#### Abstract

We propose certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzy graphs, neighbourly irregular interval-valued fuzzy graphs, neighbourly total irregular interval-valued fuzzy graphs, highly irregular interval-valued fuzzy graphs, and highly total irregular interval-valued fuzzy graphs. Some interesting properties associated with these new interval-valued fuzzy graphs are investigated, and necessary and sufficient conditions under which neighbourly irregular and highly irregular interval-valued fuzzy graphs are equivalent are obtained. We also describe the relationship between intuitionistic fuzzy graphs and interval-valued fuzzy graphs.

#### 1. Introduction

The major role of graph theory in computer applications is the development of graph algorithms. A number of algorithms are used to solve problems that are modeled in the form of graphs. These algorithms are used to solve the graph theoretical concepts, which in turn are used to solve the corresponding computer science application problems. Several computer programming languages support the graph theory concepts [1]. The main goal of such languages is to enable the user to formulate operations on graphs in a compact and natural manner. Some of these languages are SPANTREE: to find a spanning tree in the given graph, GTPL: graph theoretic language, GASP: graph algorithm software package, HINT: an extension of LISP, GRASPE: an extension of LISP, IGTS: an extension of FORTRAN, GEA: graphic extended AL-GOL, AMBIT: to manipulate digraphs, GIRL: graph information retrieval language, and FGRAAL: FORTRAN Extended graph algorithmic language [1, 2].

Zadeh [3] introduced the notion of interval-valued fuzzy sets, and Atanassov [4] introduced the concept of intuitionistic fuzzy sets as extensions of Zadeh’s fuzzy set theory [5] for representing vagueness and uncertainty. Interval-valued fuzzy set theory reflects the uncertainty by the length of the interval membership degree . In intuitionistic fuzzy set theory for every membership degree , the value denotes a measure of nondeterminacy (or undecidedness). Interval-valued fuzzy sets provide a more adequate description of vagueness than traditional fuzzy sets. It is therefore important to use interval-valued fuzzy sets in applications, such as fuzzy control. One of the computationally most intensive parts of fuzzy control is defuzzification [6]. Since interval-valued fuzzy sets are widely studied and used, we describe briefly the work of Gorzalczany on approximate reasoning [7, 8], Roy and Biswas on medical diagnosis [9], Türksen on multivalued logic [10], and Mendel on intelligent control [6].

Kauffman’s initial definition of a fuzzy graph [11] was based on Zadeh's fuzzy relations [5]. Rosenfeld [12] introduced the fuzzy analogue of several basic graph-theoretic concepts. Since then, fuzzy graph theory has been finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with differences levels of precision. Fuzzy models are becoming useful because of their aim to reduce the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. Mordeson and Peng [13] defined the concept of complement of fuzzy graph and described some operations on fuzzy graphs. In [14], the definition of complement of a fuzzy graph was modified so that the complement of the complement is the original fuzzy graph, which agrees with the crisp graph case. Ju and Wang gave the definition of interval-valued fuzzy graph in [15]. Akram et al. [16–20] introduced many new concepts including bipolar fuzzy graphs, interval-valued line fuzzy graphs, and strong intuitionistic fuzzy graphs. In this paper, we propose certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzy graphs, neighbourly irregular interval-valued fuzzy graphs, neighbourly total irregular interval-valued fuzzy graphs, highly irregular interval-valued fuzzy graphs, and highly total irregular interval-valued fuzzy graphs. Some interesting properties associated with these new interval-valued fuzzy graphs are investigated, and necessary and sufficient conditions under which neighbourly irregular and highly irregular interval-valued fuzzy graphs are equivalent are obtained. We also describe the relationship between intuitionistic fuzzy graphs and interval-valued fuzzy graphs.

We used standard definitions and terminologies in this paper. For notations, terminologies and applications are not mentioned in the paper; the readers are referred to [13, 14, 21–29].

#### 2. Preliminaries

In this section, we review some elementary concepts whose understanding is necessary to fully benefit from this paper.

By a graph , we mean a nontrivial, finite, connected, and undirected graph without loops or multiple edges. We write to mean , and if , we say that and are *adjacent*. Formally, given a graph , two vertices are said to be *neighbors, or adjacent nodes*, if . The number of vertices, the cardinality of , is called the *order of graph* and denoted by . The number of edges, the cardinality of , is called the *size of graph *and denoted by . A *path* in a graph is an alternating sequence of vertices and edges , , , , , . The path graph with vertices is denoted by . A path is sometimes denoted by . The *length* of a path in is . A path in is called a *cycle* if and . Note that path graph, , has edges and can be obtained from a cycle graph, , by removing any edge. An undirected graph is *connected* if there is a path between each pair of distinct vertices. The *neighbourhood* of a vertex in a graph is the induced subgraph of consisting of all vertices adjacent to and all edges connecting two such vertices. The neighbourhood is often denoted . The degree of vertex is the number of edges incident on or equivalently, . The set of neighbors, called a (*open*) *neighborhood* for a vertex in a graph , consists of all vertices adjacent to but not including ; that is, . When is also included, it is called a *closed neighborhood* , that is, . A *regular graph* is a graph where each vertex has the same number of neighbors, that is, all the vertices have the same closed neighbourhood degree. A connected graph is *highly irregular* if each of its vertices is adjacent only to vertices with distinct degrees. Equivalently, a graph is highly irregular if every two vertices of connected by a path of length 2 have distinct degrees. A connected graph is said to be *neighbourly irregular* if no two adjacent vertices of have the same degree. Equivalently, a connected graph is called neighbourly irregular if every two adjacent vertices of have distinct degree.

It is known that one of the best known classes of graphs is the class of regular graphs. These graphs have been studied extensively in various contexts. Regular graphs of degree and order exist with only limited, but natural, restrictions. Indeed, for integers and with , an -regular graph of order exists if and only if is even. A graph that is not regular will be called *irregular*. It is well known [30] that all nontrivial graphs, regular or irregular, must contain at least two vertices of the same degree. In a regular graph, of course, every vertex is adjacent only to vertices having the same degree. On the other hand, it is possible for a vertex in an irregular graph to be adjacent only to vertices with distinct degrees. With these observations made, we now consider graphs that are opposite, in a certain sense, to regular graphs. We consider only undirected graphs with the finite number of vertices and edges.

Applications of fuzzy relations are widespread and important, especially in the field of clustering analysis, neural networks, computer networks, pattern recognition, decision making, and expert systems. In each of these, the basic mathematical structure is that of a fuzzy graph.

*Definition 1 (see [3, 5]). *A fuzzy subset on a set is a map . A fuzzy binary relation on is a fuzzy subset on . By a fuzzy relation, we mean a fuzzy binary relation given by .

Fuzzy set theory is an extension of ordinary set theory in which to each element a real number between 0 and 1, called the membership degree, is assigned. Unfortunately, it is not always possible to give an exact degree of membership. There can be uncertainty about the membership degree because of lack of knowledge, vague information, and so forth. A possible way to overcome this problem is to use interval-valued fuzzy sets, which assign to each element a closed interval which approximates the “real,” but unknown, membership degree. The length of this interval is a measure for the uncertainty about the membership degree.

An *interval number* is an interval with . The interval is identified with the number . Let be the set of all closed subintervals of . Then, it is known that is a complete lattice with as the least element and as the greatest.

*Definition 2 (see [7]). *An interval-valued fuzzy relation in a universe is a mapping such that for all pairs .

Interval-valued fuzzy relations reflect the idea that membership grades are often not precise and the intervals represent such uncertainty.

*Definition 3 (see [15]). *By an *interval-valued fuzzy graph* of a graph , we mean a pair , where is an interval-valued fuzzy set on and is an interval-valued fuzzy relation on such that
for all .

Throughout this paper, is a crisp graph, and is an interval-valued fuzzy graph.

#### 3. Balanced Interval-Valued Fuzzy Graphs

*Definition 4. *Let be an interval-valued fuzzy graph. The *neighbourhood degree* of a vertex in is defined by , where and . Notice that , for , and for .

*Definition 5. *Let be an interval-valued fuzzy graph on . If all the vertices have the same open neighbourhood degree , then is called an *-regular interval-valued fuzzy graph*. The open neighbourhood degree of a vertex in is defined by , where and .

*Definition 6. *Let be an interval-valued fuzzy graph. The *closed neighbourhood degree* of a vertex is defined by , where
If all the vertices have the same closed neighbourhood degree , then is called a *-totally regular interval-valued fuzzy graph*.

*Example 7. *Consider a graph such that , . Let be an interval-valued fuzzy subset of , and let be an interval-valued fuzzy subset of defined by
Routine computations show that an interval-valued fuzzy graph as shown in Figure 1 is both regular and totally regular.

*Example 8. *Consider a graph such that , . Let be an interval-valued fuzzy subset of and let be an interval-valued fuzzy subset of defined by
Routine computations show that an interval-valued fuzzy graph is neither totally regular nor regular.

*Definition 9. *We define the order and size of an interval-valued fuzzy graph by

*Definition 10. *An interval-valued fuzzy graph is called *complete* if

*Example 11. *Consider a graph such that , . Let be an interval-valued fuzzy subset of and let be an interval-valued fuzzy subset of defined by

Routine computations show that is both complete and totally regular interval-valued fuzzy graph, but is not regular since .

Theorem 12. *Every complete interval-valued fuzzy graph is totally regular.*

Theorem 13. *Let be an interval-valued fuzzy graph of a graph . Then, is a constant function if and only if the following statements are equivalent:*(a)*is a regular interval-valued fuzzy graph,*(b)* is a totally regular interval-valued fuzzy graph.*

*Proof. *Suppose that is a constant function. Let and for all .

(a) (b): Assume that is an -regular interval-valued fuzzy graph. Then, and for all . So,
Thus,
Hence, is a totally regular interval-valued fuzzy graph.

(b) (a): Suppose that is a totally regular interval-valued fuzzy graph. Then,
or
or
or
Thus, is a regular interval-valued fuzzy graph. Hence, (a) and (b) are equivalent.

The converse part is obvious.

Theorem 14. *Let be an interval-valued fuzzy graph where a crisp graph is an odd cycle. Then, is a regular interval-valued fuzzy graph if and only if is a constant function.*

*Proof. *If is a constant function, say and for all ; then, and for every . Hence is a regular interval-valued fuzzy graph.

Conversely, suppose that is a -regular interval-valued fuzzy graph. Let be the edges of in that order. Let , , , , and so on. Therefore,
Thus, . So, if and incident at a vertex , then , , , , and . This shows that is a regular function.

Similarly, let , , , , and so on. Therefore,
Thus, . So, if and incident at a vertex , then , , , , and . This shows that is a constant function. Hence, is a constant function.

We state the following characterization without its proof.

Theorem 15. *Let be an interval-valued fuzzy graph where a crisp graph is an even cycle. Then, is a regular interval-valued fuzzy graph if and only if either is a constant function or alternate edges have the same membership values.*

*Definition 16. *The density of an interval-valued fuzzy graphs is , where = for and = for . An interval-valued fuzzy graph is *balanced* if ; that is, , for all subgraphs of . An interval-valued fuzzy graph is *strictly balanced* if for every , for all nonempty subgraphs.

*Example 17. *Consider the regular interval-valued fuzzy graph which is given in Example 7. Routine calculations show that and . Thus, . Consider that , , , and are a nonempty subgraphs of . Then, , , , and . It is easy to see that regular interval-valued fuzzy graph is not balanced.

*Remark 18. *Every regular interval-valued fuzzy graph may not be balanced.

*Example 19. *Consider the regular interval-valued fuzzy graph which is given in Example 11. Routine calculations show that and . Thus, . Consider , , be a nonempty subgraphs of . Then, , , and . It is easy to see that complete interval-valued fuzzy graph is balanced. is also strictly balanced.

Proposition 20. *Any complete interval-valued fuzzy graph is balanced.*

Proposition 21. *Let be a self-complementary interval-valued fuzzy graph. Then, .*

Proposition 22. *Let and be two balanced interval-valued fuzzy graphs. Then, is balanced if and only if .*

Theorem 23. *Let be a strictly balanced interval-valued fuzzy graph, and let be its complement; then .*

*Proof. *Let be a strictly balanced interval-valued fuzzy graph and its complement. Let be a nonempty subgraph of . Since is strictly balanced, for every and . In ,
for every . Dividing (16) by , we get
and dividing (17) by , we get
Then,
Multiplying both sides the above equations by 2,
Thus, and .

Now,
This completes the proof.

Corollary 24. *The complement of strictly balanced interval-valued fuzzy graph is strictly balanced. *

Theorem 25. *Let and be isomorphic interval-valued fuzzy graphs. If is balanced, then is balanced.*

#### 4. Irregularity in Interval-Valued Fuzzy Graphs

*Definition 26. *Let be an interval-valued fuzzy graph on . If there is a vertex which is adjacent to vertices with distinct neighbourhood degrees, then is called an *irregular interval-valued fuzzy graph*. That is, for all .

*Example 27. *Consider a graph such that
Let be an interval-valued fuzzy subset of , and let be an interval-valued fuzzy subset of defined by
By routine computations, we have , , and . It is clear that as shown in Figure 2 is an irregular interval-valued fuzzy graph.

*Definition 28. *Let be an interval-valued fuzzy graph. If there is a vertex which is adjacent to vertices with distinct closed neighbourhood degrees, then is called a *totally irregular interval-valued fuzzy graph*.

*Example 29. *Consider an interval-valued fuzzy graph such that
By routine computations, we have , , , , and . It is clear from calculations that as shown in Figure 3 is a totally irregular interval-valued fuzzy graph.

*Definition 30. *A connected interval-valued fuzzy graph is said to be *neighbourly irregular* if every two adjacent vertices of have distinct open neighbourhood degree.

*Example 31. *Consider an interval-valued fuzzy graph such that
By routine computations, we have , , , and . Hence, as shown in Figure 4 is neighbourly irregular.

*Definition 32. *A connected interval-valued fuzzy graph is said to be *neighbourly totally irregular* if every two adjacent vertices of have distinct closed neighbourhood degree.

*Example 33. *Consider an interval-valued fuzzy graph such that
By routine computations, we have , , , and . Hence, as shown in Figure 5 is neighbourly totally irregular.

*Definition 34. *Let be a connected interval-valued fuzzy graph. is called *highly irregular* if every vertex of is adjacent to vertices with distinct neighbourhood degrees.

*Example 35. *Consider an interval-valued fuzzy graph such that
By routine computations, we have , , , , , and . Clearly, as shown in Figure 6 is highly irregular.

*Example 36. *Consider an interval-valued fuzzy graph such that

By routine computations, we have , , , and . We see that every two adjacent vertices have distinct open neighbourhood degree. But the vertex adjacent to the vertices and has the same neighbourhood degree, that is, . Hence, as shown in Figure 7 is neighbourly irregular but not highly irregular.

*Remark 37. *A neighbourly irregular interval-valued fuzzy graph may not be highly irregular.

Theorem 38. *An interval-valued fuzzy graph is highly irregular and neighbourly irregular interval-valued fuzzy graph if and only if the neighbourhood degrees of all the vertices of are distinct.*

*Proof. *Let be an interval-valued fuzzy graph with -vertices . Assume that is highly irregular and neighbourly irregular. *Claim 1*. The neighbourhood degrees of all vertices of are distinct. Let , . Let the adjacent vertices of be with neighbourhood degrees , respectively. Then, and , since is highly irregular. Also and , since is neighbourly irregular. Hence, the neighbourhood degree of all the vertices of is distinct.

Conversely, assume that the neighbourhood degrees of all the vertices of are distinct. *Claim 2*. is highly irregular and neighbourly irregular interval-valued fuzzy graph.

Let , . Given that and , which implies that every two adjacent vertices have distinct neighbourhood degrees and to every vertex, the adjacent vertices have distinct neighbourhood degrees.

Theorem 39. *An interval-valued fuzzy graph of , where is a cycle with vertices that is neighbourly irregular and highly irregular if and only if the lower and upper membership values of the vertices between every pair of vertices are all distinct.*

*Proof. *Assume that lower and upper membership values of the vertices are all distinct. *Claim 1*. is neighbourly irregular and highly irregular interval-valued fuzzy graph.

Let . Given that and , which implies that and . That is, . Hence, is neighbourly irregular and highly irregular.

Conversely, assume that is neighbourly irregular and highly irregular. *Claim 2*. Lower and upper membership values of the vertices are all distinct.

Let , . Suppose that lower and upper membership value of any two vertices are the same. Let . Let and . Then, , since is cycle, which is a contradiction to the fact that is neighbourly irregular and highly irregular interval-valued fuzzy graph. Hence, lower membership and upper membership value of the vertices are all distinct.

*Remark 40. *A complete interval-valued fuzzy graph may not be neighbourly irregular.

*Example 41. *Consider an interval-valued fuzzy graph such that

By routine computations, we have , , and . We see that neighbourhood degree of and is not distinct. Hence, as shown in Figure 8 is not neighbourly irregular, but it is complete.

*Remark 42. *A neighbourly total irregular interval-valued fuzzy graph may not be neighbourly irregular.

*Example 43. *Consider an interval-valued fuzzy graph such that

By routine computations, we have , , , , , , , and . We see that . Hence, as shown in Figure 9 is neighbourly irregular but not a neighbourly total irregular.

Proposition 44. *If an interval-valued fuzzy graph is neighbourly irregular and is a constant function, then it is neighbourly totally irregular.*

*Proof. *Assume that is a neighbourly irregular interval-valued fuzzy graph. Then, the open neighbourhood degrees of every two adjacent vertices are distinct. Let be adjacent vertices with distinct open neighbourhood degrees and , where , . Let us assume that , where are constant and . Therefore, , , and . *Claim*. Consider that and . Suppose that, and . Consider that
Therefore, . Similarly, we consider that
Therefore, . Hence, is a neighbourly totally irregular interval-valued fuzzy graph.

Theorem 45. *If an interval-valued fuzzy graph is neighbourly totally irregular and is a constant function, then it is a neighbourly irregular interval-valued fuzzy graph.*

*Proof. *Assume that is a neighbourly total irregular interval-valued fuzzy graph. Then, the closed neighbourhood degree of every two adjacent vertices is distinct. Let and , , where and . Assume that and , where are constant and . *Claim*. Consider that .

Given that which implies and , now, we consider that
We now consider that
That is, the neighbourhood degrees of adjacent vertices of are distinct. Hence, neighbourhood degree of every pair of adjacent vertices is distinct in .

Proposition 46. *If an interval-valued fuzzy graph is neighbourly irregular and neighbourly totally irregular, then need not be a constant function.*

*Remark 47. *If is a neighbourly irregular interval-valued fuzzy graph, then interval-valued subgraph of may not be neighbourly irregular.

*Remark 48. *If is a totally irregular interval-valued fuzzy graph, then interval-valued fuzzy subgraph of may not be totally irregular.

#### 5. Relationship between IFGs and IVFGs

In 2003, Deschrijver and Kerre [31] established the relationships between some extensions of fuzzy sets. In this section, we present the relationship between extensions of fuzzy graphs. Shannon and Atanassov [32] introduced the notion of an intuitionistic fuzzy graph. Some operations on intuitionistic fuzzy graphs are discussed in [33].

*Definition 49 (see [32]). *By an *intuitionistic fuzzy graph* (IFG, for short) of a graph , we mean a pair , where is an intuitionistic fuzzy set on and is an intuitionistic fuzzy relation on such that
for all . The class of all IFGs on will be denoted by .

Theorem 50 (see [33]). *If and are intuitionistic fuzzy graphs, then , , and are intuitionistic fuzzy graphs.*

Ju and Wang introduced the notion of interval-valued fuzzy graph (IVFG, for short) in [15]. Some operations on interval-valued fuzzy graphs are discussed in [18]. The class of all IVFGs on will be denoted by .

Theorem 51 (see [19]). *If and are interval-valued fuzzy graphs, then , , and are interval-valued fuzzy graphs. *

Lemma 52. * and are complete lattices.*

Theorem 53. *The mapping
**
defined by
**
where
**
is an isomorphism between lattices and . *

*Remark 54. *From a pure mathematical point of view, Theorem 53 shows that the two concepts intuitionistic fuzzy graphs and interval-valued fuzzy graphs are equivalent.

In Figures 10 and 11, we present the relationships that exist between different models. In these figures, a double arrow between two theories means that they are equivalent, a single arrow denotes that is an extension of . In Figure 10, a dash arrow denotes that model is based on the previous model .

#### 6. Conclusions

Graph theory has several interesting applications in system analysis, operations research, computer applications, and economics. Since most of the time the aspects of graph problems are uncertain, it is nice to deal with these aspects via the methods of fuzzy systems. It is known that fuzzy graph theory has numerous applications in modern science and engineering, especially in the field of information theory, neural networks, expert systems, cluster analysis, medical diagnosis, traffic engineering, network routing, town planning, and control theory. Since interval-valued fuzzy set theory is an increasingly popular extension of fuzzy set theory where traditional []-valued membership degrees are replaced by intervals in [] that approximate the (unknown) membership degrees, specific types of interval-valued fuzzy graphs have been introduced and investigated in this paper. The natural extension of this research work is the application of interval-valued fuzzy graphs in the area of soft computing including neural networks, expert systems, database theory, and geographical information systems.

#### Acknowledgments

The authors are thankful to the referees for their valuable comments. The authors are also thankful to Professor Syed Mansoor Sarwar and Dr. Faisal Aslam for their valuable suggestions.