Certain Types of Interval-Valued Fuzzy Graphs

. We propose certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzy graphs, neighbourly irregular interval-valuedfuzzygraphs,neighbourlytotalirregularinterval-valuedfuzzygraphs,highlyirregularinterval-valuedfuzzygraphs, andhighlytotalirregularinterval-valuedfuzzygraphs.Someinterestingpropertiesassociatedwiththesenewinterval-valuedfuzzy graphsareinvestigated,andnecessaryandsufficientconditionsunderwhichneighbourlyirregularandhighlyirregularinterval-valuedfuzzygraphsareequivalentareobtained.Wealsodescribetherelationshipbetweenintuitionisticfuzzygraphsandinterval-valuedfuzzygraphs.


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 (1) SPANTREE: to find a spanning tree in the given graph, (2) GTPL: graph theoretic language, (3) GASP: graph algorithm software package, (4) HINT: an extension of LISP, (5) GRASPE: an extension of LISP, (6) IGTS: an extension of FORTRAN, (7) GEA: graphic extended AL-GOL, (8) AMBIT: to manipulate digraphs, (9) GIRL: graph information retrieval language, and (10) 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 [ 1 ,  2 ].In intuitionistic fuzzy set theory for every membership degree ( 1 ,  2 ), the value  = 1 −  1 −  2 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 intervalvalued fuzzy graph in [15].Akram et al. [16][17][18][19][20] introduced many new concepts including bipolar fuzzy graphs, intervalvalued line fuzzy graphs, and strong intuitionistic fuzzy graphs.In this paper, we propose certain types of intervalvalued 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.

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 and  ≥ 3. Note that path graph,   , has  − 1 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 V in a graph  * is the induced subgraph of  * consisting of all vertices adjacent to V and all edges connecting two such vertices.The neighbourhood is often denoted (V).The degree deg(V) of vertex V is the number of edges incident on V or equivalently, deg( 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 0 ≤  ≤  − 1, 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.
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.
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.

Balanced Interval-Valued Fuzzy Graphs
Definition 4. Let  be an interval-valued fuzzy graph.The neighbourhood degree of a vertex  in  is defined by deg Definition 6.Let  be an interval-valued fuzzy graph.The closed neighbourhood degree of a vertex  is defined by deg 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.
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 (5) 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 Thus, Hence,  is a totally regular interval-valued fuzzy graph.(b) ⇒ (a): Suppose that  is a totally regular intervalvalued fuzzy graph.Then, or or or , and so on.Therefore, , and so on.Therefore, We state the following characterization without its proof.

Corollary 24. The complement of strictly balanced intervalvalued fuzzy graph is strictly balanced.
Theorem 25.Let  1 and  2 be isomorphic interval-valued fuzzy graphs.If  2 is balanced, then  1 is balanced.

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 intervalvalued fuzzy graph.That is, deg() ̸ =  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 deg( By routine computations, we have deg    shown in Figure 3 By routine computations, we have deg(  Example 33.Consider an interval-valued fuzzy graph  such that By routine computations, we have deg By routine computations, we have deg Example 36.Consider an interval-valued fuzzy graph  such that By routine computations, we have deg [0.8, 0.9].We see that every two adjacent vertices have distinct open neighbourhood degree.But the vertex V 2 adjacent to the vertices V 1 and V 3 has the same neighbourhood degree, that is, deg(V 1 ) = deg(V 3 ).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 𝐺 is highly irregular and neighbourly irregular interval-valued fuzzy graph if and only if the neighbourhood degrees of all the vertices of 𝐺 are distinct.
Proof.Let  be an interval-valued fuzzy graph with vertices V 1 , V 2 , . . ., V  .Assume that  is highly irregular and neighbourly irregular.
, 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 deg 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 3 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 Hence,  is neighbourly irregular and highly irregular.
Conversely, assume that  is neighbourly irregular and highly irregular.same , 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 deg( and deg(V 3 ) = [0.8,1.2].We see that neighbourhood degree of V 1 and V 2 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 deg( Hence,  as shown in Figure 9 is neighbourly irregular but not a neighbourly total irregular.
Therefore, deg Therefore, deg Hence,  is a neighbourly totally irregular interval-valued fuzzy graph.

Theorem 45. If an interval-valued fuzzy graph 𝐺 is neighbourly totally irregular and [𝜇 −
,  +  ] is a constant function, then it is a neighbourly irregular interval-valued fuzzy graph.
Proof.Assume that  is a neighbourly total irregular intervalvalued fuzzy graph.Then, the closed neighbourhood degree of every two adjacent vertices is distinct.Let V  , V  ∈  and deg[ 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.

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].
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 IVFG( * ).
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 .

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 [0, 1]-valued membership degrees are replaced by intervals in [0, 1] that approximate the (unknown) membership degrees, specific types of intervalvalued 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.

Figure 7 :
Figure 7:  is neighbourly irregular but not highly irregular.

Claim 2 .
Lower and upper membership values of the vertices are all distinct.Let deg(V  ) = [  ,   ],  = 1, 2, . . ., .Suppose that lower and upper membership value of any two vertices are the

Figure 8 :
Figure 8:  is not neighbourly irregular, but it is complete.

Figure
Figure 9:  is neighbourly irregular but not neighbourly total irregular.
9:  is neighbourly irregular but not neighbourly total irregular.Assume that  is a neighbourly irregular intervalvalued fuzzy graph.Then, the open neighbourhood degrees of every two adjacent vertices are distinct.Let V  , V  ∈  be adjacent vertices with distinct open neighbourhood degrees [ 1 ,  1 ] and [ 2 ,  2 ], where  1 ̸ =  2 ,  1 ̸ =  2 .Let us assume that ( 1