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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.

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 [

Zadeh [

Kauffman’s initial definition of a fuzzy graph [

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 [

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

By a graph

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

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.

A fuzzy subset

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

An interval-valued fuzzy relation

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

By an

Throughout this paper,

Let

Let

Let

Consider a graph

Consider a graph

We define the order

An interval-valued fuzzy graph

Consider a graph

Routine computations show that

Every complete interval-valued fuzzy graph is totally regular.

Let

Suppose that

(a)

(b)

The converse part is obvious.

Let

If

Conversely, suppose that

Similarly, let

We state the following characterization without its proof.

Let

The density of an interval-valued fuzzy graphs

Consider the regular interval-valued fuzzy graph

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

Consider the regular interval-valued fuzzy graph

Any complete interval-valued fuzzy graph is balanced.

Let

Let

Let

Let

Now,

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

Let

Let

Consider a graph

Irregular interval-valued fuzzy graph.

Let

Consider an interval-valued fuzzy graph

A connected interval-valued fuzzy graph

Consider an interval-valued fuzzy graph

A connected interval-valued fuzzy graph

Consider an interval-valued fuzzy graph

Let

Consider an interval-valued fuzzy graph

Consider an interval-valued fuzzy graph

By routine computations, we have

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

An interval-valued fuzzy graph

Let

Conversely, assume that the neighbourhood degrees of all the vertices of

Let

An interval-valued fuzzy graph

Assume that lower and upper membership values of the vertices are all distinct.

Let

Conversely, assume that

Let

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

Consider an interval-valued fuzzy graph

By routine computations, we have

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

Consider an interval-valued fuzzy graph

By routine computations, we have

If an interval-valued fuzzy graph

Assume that

If an interval-valued fuzzy graph

Assume that

Given that

If an interval-valued fuzzy graph

If

If

In 2003, Deschrijver and Kerre [

By an

If

Ju and Wang introduced the notion of interval-valued fuzzy graph (IVFG, for short) in [

If

The mapping

From a pure mathematical point of view, Theorem

In Figures

Links between models.

Link between IFGs and IVFGs.

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 [

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.