\_{\mathfrak{C}} \mathbb{S}\_f \in I \dots \end{aligned}$$

**Example 1.14.** Given IVIFG *G* ¼ ð Þ *A*1, *A*<sup>2</sup> as shown in **Figure 1**. From the given IVIFG we have

*σA*<sup>1</sup> ð Þ¼ *v*<sup>1</sup> *σ*� *A*1 ð Þ *v*<sup>1</sup> , *σ*<sup>þ</sup> *A*1 ð Þ *v*<sup>1</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*3,0*:*<sup>6</sup> *σA*<sup>1</sup> ð Þ¼ *v*<sup>2</sup> *σ*� *A*1 ð Þ *v*<sup>2</sup> , *σ*<sup>þ</sup> *A*1 ð Þ *v*<sup>2</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*2,0*:*<sup>7</sup> *σ<sup>A</sup>*<sup>1</sup> ð Þ¼ *v*<sup>3</sup> *σ*� *A*1 ð Þ *v*<sup>3</sup> , *σ*<sup>þ</sup> *A*1 ð Þ *v*<sup>3</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>3</sup> *σ<sup>A</sup>*<sup>1</sup> ð Þ¼ *v*<sup>4</sup> *σ*� *A*1 ð Þ *v*<sup>4</sup> , *σ*<sup>þ</sup> *A*1 ð Þ *v*<sup>4</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*3,0*:*<sup>4</sup> *γA*1 ð Þ¼ *v*<sup>1</sup> *γ*� *A*1 ð Þ *v*<sup>1</sup> , *γ*<sup>þ</sup> *A*1 ð Þ *v*<sup>1</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>4</sup> *γA*1 ð Þ¼ *v*<sup>2</sup> *γ*� *A*1 ð Þ *v*<sup>2</sup> , *γ*<sup>þ</sup> *A*1 ð Þ *v*<sup>2</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>2</sup> *γA*1 ð Þ¼ *v*<sup>3</sup> *γ*� *A*1 ð Þ *v*<sup>3</sup> , *γ*<sup>þ</sup> *A*1 ð Þ *v*<sup>3</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*4,0*:*<sup>5</sup> *γA*1 ð Þ¼ *v*<sup>4</sup> *γ*� *A*1 ð Þ *v*<sup>4</sup> , *γ*<sup>þ</sup> *A*1 ð Þ *v*<sup>4</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*4,0*:*<sup>5</sup> *σ<sup>B</sup>*<sup>1</sup> ð Þ¼ *v*1*v*<sup>2</sup> *σ*� *B*1 ð Þ *v*1*v*<sup>2</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *v*1*v*<sup>2</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*2,0*:*<sup>5</sup> *σ<sup>B</sup>*<sup>1</sup> ð Þ¼ *v*2*v*<sup>3</sup> *σ*� *B*1 ð Þ *v*2*v*<sup>3</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *v*2*v*<sup>3</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>2</sup> *σ<sup>B</sup>*<sup>1</sup> ð Þ¼ *v*3*v*<sup>4</sup> *σ*� *B*1 ð Þ *v*3*v*<sup>4</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *v*3*v*<sup>4</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>1</sup> *σ<sup>B</sup>*<sup>1</sup> ð Þ¼ *v*4*v*<sup>1</sup> *σ*� *B*1 ð Þ *v*4*v*<sup>1</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *v*4*v*<sup>1</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*2,0*:*<sup>4</sup>

**Figure 1.** *IVIFG G.*

$$\begin{aligned} \gamma\_{B\_1}(\upsilon\_1 \upsilon\_2) &= \left[ \gamma\_{B\_1}^-(\upsilon\_1 \upsilon\_2), \gamma\_{B\_1}^+(\upsilon\_1 \upsilon\_2) \right] = [\mathbf{0}.\mathbf{1}, \mathbf{0}.\mathbf{3}] \\ \gamma\_{B\_1}(\upsilon\_2 \upsilon\_3) &= \left[ \gamma\_{B\_1}^-(\upsilon\_2 \upsilon\_3), \gamma\_{B\_1}^+(\upsilon\_2 \upsilon\_3) \right] = [\mathbf{0}.\mathbf{3}, \mathbf{0}.\mathbf{4}] \\ \gamma\_{B\_1}(\upsilon\_3 \upsilon\_4) &= \left[ \gamma\_{B\_1}^-(\upsilon\_3 \upsilon\_4), \gamma\_{B\_1}^+(\upsilon\_3 \upsilon\_4) \right] = [\mathbf{0}.\mathbf{3}, \mathbf{0}.\mathbf{4}] \\ \gamma\_{B\_1}(\upsilon\_4 \upsilon\_1) &= \left[ \gamma\_{B\_1}^-(\upsilon\_4 \upsilon\_1), \gamma\_{B\_1}^+(\upsilon\_4 \upsilon\_1) \right] = [\mathbf{0}.\mathbf{2}, \mathbf{0}.\mathbf{3}] \end{aligned}$$

To find IVIFLG *L G*ð Þ¼ ð Þ *H*, *J* of I such that

$$H = \left\{ \boldsymbol{\upsilon}\_1 \boldsymbol{\upsilon}\_2 = \mathbf{S}\_{\varepsilon\_1}, \boldsymbol{\upsilon}\_2 \boldsymbol{\upsilon}\_3 = \mathbf{S}\_{\varepsilon\_2}, \boldsymbol{\upsilon}\_3 \boldsymbol{\upsilon}\_4 = \mathbf{S}\_{\varepsilon\_3}, \boldsymbol{\upsilon}\_4 \boldsymbol{\upsilon}\_1 = \mathbf{S}\_{\varepsilon\_4} \right\} \text{ and }$$

$$J = \left\{ \mathbf{S}\_{\varepsilon\_1} \mathbf{S}\_{\varepsilon\_2}, \mathbf{S}\_{\varepsilon\_2} \mathbf{S}\_{\varepsilon\_3}, \mathbf{S}\_{\varepsilon\_2} \mathbf{S}\_{\varepsilon\_4}, \mathbf{S}\_{\varepsilon\_4} \mathbf{S}\_{\varepsilon\_1} \right\}.$$

Now, consider *A*<sup>2</sup> ¼ *σ*� *A*2 , *σ*<sup>þ</sup> *A*2 h i and *<sup>B</sup>*<sup>2</sup> <sup>¼</sup> *<sup>σ</sup>*� *B*2 , *σ*<sup>þ</sup> *B*2 h i are IVFS of H and J respectively. Then we have

*σ<sup>A</sup>*<sup>2</sup> *Se*<sup>1</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>1</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>1</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*2,0*:*<sup>5</sup> *σ<sup>A</sup>*<sup>2</sup> *Se*<sup>2</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>2</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>2</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>2</sup> *σ<sup>A</sup>*<sup>2</sup> *Se*<sup>3</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>3</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>3</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>1</sup> *σ<sup>A</sup>*<sup>2</sup> *Se*<sup>4</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>4</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>4</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*2,0*:*<sup>4</sup> *γ<sup>A</sup>*<sup>2</sup> *Se*<sup>1</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>1</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>1</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>3</sup> *γ<sup>A</sup>*<sup>2</sup> *Se*<sup>2</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>2</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>2</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*3,0*:*<sup>4</sup> *γ<sup>A</sup>*<sup>2</sup> *Se*<sup>3</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>3</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>3</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*3,0*:*<sup>4</sup> *γ<sup>A</sup>*<sup>2</sup> *Se*<sup>4</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>4</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>4</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*2,0*:*<sup>3</sup> *σ<sup>B</sup>*<sup>2</sup> *Se*1*Se*<sup>2</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>1</sup> ∧*σ*� *B*1 ð Þ *e*<sup>2</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>1</sup> ∧*σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>2</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>2</sup> *σ<sup>B</sup>*<sup>2</sup> *Se*2*Se*<sup>3</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>2</sup> ∧*σ*� *B*1 ð Þ *e*<sup>3</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>2</sup> ∧*σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>3</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>1</sup> *σ<sup>B</sup>*<sup>2</sup> *Se*3*Se*<sup>4</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>3</sup> ∧*σ*� *B*1 ð Þ *e*<sup>4</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>3</sup> ∧*σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>4</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*1,0*:*<sup>1</sup> *σ<sup>B</sup>*<sup>2</sup> *Se*2*Se*<sup>3</sup> ð Þ¼ *σ*� *B*1 ð Þ *e*<sup>4</sup> ∧*σ*� *B*1 ð Þ *e*<sup>1</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>4</sup> ∧*σ*<sup>þ</sup> *B*1 ð Þ *e*<sup>1</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*2,0*:*<sup>4</sup> *γ<sup>B</sup>*<sup>2</sup> *Se*1*Se*<sup>2</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>1</sup> ∨*γ*� *B*1 ð Þ *e*<sup>2</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>1</sup> ∨*γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>2</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*3,0*:*<sup>4</sup> *γ<sup>B</sup>*<sup>2</sup> *Se*2*Se*<sup>3</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>2</sup> ∨*γ*� *B*1 ð Þ *e*<sup>3</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>2</sup> ∨*γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>3</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*3,0*:*<sup>4</sup> *γ<sup>B</sup>*<sup>2</sup> *Se*3*Se*<sup>4</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>3</sup> ∨*γ*� *B*1 ð Þ *e*<sup>4</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>3</sup> ∨*γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>4</sup> h i <sup>¼</sup> ½ � <sup>0</sup>*:*3,0*:*<sup>4</sup> *γ<sup>B</sup>*<sup>2</sup> *Se*2*Se*<sup>3</sup> ð Þ¼ *γ*� *B*1 ð Þ *e*<sup>4</sup> ∨*γ*� *B*1 ð Þ *e*<sup>1</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>4</sup> ∨*γ*<sup>þ</sup> *B*1 ð Þ *e*<sup>1</sup> h i <sup>¼</sup> <sup>0</sup>*:*2,0*:*3�

Then L(G) of IVIFG G is shown in **Figure 2**. **Proposition 1.15.** *L G*ð Þ¼ ð Þ *A*2, *B*<sup>2</sup> is IVIFLG corresponding to IVIFG *G* ¼ ð Þ *A*1, *B*<sup>1</sup> . *Extended Intuitionistic Fuzzy Line Graphs: Theory and Properties DOI: http://dx.doi.org/10.5772/intechopen.110182*

**Figure 2.** *IVIFLG of G.*

**Definition 1.16.** A homomorphism mapping *ψ* : *G*<sup>1</sup> ! *G*<sup>2</sup> of two IVIFG *G*<sup>1</sup> ¼ ð Þ *M*1, *N*<sup>1</sup> and *G*<sup>2</sup> ¼ ð Þ *M*2, *N*<sup>2</sup> *ψ* : *V*<sup>1</sup> ! *V*<sup>2</sup> is defined as

$$\begin{aligned} \text{i. } \bar{\sigma}\_{M\_1}^-(v\_i) &\leq \sigma\_{M\_2}^-(\boldsymbol{\varphi}(v\_i)), & \sigma\_{M\_1}^+(v\_i) &\leq \sigma\_{M\_2}^+(\boldsymbol{\varphi}(v\_i)) \\ \gamma\_{M\_1}^-(v\_i) &\leq \gamma\_{M\_2}^-(\boldsymbol{\varphi}(v\_i)), & \gamma\_{M\_1}^+(v\_i) &\leq \gamma\_{M\_2}^+(\boldsymbol{\varphi}(v\_i)) \text{ for all } v\_i \in V\_1. \\ \text{i.i. } \sigma\_{N\_1}^-(v\_i v\_j) &\leq \sigma\_{N\_2}^-(\boldsymbol{\varphi}(v\_i)\boldsymbol{\varphi}(v\_j)), & \quad \sigma\_{N\_1}^+(v\_i v\_j) &\leq \sigma\_{N\_2}^+(\boldsymbol{\varphi}(v\_i)\boldsymbol{\varphi}(v\_j)) \\ \gamma\_{N\_1}^-(v\_i v\_j) &\leq \gamma\_{N\_2}^-(\boldsymbol{\varphi}(v\_i)\boldsymbol{\varphi}(v\_j)), & \quad \gamma\_{N\_1}^+(v\_i v\_j) &\leq \gamma\_{N\_2}^+(\boldsymbol{\varphi}(v\_i)\boldsymbol{\varphi}(v\_j)) \text{ for all } v\_i v\_j \in E\_1. \end{aligned}$$

**Definition 1.17.** A bijective homomorphism *ψ* : *G*<sup>1</sup> ! *G*<sup>2</sup> of IVIFG is said to be a weak vertex isomorphism, if

$$\begin{aligned} \sigma\_{\mathcal{M}\_1}(\boldsymbol{v}\_i) &= \left[\sigma\_{\mathcal{M}\_1}^-(\boldsymbol{v}\_i), \sigma\_{\mathcal{M}\_1}^+(\boldsymbol{v}\_i)\right] = \left[\sigma\_{\mathcal{M}\_2}^-(\boldsymbol{\mu}(\boldsymbol{v}\_i)), \sigma\_{\mathcal{M}\_2}^+(\boldsymbol{\mu}(\boldsymbol{v}\_i))\right] \\ \boldsymbol{\gamma}\_{\mathcal{N}\_1}(\boldsymbol{v}\_i) &= \left[\boldsymbol{\gamma}\_{\mathcal{N}\_1}^-(\boldsymbol{v}\_i), \boldsymbol{\gamma}\_{\mathcal{N}\_1}^+(\boldsymbol{v}\_i)\right] = \left[\boldsymbol{\gamma}\_{\mathcal{N}\_2}^-(\boldsymbol{\mu}(\boldsymbol{v}\_i)), \boldsymbol{\gamma}\_{\mathcal{N}\_2}^+(\boldsymbol{\mu}(\boldsymbol{v}\_i))\right], \quad \forall \boldsymbol{v}\_i \in \boldsymbol{V}\_1. \end{aligned}$$

A bijective homomorphism *ψ* : *G*<sup>1</sup> ! *G*<sup>2</sup> of IVIFG is said to be a weak line isomorphism if

$$\begin{aligned} \sigma\_{\mathcal{B}\_1}(\boldsymbol{v}\_i \boldsymbol{v}\_j) &= \left[ \sigma\_{\mathcal{B}\_1}^-(\boldsymbol{v}\_i \boldsymbol{v}\_j), \sigma\_{\mathcal{B}\_1}^+(\boldsymbol{v}\_i \boldsymbol{v}\_j) \right] = \left[ \sigma\_{\mathcal{B}\_2}^-(\boldsymbol{\varphi}(\boldsymbol{v}\_i) \boldsymbol{\varphi}(\boldsymbol{v}\_j)), \sigma\_{\mathcal{B}\_2}^+(\boldsymbol{\varphi}(\boldsymbol{v}\_i) \boldsymbol{\varphi}(\boldsymbol{v}\_j)) \right], \\ \boldsymbol{\chi}\_{\mathcal{B}\_1}(\boldsymbol{v}\_i \boldsymbol{v}\_j) &= \left[ \boldsymbol{\chi}\_{\mathcal{B}\_1}^-(\boldsymbol{v}\_i \boldsymbol{v}\_j), \boldsymbol{\chi}\_{\mathcal{B}\_1}^+(\boldsymbol{v}\_i \boldsymbol{v}\_j) \right] = \left[ \boldsymbol{\chi}\_{\mathcal{B}\_1}^-(\boldsymbol{\varphi}(\boldsymbol{v}\_i) \boldsymbol{\varphi}(\boldsymbol{v}\_j)), \boldsymbol{\chi}\_{\mathcal{B}\_1}^+(\boldsymbol{\varphi}(\boldsymbol{v}\_i) \boldsymbol{\varphi}(\boldsymbol{v}\_j)) \right] \ \forall \boldsymbol{v}\_i \boldsymbol{v}\_j \in E\_1. \end{aligned}$$

If *ψ* : *G*<sup>1</sup> ! *G*<sup>2</sup> is an isomorphism that holds Definition 1.17, then *ψ* is called a weak isomorphism of IVIFGs *G*<sup>1</sup> and *G*2.

**Proposition 1.18.** The IVIFLG *L G*ð Þ is connected graph if and only if its corresponding IVFG *G* is connected graph.

**Proof:** Assume that *L G*ð Þ is a connected IVIFLG of the IVIFG G. First, We want to show that necessary condition. Lets say G is disconnected IVIFG. Then there are at least two nodes of graph *G* which are not joined by path, say *vi* and *vj*. If we take one edge *e* in the first component of the edge set of G, then it doesn't have any edges which adjacent to edge *e* in other components. So that, the IVIFLG of graph G is disconnected and contradicts our assumption. Therefore, the IVIFG G must be connected. On the other hand, assume that IVIFG G is connected graph. Then, there is a path between each pair of nodes. This implies, edges which are adjacent in graph G are adjacent nodes in IVIFLG. As a result, every pair of nodes in IVIFLG of G are linked by a path. Therefore, the proof finished.

**Proposition 1.19.** An Interval valued line graph of star graph *K*1,*<sup>n</sup>* is a complete Interval valued graph *Kn* with *n*�vertices.

**Proof:** Consider the vertex *v*∈*V K*ð Þ 1,*<sup>n</sup>* that adjacent to all other vertices *ui* ∈*V K*ð Þ 1,*<sup>n</sup>* for *i* ¼ 1, 2⋯, *n*. Now, all the vertices in IVIFLG of *K*1,*<sup>n</sup>* are adjacent. This means, IVIFLG of *K*1,*<sup>n</sup>* is a complete graph.

**Example 1.20.** Suppose that the IVIFG *K*1,3 with *V* ¼ f g *v*, *v*1, *v*2, *v*<sup>3</sup> and *E* ¼ f*vv*1, *vv*2, *vv*<sup>3</sup> where

$$v = ([0.3, 0.5], [0.1, 0.4]), \ v\_1 = ([0.3, 0.4], [0.2, 0.5])$$

$$v\_2 = ([0.5, 0.8], [0.1, 0.2]), \ v\_3 = ([0.1, 0.3], [0.5, 0.7])$$

$$e\_1 = vv\_1 = ([0.2, 0.3], [0.3, 0.5]), \ e\_2 = vv\_2 = ([0.2, 0.5], [0.0, 0.3])$$

$$e\_3 = vv\_3 = ([0.1, 0.2], [0.3, 0.6]).$$

Then by definition of IVIFLG, the vertex sets of *L K*ð Þ 1,3 is *V* ¼ *Se*<sup>1</sup> , *Se*<sup>2</sup> , *Se*<sup>3</sup> and *Se*1*Se*<sup>2</sup> , *Se*1*Se*<sup>3</sup> , *Se*2*Se*<sup>3</sup> edge sets where

$$\begin{aligned} \mathsf{S}\_{\mathfrak{e}\_1} &= ([\mathsf{0}.2, \mathsf{0}.3], [\mathsf{0}.3, \mathsf{0}.5]), & \mathsf{S}\_{\mathfrak{e}\_2} &= ([\mathsf{0}.2, \mathsf{0}.5], [\mathsf{0}.0, \mathsf{0}.3]), \\ \mathsf{S}\_{\mathfrak{e}\_3} &= ([\mathsf{0}.1, \mathsf{0}.2], [\mathsf{0}.2, \mathsf{0}.6]), & \mathsf{S}\_{\mathfrak{e}\_1} &\mathsf{S}\_{\mathfrak{e}\_2} = ([\mathsf{0}.2, \mathsf{0}.3], [\mathsf{0}.3, \mathsf{0}.5]), \\ \mathsf{S}\_{\mathfrak{e}\_1} \mathsf{S}\_{\mathfrak{e}\_3} &= ([\mathsf{0}.2, \mathsf{0}.3], [\mathsf{0}.3, \mathsf{0}.5]), & \mathsf{S}\_{\mathfrak{e}\_2} \mathsf{S}\_{\mathfrak{e}\_3} &= ([\mathsf{0}.1, \mathsf{0}.2], [\mathsf{0}.2, \mathsf{0}.6]). \end{aligned}$$

Here *L K*ð Þ 1,3 is complete graph *K*<sup>3</sup> (**Figure 3**).

**Figure 3.** *Graphs of K*1,3 *and L K*1,3 *.*

**Proposition 1.21.** Let *L G*ð Þ be IVIFLG of IVIFG of *<sup>G</sup>*. Then *L G*<sup>∗</sup> ð Þ is a line graph of *<sup>G</sup>*<sup>∗</sup> where *<sup>G</sup>*<sup>∗</sup> <sup>¼</sup> ð Þ *<sup>V</sup>*, *<sup>E</sup>* with underlying set V.

**Proof:** Given *<sup>G</sup>* <sup>¼</sup> ð Þ *<sup>A</sup>*1, *<sup>B</sup>*<sup>1</sup> is IVIFG of *<sup>G</sup>*<sup>∗</sup> and *L G*ð Þ¼ ð Þ *<sup>A</sup>*2, *<sup>B</sup>*<sup>2</sup> is IVIFLG of *L G*<sup>∗</sup> ð Þ. Then

$$
\sigma\_{A\_2}(\mathbb{S}\_\epsilon) = \left[\sigma\_{A\_2}^-(\mathbb{S}\_\epsilon), \sigma\_{A\_2}^+(\mathbb{S}\_\epsilon)\right] = \left[\sigma\_{B\_1}^-(\varepsilon), \sigma\_{B\_1}^+(\varepsilon)\right],
$$

$$
\chi\_{A\_2}(\mathbb{S}\_\epsilon) = \left[\chi\_{A\_2}^-(\mathbb{S}\_\epsilon), \chi\_{A\_2}^+(\mathbb{S}\_\epsilon)\right] = \left[\chi\_{B\_1}^-(\varepsilon), \chi\_{B\_1}^+(\varepsilon)\right] \,\,\,\forall \epsilon \in E.
$$

This implies, *Se* ∈ *H* ¼ f g*e* ∪ *ue* f g , *ve* : *e*∈*E*, *ue* f g , *ve* ∈*V*&*e* ¼ *ueve* if and only if *e*∈*E:*

$$\begin{split} \sigma\_{B\_{2}}\left(\mathcal{S}\_{\mathbf{c}}\mathcal{S}\_{f}\right) &= \left[\sigma\_{B\_{2}}^{-}\left(\mathcal{S}\_{\mathbf{c}}\mathcal{S}\_{f}\right), \sigma\_{B\_{2}}^{+}\left(\mathcal{S}\_{\mathbf{c}}\mathcal{S}\_{f}\right)\right] = \left[\sigma\_{B\_{1}}^{-}(\mathbf{e}) \wedge \sigma\_{B\_{1}}^{-}(f), \sigma\_{B\_{1}}^{+}(\mathbf{e}) \wedge \sigma\_{B\_{1}}^{+}(f)\right] \\ \chi\_{B\_{2}}\left(\mathcal{S}\_{\mathbf{c}}\mathcal{S}\_{f}\right) &= \left[\chi\_{B\_{2}}^{-}\left(\mathcal{S}\_{\mathbf{c}}\mathcal{S}\_{f}\right), \chi\_{B\_{2}}^{+}\left(\mathcal{S}\_{\mathbf{c}}\mathcal{S}\_{f}\right)\right] = \left[\chi\_{B\_{1}}^{-}(\mathbf{e}) \vee \chi\_{B\_{1}}^{-}(f), \chi\_{B\_{1}}^{+}(\mathbf{e}) \vee \chi\_{B\_{1}}^{+}(f)\right] \\ \forall \mathbf{S}\_{\mathbf{c}}\mathcal{S}\_{f} &\in f, \end{split}$$

where *<sup>J</sup>* <sup>¼</sup> *SeSf* <sup>j</sup> *Se*∩*Sf* ∉ ∅,*e*, *<sup>f</sup>* <sup>∈</sup> *<sup>E</sup>*&*<sup>e</sup>* <sup>∉</sup> *<sup>f</sup>* � �*:* Hence, *L G*<sup>∗</sup> ð Þ is a line graph of *<sup>G</sup>*<sup>∗</sup> . **Proposition 1.22.** Let *L G*ð Þ¼ ð Þ *<sup>A</sup>*2, *<sup>B</sup>*<sup>2</sup> be IVIFLG of *L G*<sup>∗</sup> ð Þ. Then *L G*ð Þ is also IVIFLG of some IVIFG *G* ¼ ð Þ *A*1, *B*<sup>1</sup> iff

$$\begin{split} \text{i. i. } \sigma\_{B\_{2}} \left( \mathcal{S}\_{\epsilon} \mathcal{S}\_{f} \right) &= \left[ \sigma\_{B\_{2}}^{-} \left( \mathcal{S}\_{\epsilon} \mathcal{S}\_{f} \right), \sigma\_{B\_{2}}^{+} \left( \mathcal{S}\_{\epsilon} \mathcal{S}\_{f} \right) \right] = \left[ \sigma\_{A\_{2}}^{-} \left( \mathcal{S}\_{\epsilon} \right) \wedge \sigma\_{A\_{2}}^{-} \left( \mathcal{S}\_{f} \right), \sigma\_{A\_{2}}^{+} \left( \mathcal{S}\_{\epsilon} \right) \wedge \sigma\_{A\_{2}}^{+} \left( \mathcal{S}\_{f} \right) \right], \\\\ \text{ii i. } \chi\_{B\_{2}} \left( \mathcal{S}\_{\epsilon} \mathcal{S}\_{f} \right) &= \left[ \chi\_{B\_{2}}^{-} \left( \mathcal{S}\_{\epsilon} \mathcal{S}\_{f} \right), \chi\_{B\_{2}}^{+} \left( \mathcal{S}\_{\epsilon} \mathcal{S}\_{f} \right) \right] = \left[ \chi\_{A\_{2}}^{-} \left( \mathcal{S}\_{\epsilon} \right) \vee \chi\_{A\_{2}}^{-} \left( \mathcal{S}\_{f} \right), \chi\_{A\_{2}}^{+} \left( \mathcal{S}\_{\epsilon} \right) \vee \chi\_{A\_{2}}^{+} \left( \mathcal{S}\_{f} \right) \right] \\ \text{ } \forall \mathcal{S}\_{\epsilon}, \mathcal{S}\_{f} \in H, \mathcal{S}\_{\epsilon} \mathcal{S}\_{f} \in \mathcal{J}. \end{split}$$

**Proof:** Suppose both conditions ð Þ*i* and ð Þ *ii* are satisfied. i.e., *σ*� *<sup>B</sup>*<sup>2</sup> *SeSf* � � <sup>¼</sup> *<sup>σ</sup>*� *A*2 ð Þ *Se* ∧*σ*� *<sup>A</sup>*<sup>2</sup> *Sf* � �, *σ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *SeSf* � � <sup>¼</sup> *<sup>σ</sup>*<sup>þ</sup> *A*2 ð Þ *Se* ∧*σ*<sup>þ</sup> *<sup>A</sup>*<sup>2</sup> *Sf* � �, *γ*� *<sup>B</sup>*<sup>2</sup> *SeSf* � � <sup>¼</sup> *<sup>γ</sup>*� *A*2 ð Þ *Se* ∨*γ*� *<sup>A</sup>*<sup>2</sup> *Sf* � � and *γ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *SeSf* � � <sup>¼</sup> *<sup>γ</sup>*<sup>þ</sup> *A*2 ð Þ *Se* ∨*γ*<sup>þ</sup> *<sup>A</sup>*<sup>2</sup> *Sf* � � for all *SeSf* ∈*W*. For every *e*∈ *E* we define *σ*� *A*2 ð Þ¼ *Se σ*� *A*1 ð Þ*e* , *σ*<sup>þ</sup> *A*2 ð Þ¼ *Se σ*<sup>þ</sup> *A*1 ð Þ*e* , *γ*� *A*2 ð Þ¼ *Se γ*� *A*1 ð Þ*e* and *γ*<sup>þ</sup> *A*2 ð Þ¼ *Se γ*<sup>þ</sup> *A*1 ð Þ*e* . Then

$$
\sigma\_{B\_2}^- \left( \mathbf{S}\_{\epsilon} \mathbf{S}\_f \right) = \left[ \sigma\_{B\_2}^- \left( \mathbf{S}\_{\epsilon} \mathbf{S}\_f \right), \sigma\_{B\_2}^+ \left( \mathbf{S}\_{\epsilon} \mathbf{S}\_f \right) \right]
$$

$$
= \left[ \sigma\_{A\_2}^- \left( \mathbf{S}\_{\epsilon} \right) \wedge \sigma\_{A\_2}^- \left( \mathbf{S}\_f \right), \sigma\_{A\_2}^+ \left( \mathbf{S}\_{\epsilon} \right) \wedge \sigma\_{A\_2}^+ \left( \mathbf{S}\_f \right) \right]
$$

$$
= \left[ \sigma\_{B\_1}^- (\epsilon) \wedge \sigma\_{B\_1}^- (f), \sigma\_{B\_1}^+ (\epsilon) \wedge \sigma\_{B\_1}^+ (f) \right].
$$

$$
\chi\_{B\_2}^- \left( \mathbf{S}\_{\epsilon} \mathbf{S}\_f \right) = \left[ \chi\_{B\_2}^- \left( \mathbf{S}\_{\epsilon} \mathbf{S}\_f \right), \chi\_{B\_2}^+ \left( \mathbf{S}\_{\epsilon} \mathbf{S}\_f \right) \right]
$$

$$
= \left[ \chi\_{A\_2}^- (\mathbf{S}\_{\epsilon}) \vee \chi\_{A\_2}^- \left( \mathbf{S}\_f \right), \chi\_{A\_2}^+ (\mathbf{S}\_{\epsilon}) \vee \chi\_{A\_2}^+ \left( \mathbf{S}\_f \right) \right]
$$

$$
= \left[ \chi\_{B\_1}^- (\epsilon) \vee \chi\_{B\_1}^- (f), \chi\_{B\_1}^+ (\epsilon) \vee \chi\_{B\_1}^+ (f) \right].
$$

We know that IVIFS *A*<sup>1</sup> ¼ *σ*� *A*1 , *σ*<sup>þ</sup> *A*1 h i, *<sup>γ</sup>*� *A*1 , *γ*<sup>þ</sup> *A*1 � � h i yields the properties

**77**

$$\begin{aligned} \sigma\_{B\_1}^-(\upsilon\_i v\_j) &\leq \sigma\_{A\_1}^-(\upsilon\_i) \wedge \sigma\_{A\_1}^-(\upsilon\_j) \\ \sigma\_{B\_1}^+(\upsilon\_i v\_j) &\leq \sigma\_{A\_1}^+(\upsilon\_i) \wedge \sigma\_{A\_1}^+(\upsilon\_j) \\ \chi\_{B\_1}^-(\upsilon\_i v\_j) &\leq \chi\_{A\_1}^-(\upsilon\_i) \vee \chi\_{A\_1}^-(\upsilon\_j) \\ \chi\_{B\_1}^+(\upsilon\_i v\_j) &\leq \chi\_{A\_1}^+(\upsilon\_i) \vee \chi\_{A\_1}^+(\upsilon\_j) \end{aligned}$$

will suffice. From definition of IVIFLG the converse of this statement is well known.

**Proposition 1.23.** An IVIFLG is always a strong IVIFG.

**Proof:** It is straightforward from the definition, therefore it is omitted. **Proposition 1.24.** Let *G*<sup>1</sup> and *G*<sup>2</sup> IVIFGs of *G*<sup>∗</sup> <sup>1</sup> and *G*<sup>∗</sup> <sup>2</sup> respectively. If the mapping *<sup>ψ</sup>* : *<sup>G</sup>*<sup>1</sup> ! *<sup>G</sup>*<sup>2</sup> is a weak isomorphism, then *<sup>ψ</sup>* : *<sup>G</sup>*<sup>∗</sup> <sup>1</sup> ! *<sup>G</sup>*<sup>∗</sup> <sup>2</sup> is isomorphism map. **Proof:** Suppose *ψ* : *G*<sup>1</sup> ! *G*<sup>2</sup> is a weak isomorphism. Then

$$v \in V\_1 \Leftrightarrow \psi(v) \in V\_2 \text{ and }$$

$$uv \in E\_1 \Leftrightarrow \psi(u)\psi(v) \in E\_2.$$

Hence the proof.

**Theorem 1.25.** Let *<sup>G</sup>*<sup>∗</sup> <sup>¼</sup> ð Þ *<sup>V</sup>*, *<sup>E</sup>* is connected graph and consider that *L G*ð Þ¼ ð Þ *A*2, *B*<sup>2</sup> is IVIFLG corresponding to IVIFG *G* ¼ ð Þ *A*1, *B*<sup>1</sup> . The,

1. there exists a map *<sup>ψ</sup>* : *<sup>G</sup>* ! *L G*ð Þ which is a weak isomorphism if and only if *<sup>G</sup>*<sup>∗</sup> is a cyclic graph with

$$\begin{aligned} \sigma\_{A\_1}(\boldsymbol{v}) &= \left[\sigma\_{A\_1}^-(\boldsymbol{v}), \sigma\_{A\_1}^+(\boldsymbol{v})\right] = \left[\sigma\_{B\_1}^-(\boldsymbol{e}), \sigma\_{B\_1}^+(\boldsymbol{e})\right], \\\\ \chi\_{A\_1}(\boldsymbol{v}) &= \left[\boldsymbol{\chi}\_{A\_1}^-(\boldsymbol{v}), \boldsymbol{\chi}\_{A\_1}^+(\boldsymbol{v})\right] = \left[\boldsymbol{\chi}\_{B\_1}^-(\boldsymbol{e}), \boldsymbol{\chi}\_{B\_1}^+(\boldsymbol{e})\right], \\\\ \text{such that } \boldsymbol{A}\_1 &= \left(\left[\sigma\_{A\_1}^-, \sigma\_{A\_1}^+\right], \left[\boldsymbol{\chi}\_{A\_1}^-, \boldsymbol{\chi}\_{A\_1}^+\right]\right) \& \& B\_1 = \left(\left[\sigma\_{B\_1}^-, \sigma\_{B\_1}^+\right], \left[\boldsymbol{\chi}\_{B\_1}^-, \boldsymbol{\chi}\_{B\_1}^+\right]\right), \\ \forall \boldsymbol{v} \in V, \boldsymbol{e} \in E. \end{aligned}$$

2.The map *ψ* is isomorphism if *ψ* : *G* ! *L G*ð Þ is a weak isomorphism.

**Proof:** Consider *ψ* : *G* ! *L G*ð Þ is a weak isomorphism. Then we have

*σ<sup>A</sup>*<sup>1</sup> ð Þ¼ *vi σ*� *A*1 ð Þ *vi* , *σ*<sup>þ</sup> *A*1 ð Þ *vi* h i <sup>¼</sup> *<sup>σ</sup>*� *A*2 ð Þ *ψ*ð Þ *vi* , *σ*<sup>þ</sup> *A*2 ð Þ *ψ*ð Þ *vi* h i *γB*1 ð Þ¼ *vi γ*� *B*1 ð Þ *vi* , *γ*<sup>þ</sup> *B*1 ð Þ *vi* h i <sup>¼</sup> *<sup>γ</sup>*� *B*2 ð Þ *ψ*ð Þ *vi* , *γ*<sup>þ</sup> *B*2 ð Þ *ψ*ð Þ *vi* h i ∀*vi* ∈*V: σ<sup>B</sup>*<sup>1</sup> *vivj* � � <sup>¼</sup> *<sup>σ</sup>*� *<sup>B</sup>*<sup>1</sup> *vivj* � �, *σ*<sup>þ</sup> *<sup>B</sup>*<sup>1</sup> *vivj* � � h i <sup>¼</sup> *<sup>σ</sup>*� *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ vj* � � � � , *σ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ vj* � � � � h i *γ<sup>B</sup>*<sup>1</sup> *vivj* � � <sup>¼</sup> *<sup>γ</sup>*� *<sup>B</sup>*<sup>1</sup> *vivj* � �, *γ*<sup>þ</sup> *<sup>B</sup>*<sup>1</sup> *vivj* � � h i <sup>¼</sup> *<sup>γ</sup>*� *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ vj* � � � � , *γ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ vj* � � � � h i <sup>∀</sup>*vivj* <sup>∈</sup> *<sup>E</sup>:*

This follows that *<sup>G</sup>*<sup>∗</sup> <sup>¼</sup> ð Þ *<sup>V</sup>*, *<sup>E</sup>* is a cyclic from Proposition 1.24.

Now let *<sup>v</sup>*1*v*2*v*3⋯*vnv*<sup>1</sup> be a cycle of *<sup>G</sup>*<sup>∗</sup> where vertices set *<sup>V</sup>* <sup>¼</sup> f g *<sup>v</sup>*1, *<sup>v</sup>*2, <sup>⋯</sup>, *vn* and edges set *E* ¼ f g *v*1*v*2, *v*2*v*3, ⋯, *vnv*<sup>1</sup> . Then we have IVIFS

*Extended Intuitionistic Fuzzy Line Graphs: Theory and Properties DOI: http://dx.doi.org/10.5772/intechopen.110182*

$$
\sigma\_{A\_1}(\upsilon\_i) = \left[\sigma\_{A\_1}^-(\upsilon\_i), \sigma\_{A\_1}^+(\upsilon\_i)\right] = \left[t\_i^-, t\_i^+\right]
$$

$$
\chi\_{A\_1}(\upsilon\_i) = \left[\chi\_{A\_1}^-(\upsilon\_i), \chi\_{A\_1}^+(\upsilon\_i)\right] = \left[f\_i^-, f\_i^+\right].
$$

and

$$
\sigma\_{B\_1}(v\_i v\_{i+1}) = \left[\sigma\_{B\_1}^-(v\_i v\_{i+1}), \sigma\_{B\_1}^+(v\_i v\_{i+1})\right] = \left[l\_i^-, l\_i^+\right],
$$

$$
\chi\_{B\_1}(v\_i v\_{i+1}) = \left[\chi\_{B\_1}^-(v\_i v\_{i+1}), \chi\_{B\_1}^+(v\_i v\_{i+1})\right] = \left[q\_i^-, q\_i^+\right],
$$

where *i* ¼ 1, 2, ⋯, *n* and *vn*þ<sup>1</sup> ¼ *v*1*:* Thus, for *t* � <sup>1</sup> ¼ *t* � *<sup>n</sup>*þ1, *t* þ <sup>1</sup> ¼ *t* þ *<sup>n</sup>*þ<sup>1</sup>, *<sup>f</sup>* � <sup>1</sup> ¼ *f* � *<sup>n</sup>*þ<sup>1</sup>, *<sup>f</sup>* þ <sup>1</sup> ¼ *f* � *n*þ1

$$\begin{aligned} \mathfrak{l}\_i^- &\le t\_i^- \wedge t\_{i+1}^-,\\ \mathfrak{l}\_i^+ &\le t\_i^+ \wedge t\_{i+1}^+,\\ \mathfrak{q}\_i^- &\le f\_i^- \vee f\_{i+1}^- \\ \mathfrak{q}\_i^+ &\le f\_i^+ \vee f\_{i+1}^+. \end{aligned} \tag{1}$$

Now

$$H = \{ \mathbb{S}\_{\varepsilon\_i} : i = 1, 2, \dots, n \} \text{ and } J = \{ \mathbb{S}\_{\varepsilon\_i} \mathbb{S}\_{\varepsilon\_{i+1}} : i = 1, 2, \dots, n - 1 \mid \}.$$

And also,

*σ<sup>A</sup>*<sup>2</sup> *Sei* ð Þ¼ *σ*� *<sup>A</sup>*<sup>2</sup> *Sei* ð Þ, *σ*<sup>þ</sup> *<sup>A</sup>*<sup>2</sup> *Sei* ð Þ h i ¼ *σ*� *B*1 ð Þ *ei* , *σ*<sup>þ</sup> *B*1 ð Þ *ei* h i ¼ *σ*� *B*1 ð Þ *vivi*þ<sup>1</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *vivi*þ<sup>1</sup> h i ¼ *ι* � *<sup>i</sup>* , *ι* þ *i* � � *γ<sup>A</sup>*<sup>2</sup> *Sei* ð Þ¼ *γ*� *<sup>A</sup>*<sup>2</sup> *Sei* ð Þ, *γ*<sup>þ</sup> *<sup>A</sup>*<sup>2</sup> *Sei* ð Þ h i ¼ *γ*� *B*1 ð Þ *ei* , *γ*<sup>þ</sup> *B*1 ð Þ *ei* h i ¼ *γ*� *B*1 ð Þ *vivi*þ<sup>1</sup> , *γ*<sup>þ</sup> *B*1 ð Þ *vivi*þ<sup>1</sup> h i ¼ *q*� *<sup>i</sup>* , *q*<sup>þ</sup> *i* � � *σ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *Sei Sei*þ<sup>1</sup> � � <sup>¼</sup> *min <sup>σ</sup>*<sup>þ</sup> *B*1 ð Þ*e* , *σ*<sup>þ</sup> *B*1 ð Þ *ei*þ<sup>1</sup> n o ¼ *min σ*<sup>þ</sup> *B*1 ð Þ *vivi*þ<sup>1</sup> , *σ*<sup>þ</sup> *B*1 ð Þ *vi*þ<sup>1</sup>*vi*þ<sup>2</sup> n o ¼ *min ι* þ *<sup>i</sup>* , *ι* þ *i*þ1 � � *σ*� *<sup>B</sup>*<sup>2</sup> *Sei Sei*þ<sup>1</sup> � � <sup>¼</sup> *min <sup>σ</sup>*� *B*1 ð Þ*e* , *σ*� *B*1 ð Þ *ei*þ<sup>1</sup> n o ¼ *min σ*� *B*1 ð Þ *vivi*þ<sup>1</sup> , *σ*� *B*1 ð Þ *vi*þ<sup>1</sup>*vi*þ<sup>2</sup> n o ¼ *min ι* � *<sup>i</sup>* , *ι* � *i*þ1 � �

$$\begin{split} \gamma\_{B\_2}^+ \left( \mathbf{S}\_{\boldsymbol{e}\_i} \mathbf{S}\_{\boldsymbol{e}\_{i+1}} \right) &= \max \left\{ \gamma\_{B\_1}^+ (\boldsymbol{e}), \gamma\_{B\_1}^+ (\boldsymbol{e}\_{i+1}) \right\} \\ &= \max \left\{ \gamma\_{B\_1}^+ (\boldsymbol{v}\_i \boldsymbol{v}\_{i+1}), \gamma\_{B\_1}^+ (\boldsymbol{v}\_{i+1} \boldsymbol{v}\_{i+2}) \right\} \\ &= \max \left\{ q\_i^+, q\_{i+1}^+ \right\} \\ \gamma\_{B\_2}^- \left( \mathbf{S}\_{\boldsymbol{e}\_i} \mathbf{S}\_{\boldsymbol{e}\_{i+1}} \right) &= \max \left\{ \gamma\_{B\_1}^- (\boldsymbol{e}), \gamma\_{B\_1}^- (\boldsymbol{e}\_{i+1}) \right\} \\ &= \max \left\{ \gamma\_{B\_1}^- (\boldsymbol{v}\_i \boldsymbol{v}\_{i+1}), \gamma\_{B\_1}^- (\boldsymbol{v}\_{i+1} \boldsymbol{v}\_{i+2}) \right\} \\ &= \max \left\{ q\_i^-, q\_{i+1}^- \right\} \end{split}$$

where *vn*þ<sup>1</sup> ¼ *v*1, *vn*þ<sup>2</sup> ¼ *v*2, *ι* þ <sup>1</sup> ¼ *ι* þ *<sup>n</sup>*þ<sup>1</sup>, *<sup>ι</sup>* � <sup>1</sup> ¼ *ι* � *<sup>n</sup>*þ1, *q*<sup>þ</sup> *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>ι</sup>* þ <sup>1</sup> , , *q*� *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>q</sup>*� <sup>1</sup> , and *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>n</sup>*.*<sup>ψ</sup>* : *<sup>V</sup>* ! *<sup>H</sup>* is bijective map since *<sup>ψ</sup>* : *<sup>G</sup>*<sup>∗</sup> ! *L G*<sup>∗</sup> ð Þ is isomorphism. And also, *ψ* preserves adjacency. So that *ψ* persuades an alternative *τ* of 1, 2, f g ⋯, *n* which *ψ*ð Þ¼ *vi Se<sup>τ</sup>*ð Þ*<sup>i</sup>* and for *ei* ¼ *vivi*þ<sup>1</sup> then *ψ*ð Þ *vi ψ*ð Þ¼ *vi*þ<sup>1</sup> *Se<sup>τ</sup>*ð Þ*<sup>i</sup> Se<sup>τ</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> , *i* ¼ 1, 2, ⋯, *n* � 1. Now

$$t\_i^- = \sigma\_{A\_1}^-(\upsilon\_i) \le \sigma\_{A\_2}^-(\psi(v\_i)) = \sigma\_{A\_2}^-\left(\mathcal{S}\_{\epsilon\_{\tau(i)}}\right) = \iota\_{\tau(i)}^-,$$

$$t\_i^+ = \sigma\_{A\_1}^+(\upsilon\_i) \le \sigma\_{A\_2}^+(\psi(v\_i)) = \sigma\_{A\_2}^+\left(\mathcal{S}\_{\epsilon\_{\tau(i)}}\right) = \iota\_{\tau(i)}^+,$$

$$f\_i^- = \chi\_{A\_1}^-(\upsilon\_i) \le \chi\_{A\_2}^-(\psi(v\_i)) = \chi\_{A\_2}^-\left(\mathcal{S}\_{\epsilon\_{\tau(i)}}\right) = q\_{\tau(i)}^-,$$

$$f\_i^+ = \chi\_{A\_1}^+(\upsilon\_i) \le \chi\_{A\_2}^+(\psi(v\_i)) = \chi\_{A\_2}^+\left(\mathcal{S}\_{\epsilon\_{\tau(i)}}\right) = q\_{\tau(i)}^+.$$

And let *ei* ¼ *vivi*þ1,

*ι* � *<sup>i</sup>* ¼ *σ*� *B*1 ð Þ *vivi*þ<sup>1</sup> ≤*σ*� *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ*ð Þ¼ *vi*þ<sup>1</sup> *σ*� *<sup>B</sup>*<sup>2</sup> *Se<sup>τ</sup>*ð Þ*<sup>i</sup> Se<sup>τ</sup>*ð*i*þ<sup>1</sup> � � � � ¼ *min σ*� *<sup>B</sup>*<sup>1</sup> *eτ*ð Þ*<sup>i</sup>* � �, *σ*� *<sup>B</sup>*<sup>1</sup> *eτ*ð Þ *<sup>i</sup>*þ<sup>1</sup> n o � � <sup>¼</sup> *min <sup>ι</sup>* � *<sup>τ</sup>*ð Þ*<sup>i</sup>* , *<sup>ι</sup>* � *τ*ð Þ *i*þ1 n o *ι* þ *<sup>i</sup>* ¼ *σ*<sup>þ</sup> *B*1 ð Þ *vivi*þ<sup>1</sup> ≤*σ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ*ð Þ¼ *vi*þ<sup>1</sup> *σ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *Se<sup>τ</sup>*ð Þ*<sup>i</sup> Se<sup>τ</sup>*ð*i*þ<sup>1</sup> � � � � ¼ *min σ*<sup>þ</sup> *<sup>B</sup>*<sup>1</sup> *e<sup>τ</sup>*ð Þ*<sup>i</sup>* � �, *σ*<sup>þ</sup> *<sup>B</sup>*<sup>1</sup> *e<sup>τ</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> n o � � <sup>¼</sup> *min <sup>ι</sup>* þ *<sup>τ</sup>*ð Þ*<sup>i</sup>* , *<sup>ι</sup>* þ *τ*ð Þ *i*þ1 n o *q*� *<sup>i</sup>* ¼ *γ*� *B*1 ð Þ *vivi*þ<sup>1</sup> ≤*γ*� *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ*ð Þ¼ *vi*þ<sup>1</sup> *γ*� *<sup>B</sup>*<sup>2</sup> *Se<sup>τ</sup>*ð Þ*<sup>i</sup> Se<sup>τ</sup>*ð*i*þ<sup>1</sup> � � � � ¼ *max γ*� *<sup>B</sup>*<sup>1</sup> *e<sup>τ</sup>*ð Þ*<sup>i</sup>* � �, *γ*� *<sup>B</sup>*<sup>1</sup> *e<sup>τ</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> n o � � <sup>¼</sup> *max q*� *<sup>τ</sup>*ð Þ*<sup>i</sup>* , *<sup>q</sup>*� *τ*ð Þ *i*þ1 n o *q*<sup>þ</sup> *<sup>i</sup>* ¼ *γ*<sup>þ</sup> *B*1 ð Þ *vivi*þ<sup>1</sup> ≤*γ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *ψ*ð Þ *vi ψ*ð Þ¼ *vi*þ<sup>1</sup> *γ*<sup>þ</sup> *<sup>B</sup>*<sup>2</sup> *Se<sup>τ</sup>*ð Þ*<sup>i</sup> Se<sup>τ</sup>*ð*i*þ<sup>1</sup> � � � � ¼ *max γ*<sup>þ</sup> *<sup>B</sup>*<sup>1</sup> *e<sup>τ</sup>*ð Þ*<sup>i</sup>* � �, *γ*<sup>þ</sup> *<sup>B</sup>*<sup>1</sup> *e<sup>τ</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> n o � � ¼ *max q*<sup>þ</sup> *<sup>τ</sup>*ð Þ*<sup>i</sup>* , *<sup>q</sup>*<sup>þ</sup> *τ*ð Þ *i*þ1 n o *for i* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>n</sup>:*

Which implies,

$$\begin{aligned} t\_i^- \le l\_{\tau(i)}^-, \quad t\_i^+ \le l\_{\tau(i)}^+ \\ f\_i^- \le q\_{\tau(i)}^-, \quad f\_i^+ \le q\_{\tau(i)}^+ \end{aligned} \tag{2}$$

and

$$\begin{aligned} \boldsymbol{h}\_{i}^{-} &\leq \min\left\{\boldsymbol{l}\_{\tau(i)}^{-}, \boldsymbol{l}\_{\tau(i+1)}^{-}\right\}, \quad \boldsymbol{l}\_{i}^{+} \leq \min\left\{\boldsymbol{l}\_{\tau(i)}^{+}, \boldsymbol{l}\_{\tau(i+1)}^{+}\right\} \\\boldsymbol{q}\_{i}^{-} &\leq \max\left\{\boldsymbol{q}\_{\tau(i)}^{-}, \boldsymbol{q}\_{\tau(i+1)}^{-}\right\}, \quad \boldsymbol{q}\_{i}^{+} \leq \max\left\{\boldsymbol{q}\_{\tau(i)}^{+}, \boldsymbol{q}\_{\tau(i+1)}^{+}\right\}. \end{aligned} \tag{3}$$

Thus from the above equations, we obtain *ι* � *<sup>i</sup>* ≤*ι* � *<sup>τ</sup>*ð Þ*<sup>i</sup>* , *<sup>ι</sup>* þ *<sup>i</sup>* ≤*ι* þ *<sup>τ</sup>*ð Þ*<sup>i</sup>* , *<sup>q</sup>*� *<sup>i</sup>* ≤*q*� *<sup>τ</sup>*ð Þ*<sup>i</sup>* and *<sup>q</sup>*<sup>þ</sup> *<sup>i</sup>* ≤*q*<sup>þ</sup> *τ*ð Þ*i* . and also *ι* � *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>≤</sup>*<sup>ι</sup>* � *τ τ*ð Þ ð Þ*<sup>i</sup>* , *<sup>ι</sup>* þ *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>≤</sup>*<sup>ι</sup>* þ *τ τ*ð Þ ð Þ*<sup>i</sup>* , *<sup>q</sup>*� *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>≤</sup>*q*� *τ τ*ð Þ ð Þ*<sup>i</sup>* and *<sup>q</sup>*<sup>þ</sup> *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>≤</sup>*q*<sup>þ</sup> *τ τ*ð Þ ð Þ*<sup>i</sup>* . By proceeding this process, we get

$$\begin{aligned} \mathfrak{l}\_i^- \le \mathfrak{l}\_{\tau(i)}^- \le \cdots \le \mathfrak{l}\_{\tau^k(i)}^- \le \mathfrak{l}\_i^- \\\mathfrak{l}\_i^+ \le \mathfrak{l}\_{\tau(i)}^+ \le \cdots \le \mathfrak{l}\_{\tau^k(i)}^+ \le \mathfrak{l}\_i^+ \\\mathfrak{q}\_i^- \le \mathfrak{q}\_{\tau(i)}^- \le \cdots \le \mathfrak{q}\_{\tau^k(i)}^- \le \mathfrak{q}\_i^- \\\mathfrak{q}\_i^+ \le \mathfrak{q}\_{\tau(i)}^+ \le \cdots \le \mathfrak{q}\_{\tau^k(i)}^+ \le \mathfrak{q}\_i^+ \end{aligned}$$

where *τ<sup>k</sup>*þ<sup>1</sup> is the identity function. It follows *ι* � *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>ι</sup>* � *τ τ*ð Þ ð Þ*<sup>i</sup>* , *<sup>ι</sup>* þ *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>ι</sup>* þ *τ τ*ð Þ ð Þ*<sup>i</sup>* , *<sup>q</sup>*� *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>q</sup>*� *τ τ*ð Þ ð Þ*i* and *q*<sup>þ</sup> *<sup>τ</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>q</sup>*<sup>þ</sup> *τ τ*ð Þ ð Þ*<sup>i</sup>* . Again, from Eq. (3), we get

$$\begin{aligned} \mathfrak{l}\_i^- \le \mathfrak{l}\_{\mathfrak{r}(i+1)}^- = \mathfrak{l}\_{i+1}^-, \ \mathfrak{l}\_i^+ \le \mathfrak{l}\_{\mathfrak{r}(i+1)}^+ = \mathfrak{l}\_{i+1}^+ \\\mathfrak{q}\_i^- \le \mathfrak{q}\_{\mathfrak{r}(i+1)}^- = \mathfrak{q}\_{i+1}^-, \mathfrak{q}\_i^+ \le \mathfrak{q}\_{\mathfrak{r}(i+1)}^+ = \mathfrak{q}\_{i+1}^-. \end{aligned}$$

This implies for all *i* ¼ 1, 2, ⋯, *n*, *ι* � *<sup>i</sup>* ¼ *ι* � <sup>1</sup> , *ι* þ *<sup>i</sup>* ¼ *ι* þ <sup>1</sup> , *q*� *<sup>i</sup>* ¼ *q*� <sup>1</sup> and *q*<sup>þ</sup> *<sup>i</sup>* ¼ *q*þ1. Thus, from Eqs. (1) and (2) we obtain

$$\begin{aligned} \mathfrak{l}\_1^- &= \cdots = \mathfrak{l}\_n^- = \mathfrak{t}\_1^- = \cdots = \mathfrak{t}\_n^- \\ \mathfrak{l}\_1^+ &= \cdots = \mathfrak{l}\_n^+ = \mathfrak{t}\_1^+ = \cdots = \mathfrak{t}\_n^+ \\ \mathfrak{q}\_1^- &= \cdots = \mathfrak{q}\_n^- = \mathfrak{f}\_1^- = \cdots = \mathfrak{f}\_n^- \\ \mathfrak{q}\_1^+ &= \cdots = \mathfrak{q}\_n^+ = \mathfrak{f}\_1^+ = \cdots = \mathfrak{f}\_n^+. \end{aligned}$$

As a result, the proof.

**Theorem 1.26.** Let *G* be connected simple IVIFG, then IVIFLG of G is a path graph if and only if *G* is path graph.

**Proof:** Suppose that G is a path IVIFG with ∣*V G*ð Þ∣ ¼ *k*. Thus, G is a path *Pk* with length *k* and ∣*E G*ð Þ∣ ¼ *k* � 1. Since the vertices set of IVIFLG *L G*ð Þ is an edge sets of G, clearly *L G*ð Þ is a path with ∣*VLG* ð Þ ð Þ ∣ ¼ *k* � 1 graph and ∣*ELG* ð Þ ð Þ ∣ ¼ *k* � 2. Implies that *L G*ð Þ is a path graph. On the other hand, assume *L G*ð Þ is a path. Then every degree of vertex *vi* ∈ *G* is can't be greater than two. If there is a vertex *vi* ∈ *G* is greater than two, then an edge *e* which incident to *vi* ∈ *G* would form a complete sub-graph of IVIFLG *L G*ð Þ of more than two vertices. As a result, the IVIFG *G* must be either path graph or cyclic. But, *G* can't be the cyclic graph since a line graph of the cyclic graph is the cyclic graph. The proof is finished.

### **3. Conclusion**

In this chapter, we introduced interval-valued intuitionistic fuzzy line graphs (IVIFLG) and investigated their results. In addition, we developed many theorems, and propositions related to IVIFLG with proof. Moreover, some remarkable properties of isomorphic properties, strong IVIFLG, and complete IVIFLG have been investigated, and the proposed concepts are illustrated with the examples.

### **Acknowledgements**

The authors do thankful to the editor for giving an opportunity to submit our book chapter.

### **Competing interest**

The authors declared that they have no competing interests.

## **Author details**

Venkata Naga Srinivasa Rao Repalle<sup>1</sup> \*†, Keneni Abera Tola2† and Maamo Abebe Ashebo1†


© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Extended Intuitionistic Fuzzy Line Graphs: Theory and Properties DOI: http://dx.doi.org/10.5772/intechopen.110182*

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### **Chapter 6**
