JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 857070 10.1155/2013/857070 857070 Research Article Certain Types of Interval-Valued Fuzzy Graphs http://orcid.org/0000-0001-7217-7962 Akram Muhammad 1 Alshehri Noura Omair 2 Dudek Wieslaw A. 3 Lin Chong 1 Punjab University College of Information Technology University of the Punjab Old Campus Lahore 54000 Pakistan pu.edu.pk 2 Department of Mathematics Faculty of Sciences (Girls) King Abdulaziz University Jeddah Saudi Arabia kau.edu.sa 3 Institute of Mathematics and Computer Science Wroclaw University of Technology Wybrzeze Wyspianskiego 27 50-370 Wroclaw Poland pwr.wroc.pl 2013 28 10 2013 2013 29 07 2013 31 08 2013 2013 Copyright © 2013 Muhammad Akram et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . 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  introduced the notion of interval-valued fuzzy sets, and Atanassov  introduced the concept of intuitionistic fuzzy sets as extensions of Zadeh’s fuzzy set theory  for representing vagueness and uncertainty. Interval-valued fuzzy set theory reflects the uncertainty by the length of the interval membership degree [μ1,μ2]. In intuitionistic fuzzy set theory for every membership degree (μ1,μ2), the value π=1-μ1-μ2 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 . 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 , Türksen on multivalued logic , and Mendel on intelligent control .

Kauffman’s initial definition of a fuzzy graph  was based on Zadeh's fuzzy relations . Rosenfeld  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  defined the concept of complement of fuzzy graph and described some operations on fuzzy graphs. In , 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 . Akram et al.  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, 2129].

2. Preliminaries

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

By a graph G*=(V,E), we mean a nontrivial, finite, connected, and undirected graph without loops or multiple edges. We write xyE to mean (x,y)E, and if e=xyE, we say that x and y are adjacent. Formally, given a graph G*=(V,E), two vertices x,yV are said to be neighbors, or adjacent nodes, if xyE. The number of vertices, the cardinality of V, is called the order of graph and denoted by |V|. The number of edges, the cardinality of E, is called the size of graph and denoted by |E|. A path in a graph G* is an alternating sequence of vertices and edges v0, e1, v1, e2,,vn-1, en, vn. The path graph with n vertices is denoted by Pn. A path is sometimes denoted by Pn:v0v1vn(n>0). The length of a path Pn in G* is n. A path Pn:v0v1vn in G* is called a cycle if v0=vn and n3. Note that path graph, Pn, has n-1 edges and can be obtained from a cycle graph, Cn, by removing any edge. An undirected graph G* is connected if there is a path between each pair of distinct vertices. The neighbourhood of a vertex v in a graph G* is the induced subgraph of G* consisting of all vertices adjacent to v and all edges connecting two such vertices. The neighbourhood is often denoted N(v). The degree deg(v) of vertex v is the number of edges incident on v or equivalently, deg(v)=|N(v)|. The set of neighbors, called a (open) neighborhood  N(v) for a vertex v in a graph G*, consists of all vertices adjacent to v but not including v; that is, N(v)={uVvuE}. When v is also included, it is called a closed neighborhood  N[v], that is, N[v]=N(v){v}. 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 G* is highly irregular if every two vertices of G* 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 G* have the same degree. Equivalently, a connected graph G* is called neighbourly irregular if every two adjacent vertices of G* 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 r and order n exist with only limited, but natural, restrictions. Indeed, for integers r and n with 0rn-1, an r-regular graph of order n exists if and only if nr is even. A graph that is not regular will be called irregular. It is well known  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 [<xref ref-type="bibr" rid="B36">3</xref>, <xref ref-type="bibr" rid="B34">5</xref>]).

A fuzzy subset μ on a set X is a map μ:X[0,1]. A fuzzy binary relation on X is a fuzzy subset μ on X×X. By a fuzzy relation, we mean a fuzzy binary relation given by μ:X×X[0,1].

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  D is an interval [a-,a+] with 0a-a+1. The interval [a,a] is identified with the number a[0,1]. Let D[0,1] be the set of all closed subintervals of [0,1]. Then, it is known that (D[0,1],,,) is a complete lattice with [0,0] as the least element and [1,1] as the greatest.

Definition 2 (see [<xref ref-type="bibr" rid="B17">7</xref>]).

An interval-valued fuzzy relation R in a universe X×Y is a mapping R:X×YD[0,1] such that R(x,y)=[R-(x,y),R+(x,y)]D[0,1] for all pairs (x,y)X×Y.

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

Definition 3 (see [<xref ref-type="bibr" rid="B19">15</xref>]).

By an interval-valued fuzzy graph  G of a graph G*, we mean a pair G=(A,B), where A=[μA-,μA+] is an interval-valued fuzzy set on V and B=[μB-,μB+] is an interval-valued fuzzy relation on E such that (1)μB-(xy)min(μA-(x),μA-(y)),μB+(xy)min(μA+(x),μA+(y)), for all xyE.

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

3. Balanced Interval-Valued Fuzzy Graphs Definition 4.

Let G be an interval-valued fuzzy graph. The neighbourhood degree of a vertex x in G is defined by deg(x)=[degμ-(x),degμ+(x)], where degμ-(x)=yN(x)μA-(y) and degμ+(x)=yN(x)μA+(y). Notice that μB-(xy)>0, μB+(xy)>0 for xyE, and μB-(xy)=μB+(xy)=0 for xyE.

Definition 5.

Let G=(A,B) be an interval-valued fuzzy graph on G*. If all the vertices have the same open neighbourhood degree n, then G is called an n-regular interval-valued fuzzy graph. The open neighbourhood degree of a vertex x in G is defined by deg(x)=[degμ-(x),degμ+(x)], where degμ-(x)=yN(x)μA-(y) and degμ+(x)=yN(x)μA+(y).

Definition 6.

Let G be an interval-valued fuzzy graph. The closed neighbourhood degree of a vertex x is defined by deg[x]=[degμ-[x],degμ+[x]], where (2)degμ-[x]=degμ-(x)+μA-(x),degμ+[x]=degμ+(x)+μA+(x). If all the vertices have the same closed neighbourhood degree m, then G is called a m-totally regular interval-valued fuzzy graph.

Example 7.

Consider a graph G* such that V={a,b,c,d}, E={ab,bc,cd,ad}. Let A be an interval-valued fuzzy subset of V, and let B be an interval-valued fuzzy subset of EV×V defined by (3)[abcdμA-0.30.30.30.3μA+0.50.50.50.5],[abbccddaμB-0.10.10.10.1μB+0.20.40.20.4]. Routine computations show that an interval-valued fuzzy graph G as shown in Figure 1 is both regular and totally regular.

G    is regular and totally regular.

Example 8.

Consider a graph G* such that V={v1,v2,v3}, E={v1v2,v1v3}. Let A be an interval-valued fuzzy subset of V and let B be an interval-valued fuzzy subset of E defined by (4)μA-(v1)=0.4,μA-(v2)=0.7,μA-(v3)=0.6,μA+(v1)=0.4,μA+(v2)=0.8,μA+(v3)=0.7,μB-(v1v2)=0.2,μB-(v1v3)=0.2;μB+(v1v2)=0.3,μB+(v1v3)=0.4. Routine computations show that an interval-valued fuzzy graph G is neither totally regular nor regular.

Definition 9.

We define the order O(G) and size S(G) of an interval-valued fuzzy graph G=(A,B) by (5)O(G)=xV1+μA+(x)-μA-(x)2,S(G)=xyE1+μB+(xy)-μB-(xy)2.

Definition 10.

An interval-valued fuzzy graph G=(A,B) is called complete if (6)μB-(xy)=min(μA-(x),μA-(y)),μB+(xy)=max(μA+(x),μA+(y))x,yV.

Example 11.

Consider a graph G* such that V={x,y,z}, E={xy,yz,zx}. Let A be an interval-valued fuzzy subset of V and let B be an interval-valued fuzzy subset of E defined by (7)μA-(x)=0.3,μA-(y)=0.4,μA-(z)=0.5,μA+(x)=0.5,μA+(y)=0.7,μA+(z)=0.6,μB-(xy)=0.3,μB-(yz)=0.4,μB-(zx)=0.3,μB+(xy)=0.5,μB+(yz)=0.6,μB+(zx)=0.5.

Routine computations show that G is both complete and totally regular interval-valued fuzzy graph, but G is not regular since deg(x)deg(z)deg(y).

Theorem 12.

Every complete interval-valued fuzzy graph is totally regular.

Theorem 13.

Let G=(A,B) be an interval-valued fuzzy graph of a graph G*. Then, A=[μA-,μA+] is a constant function if and only if the following statements are equivalent:

G  is a regular interval-valued fuzzy graph,

G is a totally regular interval-valued fuzzy graph.

Proof.

Suppose that A=[μA-,μA+] is a constant function. Let μA-(x)=c1 and μA+(x)=c2 for all xV.

(a) (b): Assume that G is an n-regular interval-valued fuzzy graph. Then, degμ-(x)=n1 and degμ+(x)=n2 for all xV. So, (8)degμ-[x]=degμ-(x)+μA-(x),degμ+[x]=degμ+(x)+μA+(x),xV. Thus, (9)degμ-[x]=n1+c1,degμ+[x]=n2+c2,xV. Hence, G is a totally regular interval-valued fuzzy graph.

(b) (a): Suppose that G is a totally regular interval-valued fuzzy graph. Then, (10)degμ-[x]=k1,degμ+[x]=k2,xV, or (11)degμ-(x)+μA-(x)=k1,degμ+(x)+μA+(x)=k2,xV, or (12)degμ-(x)+c1=k1,degμ+(x)+c2=k2,xV, or (13)degμ-(x)=k1-c1,degμ+(x)=k2-c2,xV. Thus, G is a regular interval-valued fuzzy graph. Hence, (a) and (b) are equivalent.

The converse part is obvious.

Theorem 14.

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

Proof.

If B=[μB-,μB+] is a constant function, say μB-=c1 and μB+=c2 for all xyE; then, degμ-(x)=2c1 and degμ+(x)=2c2 for every xV. Hence G is a regular interval-valued fuzzy graph.

Conversely, suppose that G is a (k1,k2)-regular interval-valued fuzzy graph. Let e1,e2,,e2n+1 be the edges of G in that order. Let μB-(e1)=c1, μB-(e2)=k1-c1, μB-(e3)=k1-(k1-c1)=c1, μB-(e4)=k1-c1, and so on. Therefore, (14)μB-(ei)={c1,if  i  is  odd,k1-c1,if  i  is  even. Thus, μB-(e1)=μB-(e2n+1)=c1. So, if e1 and e2n+1 incident at a vertex v1, then degμ-(v1)=k1, degμ-(e1)+degμ-(e2n+1)=k1, c1+c1=k1, 2c1=k1, and c1=k1/2. This shows that μB- is a regular function.

Similarly, let μB+(e1)=c2, μB+(e2)=k2-c2, μB+(e3)=k2-(k2-c2)=c2, μB+(e4)=k2-c2, and so on. Therefore, (15)μB+(ei)={c2,if  i  is  odd,k2-c2,if  i  is  even. Thus, μB+(e2)=μB+(e2n+1)=c2. So, if e2 and e2n incident at a vertex v2, then degμ+(v2)=k2, degμ+(e2)+degμ+(e2n+1)=k2, c2+c2=k2, 2c2=k2, and c2=k2/2. This shows that μB+ is a constant function. Hence, B=[μB-,μB+] is a constant function.

We state the following characterization without its proof.

Theorem 15.

Let G be an interval-valued fuzzy graph where a crisp graph G* is an even cycle. Then, G is a regular interval-valued fuzzy graph if and only if either B=[μB-,μB+] is a constant function or alternate edges have the same membership values.

Definition 16.

The density of an interval-valued fuzzy graphs G is D(G)=(D-(G),D+(G)), where D-(G) = (2x,yV(μB-(xy)))/(x,yV(μA-(x)μA-(y))) for x,yV and D+(G) = (2x,yV(μB+(xy)))/(x,yV(μA+(x)μA+(y))) for x,yV. An interval-valued fuzzy graph G is balanced if D(H)D(G); that is, D-(H)D-(G), D+(H)D+(G) for all subgraphs H of G. An interval-valued fuzzy graph G is strictly balanced if for every x,yV, D(H)=D(G) for all nonempty subgraphs.

Example 17.

Consider the regular interval-valued fuzzy graph G which is given in Example 7. Routine calculations show that D-(G)=0.67 and D+(G)=1.2. Thus, D(G)=(0.67,1.2). Consider that H1={a,b,c}, H2={a,c}, H3={a,d}, and H4={b,c} are a nonempty subgraphs of G. Then, D(H1)=(0.67,1.2), D(H2)=(0,0), D(H3)=(0.67,1.6), and D(H4)=(0.67,1.6). 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 G which is given in Example 11. Routine calculations show that D-(G)=2 and D+(G)=2. Thus, D(G)=(2,2). Consider H1={x,y}, H2={x,z}, H3={y,z} be a nonempty subgraphs of G. Then, D(H1)=(2,2), D(H2)=(2,2), and D(H3)=(2,2). It is easy to see that complete interval-valued fuzzy graph is balanced. G is also strictly balanced.

Proposition 20.

Any complete interval-valued fuzzy graph is balanced.

Proposition 21.

Let G be a self-complementary interval-valued fuzzy graph. Then, D(G)=(1,1).

Proposition 22.

Let G1 and G2 be two balanced interval-valued fuzzy graphs. Then, G1×G2 is balanced if and only if D(G1)=D(G2)=D(G1×G2).

Theorem 23.

Let G be a strictly balanced interval-valued fuzzy graph, and let G¯ be its complement; then D(G)+D(G¯)=(2,2).

Proof.

Let G be a strictly balanced interval-valued fuzzy graph and G¯ its complement. Let H be a nonempty subgraph of G. Since G is strictly balanced, D(G)=D(H) for every HG and x,yV. In G¯, (16)μB-(xy)¯=μA-(x)μA-(y)-μB-(xy),(17)μB+(xy)¯=μA+(x)μA+(y)-μB+(xy), for every x,yV. Dividing (16) by μA-(x)μA-(y), we get (18)μB-(xy)¯μA-(x)μA-(y)=1-μB-(xy)μA-(x)μA-(y),for  every  x,yV, and dividing (17) by μA+(x)μA+(y), we get (19)μB+(xy)¯μA+(x)μA+(y)=1-μB+(xy)μA+(x)μA+(y),for  every  x,yV. Then, (20)x,yVμB-(xy)¯μA-(x)μA-(y)=1-x,yVμB-(xy)μA-(x)μA-(y),for  every  x,yV,x,yVμB+(xy)¯μA+(x)μA+(y)=1-x,yVμB+(xy)μA+(x)μA+(y),for  every  x,yV. Multiplying both sides the above equations by 2, (21)2x,yVμB-(xy)¯μA-(x)¯μA-(y)¯=2-2x,yVμB-(xy)μA-(x)μA-(y),for  every  x,yV,2x,yVμB+(xy)¯μA+(x)¯μA+(y)¯=2-2x,yVμB+(xy)μA+(x)μA+(y),for  every  x,yV. Thus, D-(G¯)=2-D-(G) and D+(G¯)=2-D+(G).

Now, (22)D(G)+D(G¯)=(D-(G),D+(G))+(D-(G¯),D+(G¯))=(D-(G)+D-(G¯)),(D+(G)+D+(G¯))=(2,2). This completes the proof.

Corollary 24.

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

Theorem 25.

Let G1 and G2 be isomorphic interval-valued fuzzy graphs. If G2 is balanced, then G1 is balanced.

4. Irregularity in Interval-Valued Fuzzy Graphs Definition 26.

Let G be an interval-valued fuzzy graph on G*. If there is a vertex which is adjacent to vertices with distinct neighbourhood degrees, then G is called an irregular interval-valued fuzzy graph. That is, deg(x)n for all xV.

Example 27.

Consider a graph G* such that (23)V={v1,v2,v3},E={v1v2,v2v3,v1v3}. Let A be an interval-valued fuzzy subset of V, and let B be an interval-valued fuzzy subset of EV×V defined by (24)[v1v2v3μA-0.20.20.3μA+0.60.70.4],[v1v2v1v3v2v3μB-0.10.10.2μB+0.20.20.3]. By routine computations, we have deg(v1)=[0.5,1.1], deg(v2)=[0.5,1.0], and deg(v3)=[0.4,1.3]. It is clear that G as shown in Figure 2 is an irregular interval-valued fuzzy graph.

Irregular interval-valued fuzzy graph.

Definition 28.

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

Example 29.

Consider an interval-valued fuzzy graph G such that (25)V={v1,v2,v3,v4,v5},E={v1v2,v2v3,v2v4,v3v1,v3v4,v4v1,v4v5}. By routine computations, we have deg[v1]=[1.4,2.4], deg[v2]=[1.4,2.4], deg[v3]=[1.4,2.4], deg[v4]=[1.6,2.6], and deg[v5]=[0.6,0.8]. It is clear from calculations that G as shown in Figure 3 is a totally irregular interval-valued fuzzy graph.

G is totally irregular.

Definition 30.

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

Example 31.

Consider an interval-valued fuzzy graph G such that (26)V={v1,v2,v3,v4},E={v1v2,v2v3,v3v4,v4v1}. By routine computations, we have deg(v1)=[0.8,1.2], deg(v2)=[0.6,1.0], deg(v3)=[0.8,1.2], and deg(v4)=[0.6,1.0]. Hence, G as shown in Figure 4 is neighbourly irregular.

G is neighbourly irregular.

Definition 32.

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

Example 33.

Consider an interval-valued fuzzy graph G such that (27)V={v1,v2,v3,v4},E={v1v2,v2v3,v3v4,v4v1}. By routine computations, we have deg[v1]=[1.1,1.6], deg[v2]=[0.9,1.8], deg[v3]=[1.0,1.7], and deg[v4]=[0.9,1.8]. Hence, G as shown in Figure 5 is neighbourly totally irregular.

G is neighbourly totally irregular.

Definition 34.

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

Example 35.

Consider an interval-valued fuzzy graph G such that (28)V={v1,v2,v3,v4,v5,v6},E={v1v2,v2v3,v2v6,v3v4,v3v5,v4v5,v5v1}. By routine computations, we have deg(v1)=[0.4,0.8], deg(v2)=[0.6,1.7], deg(v3)=[0.9,1.3], deg(v4)=[0.6,1.1], deg(v5)=[1.0,1.8], and deg(v6)=[0.1,0.4]. Clearly, G as shown in Figure 6 is highly irregular.

G is highly irregular.

Example 36.

Consider an interval-valued fuzzy graph G such that (29)V={v1,v2,v3,v4},E={v1v2,v2v3,v3v4,v4v1}.

By routine computations, we have deg(v1)=[0.6,1.0], deg(v2)=[0.8,0.9], deg(v3)=[0.6,1.0], and deg(v4)=[0.8,0.9]. We see that every two adjacent vertices have distinct open neighbourhood degree. But the vertex v2 adjacent to the vertices v1 and v3 has the same neighbourhood degree, that is, deg(v1)=deg(v3). Hence, G as shown in Figure 7 is neighbourly irregular but not highly irregular.

G 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 G be an interval-valued fuzzy graph with n-vertices v1,v2,,vn. Assume that G is highly irregular and neighbourly irregular.

Claim  1. The neighbourhood degrees of all vertices of G are distinct. Let deg(vi)=[ki,li], i=1,2,,n. Let the adjacent vertices of v1 be v2,v3,,vn with neighbourhood degrees [k2,l2],[k3,l3],,[kn,ln], respectively. Then, k2k3kn and l2l3ln, since G is highly irregular. Also k1k2k3kn and l1l2l3ln, since G is neighbourly irregular. Hence, the neighbourhood degree of all the vertices of G is distinct.

Conversely, assume that the neighbourhood degrees of all the vertices of G are distinct.

Claim  2. G is highly irregular and neighbourly irregular interval-valued fuzzy graph.

Let deg(vi)=[ki,li], i=1,2,,n. Given that k1k2k3kn and l1l2l3ln, 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 G of G*, where G* 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. G is neighbourly irregular and highly irregular interval-valued fuzzy graph.

Let vi,vj,vkV. Given that μA-(vi)μA-(vj)μA-(vk) and μA+(vi)μA+(vj)μA+(vk), which implies that xN(x)μA-(vi)xN(x)μA-(vj)xN(x)μA-(vk) and xN(x)μA+(vi)xN(x)μA+(vj)xN(x)μA+(vk). That is, deg(vi)deg(vj)deg(vk). Hence, G is neighbourly irregular and highly irregular.

Conversely, assume that G is neighbourly irregular and highly irregular.

Claim  2. Lower and upper membership values of the vertices are all distinct.

Let deg(vi)=[ki,li], i=1,2,,n. Suppose that lower and upper membership value of any two vertices are the same. Let v1,v2V. Let μA-(v1)=μA-(v2) and μA+(v1)=μA+(v2). Then, deg(v1)=deg(v2), since G* is cycle, which is a contradiction to the fact that G 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 G such that (30)V={v1,v2,v3},E={v1v2,v2v3,v1v3}.

By routine computations, we have deg(v1)=[0.6,1.3], deg(v2)=[0.6,1.3], and deg(v3)=[0.8,1.2]. We see that neighbourhood degree of v1 and v2 is not distinct. Hence, G as shown in Figure 8 is not neighbourly irregular, but it is complete.

G 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 G such that (31)V={v1,v2,v3,v4},E={v1v2,v2v3,v3v4,v4v1}.

By routine computations, we have deg(v1)=[0.7,0.9], deg(v2)=[0.8,1.1], deg(v3)=[0.7,0.9], deg(v4)=[0.8,1.1], deg[v1]=[1.1,1.5], deg[v2]=[1.1,1.5], deg[v3]=[1.1,1.4], and deg[v4]=[1.2,1.6]. We see that deg[v1]=deg[v2]. Hence, G as shown in Figure 9 is neighbourly irregular but not a neighbourly total irregular.

G is neighbourly irregular but not neighbourly total irregular.

Proposition 44.

If an interval-valued fuzzy graph G is neighbourly irregular and [μA-,μA+] is a constant function, then it is neighbourly totally irregular.

Proof.

Assume that G is a neighbourly irregular interval-valued fuzzy graph. Then, the open neighbourhood degrees of every two adjacent vertices are distinct. Let vi,vjV be adjacent vertices with distinct open neighbourhood degrees [k1,l1] and [k2,l2], where k1k2, l1l2. Let us assume that (μ1(vi),ν1(vi))=(μ1(vj),ν1(vj))=[c1,c2], where c1,c2 are constant and c1,c2[0,1]. Therefore, degμ-[vi]=degμ-(vi)+μ1(vi)=k1+c1, degμ+[vi]=degμ+(vi)+ν1(vi)=l1+c2degμ-[vj]=degμ-(vj)+μ1(vj)=k2+c1, and degμ+[vj]=degμ+(vj)+ν1(vj)=l2+c2.

Claim. Consider that  degμ-[vi]degμ-[vj] and degμ+[vi]degμ+[vj]. Suppose that, degμ-[vi]=degμ-[vj] and degμ+[vi]=degμ+[vj]. Consider that (32)degμ-[vi]=degμ-[vj],k1+c1=k2+c1,k1-k2=c1-c1=0,k1=k2,which  is  a  contradiction  to  k1k2. Therefore, degμ-[vi]degμ-[vj]. Similarly, we consider that (33)degμ+[vi]=degμ+[vj],l1+c2=l2+c2,l1-l2=c2-c2=0,l1=l2,which  is  a  contradiction  to  l1l2. Therefore, degμ+[vi]degμ+[vj]. Hence, G is a neighbourly totally irregular interval-valued fuzzy graph.

Theorem 45.

If an interval-valued fuzzy graph G is neighbourly totally irregular and [μA-,μA+] is a constant function, then it is a neighbourly irregular interval-valued fuzzy graph.

Proof.

Assume that G is a neighbourly total irregular interval-valued fuzzy graph. Then, the closed neighbourhood degree of every two adjacent vertices is distinct. Let vi,vjV and deg[vi]=[k1,l1], deg[vj]=[k2,l2], where k1k2 and l1l2. Assume that (μ1(vi),ν1(vi))=[c1,c2] and (μ1(vj),ν1(vj))=[c1,c2], where c1,c2[0,1] are constant and deg[vi]deg[vj].

Claim. Consider that deg(vi)deg(vj).

Given that deg[vi]deg[vj] which implies degμ-[vi]degμ-[vj] and degμ+[vi]degμ+[vj], now, we consider that (34)degμ-[vi]degμ-[vj],k1+c1k2+c1,k1k2. We now consider that (35)degμ+[vi]degμ+[vj],l1+c2l2+c2,l1l2. That is, the neighbourhood degrees of adjacent vertices of G are distinct. Hence, neighbourhood degree of every pair of adjacent vertices is distinct in G.

Proposition 46.

If an interval-valued fuzzy graph G is neighbourly irregular and neighbourly totally irregular, then [μA-,μA+] need not be a constant function.

Remark 47.

If G is a neighbourly irregular interval-valued fuzzy graph, then interval-valued subgraph H=(A,B) of G may not be neighbourly irregular.

Remark 48.

If G is a totally irregular interval-valued fuzzy graph, then interval-valued fuzzy subgraph H=(A,B) of G may not be totally irregular.

5. Relationship between IFGs and IVFGs

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

Definition 49 (see [<xref ref-type="bibr" rid="B30">32</xref>]).

By an intuitionistic fuzzy graph (IFG, for short) G of a graph G*, we mean a pair G=(A,B), where A=(μA,νA) is an intuitionistic fuzzy set on V and B=(μB,νB) is an intuitionistic fuzzy relation on E such that (36)μB(xy)min(μA(x),μA(y)),νB(xy)max(νA(x),νA(y)), for all xyE. The class of all IFGs on G* will be denoted by 𝒢(G*).

Theorem 50 (see [<xref ref-type="bibr" rid="B26">33</xref>]).

If G1 and G2 are intuitionistic fuzzy graphs, then G1G2, G1G2, and G1¯ are intuitionistic fuzzy graphs.

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

Theorem 51 (see [<xref ref-type="bibr" rid="B4">19</xref>]).

If G1 and G2 are interval-valued fuzzy graphs, then G1G2, G1G2, and G1¯ are interval-valued fuzzy graphs.

Lemma 52.

( 𝒱 𝒢 ( G * ) , , ) and (𝒢(G*),,) are complete lattices.

Theorem 53.

The mapping (37)ψ1:𝒱𝒢(G*)𝒢(G*) defined by (38)ψ1(B)=({(x,μA-(x),1-μA+(x))xV},{(xy,μB-(xy),1-μB+(xy))xyE}), where (39)B=({(x,μA-(x),μA+(x))xV},{(xy,μB-(xy),μB+(xy))xyE}), is an isomorphism between lattices (𝒱𝒢(G*),,) and (𝒢(G*),,).

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 XY denotes that Y is an extension of X. In Figure 10, a dash arrow XY denotes that model Y is based on the previous model X.

Links between models.

Link between IFGs and IVFGs.

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 [0,1]-valued membership degrees are replaced by intervals in [0,1] 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.

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