An algorithm for computing certain metrics in intuitionistic fuzzy graphs

1 Introduction

Presently, science and technology are featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle types of systems containing elements of uncertainty. A large number of these models is based on an extension of the ordinary set theory, namely, fuzzy sets. Fuzzy set was introduced by Zadeh in 1965. A fuzzy set gives the degree of membership of an object in a given set. In 1983, Atanassov [5] extended this idea and introduced the concept of intuitionistic fuzzy set. He added a new component, degree of non-membership, in the definition of a fuzzy set with the requirement that sum of two degrees must be less or equal to one. Several applications of intuitionistic fuzzy sets have been discussed in [12, 23].

Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. Kaufmann’s initial definition of a fuzzy graph [12] was based on Zadeh’s fuzzy relations [25]. The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. Later on, Bhattacharya [6] gave some remarks on fuzzy graphs and fuzzy hypergraphs were introduced by Mordeson and Nair [15]. The complement of a fuzzy graph was defined by Mordeson [15] and further studied by Sunitha and Vijayakumar [20]. Bhutani and Rosenfeld introduced the concept of M-strong fuzzy graphs in [8] and studied some of their properties. The concept of strong arcs in fuzzy graphs was discussed in [7]. Tom and Sunitha [23] introduced the concept of sum distance in fuzzy graphs and studied some of its properties. Atanassov [5] introduced the concept of intuitionistic fuzzy graphs and intuitionistic fuzzy relations. Shannon and Atanassov [21] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs, and investigated some of their properties in [22].Parvathi et al. defined operations on intuitionistic fuzzy graphs in [17]. Karunambigai et al. used intuitionistic fuzzy graphs to find shortest paths in networks [11] and discussed self-centered intuitionistic fuzzy graphs [10]. Akram et al. [1–4] introduced many new concepts, including strong intuitionistic fuzzy graphs, intuitionistic fuzzy trees, intuitionistic fuzzy hypergraphs and intuitionistic fuzzy digraphs in decision support systems. Nowadays intuitionistic fuzzy graphs are playing a substantial role in chemistry, economics, computer science, engineering, medicine and decision making problems [9]. The advantages of intuitionistic fuzzy sets and graphs are that they give more accuracy into the problems and reduce the cost of implementation and improve efficiency. In this article, we introduce the concepts of certain metrics in intuitionistic fuzzy graphs and investigate some of their interesting properties. We present an algorithm for computing the sum distance matrix, eccentricity of the vertices, radius and diameter in intuitionistic fuzzy graphs. We calculate time complexity of our proposed algorithm. We also present an example of the intuitionistic fuzzy graph in decision making system.

We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [16, 19].

2 Preliminaries

In this section, we review some definitions that are necessary for this paper.

A graph is an ordered pair G^* = (V, E), where V is the set of vertices and E ⊆ V × V is the set of edges. A graph is called simple if it has no multiple edges and loops. Let G^* be a finite and simple graph, the distance between any two distinct vertices x, y of G^* is defined as the shortest length between them. It is denoted by d (x, y). A fuzzy subset [24, 25] μ on a set X is a map μ : X → [0, 1]. A fuzzy binary relation on X is a fuzzy subset ν on X × X. By a fuzzy relation we mean a fuzzy binary relation given by ν : X × X → [0, 1]. A fuzzy graph [12, 19] is a pair G = (V, μ, ν) where, μ is a fuzzy set on V and ν is a fuzzy set on E such that ν (xy) ≤ μ (x) ∧ μ (y) for all x, y ∈ V. Note that ν (xy) =0 for all xy ∈ V × V - E. An intuitionistic fuzzy set [5] is an object of the form A = {(x, μ _A  (x) , ν _A  (x)) ∣ x ∈ V}, where the mappings μ _A  : V → [0, 1] and ν _A  : V → [0, 1] denote, respectively, the degree ofmembership and non-membership of each element x ∈ V such that μ _A  (x) + ν _A  (x) ≤1 for allx ∈ V. An intuitionistic fuzzy relation [5] on a non-empty set V is a mapping B = (μ _B , ν _B ) : V × V → [0, 1] × [0, 1] such that μ _B  (xy) + ν _B  (xy) ≤1 for all xy ∈ V × V. An intuitionistic fuzzy graph [1] a crisp graph G^* = (V, E) is an ordered pair G = (A, B), where the functions μ _A  : V → [0, 1] and ν _A  : V → [0, 1], denote the degree of membership and non-membership of the element x ∈ V, respectively such that μ _A  (x) + ν _A  (x) ≤1 for all x ∈ V, and the membership and nonmembership functions of the edge xy belongs to V × V are μ _B  : V × V → [0, 1] and ν _B  : V × V → [0, 1] defined as μ _B  (xy) ≤ min {μ _A  (x) , μ _A  (y)}, ν _B  (xy) ≥ max {ν _A  (x) , ν _A  (y)} such that μ _B  (xy) + ν _B  (xy) ≤1 for all xy ∈ E. An intuitionistic fuzzy graph G = (A, B) is said to be complete intuitionistic fuzzy graph [1] if μ _B  (xy) = min {μ _A  (x) , μ _A  (y)}, ν _B  (xy) = max {ν _A  (x) , ν _A  (y)} for all x, y ∈ V. An intuitionistic fuzzy path [16] is a sequence of distinct vertices x₁ - x₂ - x₃ - . . . - x _n which satisfy either one of the following conditions:

μ _B  (x _i x _j ) >0 and μ _B  (x _i x _j ) =0

There are some fundamental properties of ordered pairs which are used throughout the paper. For any two ordered pairs (a, b) and (c, d), where a, b, c and d are real numbers:

(a, b) = (c, d) ⇔ a = b and c = d

Throughput this paper, we use G^* as a crisp graph and G as an intuitionistic fuzzy graph.

3 Certain metrics in intuitionistic fuzzy graphs

Definition 3.1. Let G = (A, B) be an intuitionistic fuzzy graph where A = (μ _A , ν _A ) is an intuitionistic set on V and B = (μ _B , ν _B ) is an intuitionistic fuzzy relation on V. For any path P: x₁ - x₂ - . . . - x _n , the μ _B -length of P is defined as the sum of μ _B weights (membership values) of edges i.e., l μ B ( P ) = ∑ i = 2 n μ B ( x i - 1 x i ) and the ν _B -length of P is defined as the sum of ν _B weights (non-membership values) of edges i.e., l ν B ( P ) = ∑ i = 2 n ν B ( x i - 1 x i ). For any two vertices x, y in G, let ℙ = { P i , P i is an x - y path, i = 1, 2, 3, . . .} be the set of all paths from x to y. Then μ _B -sum distance of the path x - y, denoted by d _{μ B}  (x, y) and defined as d μ B ( x , y ) = min { l μ B ( P i ) : P i ∈ ℙ , i = 1 , 2 , 3 , . . . } and the ν _B -sum distance of x - y, denoted by d _{ν B}  (x, y) and defined as d ν B ( x , y ) = max { l ν B ( P i ) : P i ∈ ℙ , i = 1 , 2 , 3 , . . . }. The sum distance of the path x - y, denoted by d (x, y), and is defined as a pair d (x, y) = (d _{μ B}  (x, y) , d _{ν B}  (x, y)).

Remark. If μ _B  (xy) =1 and ν _B  (xy) =0, for all (x, y) ∈ V × V, then d (x, y) is the distance between x and y path in a crisp graph.

Theorem 3.2. For an intuitionistic fuzzy graphG = (A, B), d = (d _{μ B} , d _{ν B} )   : V × V → [0, 1] × [0, 1]defines a metric on V, with the following conditions:

d (x, y) ≥ (0, 0),

Proof. It is very clear from the definition of d _{μ B} and d _{ν B} that d μ B ( x , y ) ≥ 0 and d ν B ( x , y ) ≥ 0 for all x , y ∈ V ⇒ ( d μ B ( x , y ) , d ν B ( x , y ) ) ≥ ( 0 , 0 ) ⇒ d ( x , y ) ≥ ( 0 , 0 ) .

If x = y, then d (x, y) = d (x, x) = (d _{μ B}  (x, x) , d _{ν B}  (x, x)) = (0, 0). The inverse of any x - y path is a y - x path and vice versa of same μ _B -sum distance and ν _B -sum distance. Let P₁, Q₁ be x - y paths and P₂, Q₂ be y - z paths such that ( d μ B ( x , y ) , d ν B ( x , y ) ) = ( L ( P 1 ) , L ( Q 1 ) ) ( d μ B ( y , z ) , d ν B ( y , z ) ) = ( L ( P 2 ) , L ( Q 2 ) ) .

That is, d ( x , y ) = ( L ( P 1 ) , L ( Q 1 ) ) and d ( y , z ) = ( L ( P 2 ) , L ( Q 2 ) ) .

The paths obtained from P₁ followed by P₂ and Q₁ followed by Q₂ are x - z walks which contains only one path whose lengths are not more than d _{μ B}  (x, y) + d _{μ B}  (y, z) and d _{ν B}  (x, y) + d _{ν B}  (y, z), we can write ( d μ B ( x , z ) , d s ν B ( x , z ) ) ≤ ( d μ B ( x , y ) + d μ B ( y , z ) , d s ν B ( x , y ) + d s ν B ( y , z ) ) . ( d μ B ( x , z ) , d ν B ( x , z ) ) ≤ ( d μ B ( x , y ) , d ν B ( x , y ) ) + ( d μ B ( y , z ) , d ν B ( y , z ) ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) . □

Definition 3.3. Let G = (A, B) be a connected intuitionistic fuzzy graph. Then μ _B -eccentricity of a vertex x is the μ _B -sum distance to a vertex farthermost from x, i.e., e _{μ B}  (x) = max {d _{sμ B}  (x, y) : y ∈ V} and the ν _B -eccentricity of X is defined as e _{ν B}  (x) = max {d _{sν B}  (x, y) : y ∈ V}. The eccentricityof x is e (x) = (e _{μ B}  (x) , e _{ν B}  (x)). The μ _B  - radius and ν _B  - radius are the minimum of all μ _B -eccentricities and ν _B -eccentricities, respectively, r _{μ B}  (G) = min {e _{μ B}  (x) : x ∈ V} , r _{ν B}  (G) = min {e _{ν B}  (x) : x ∈ V} . The radius of the intuitionistic fuzzy graph G is r (G) = (r _{μ B}  (G) , r _{ν B}  (G)). The μ _B -diameter and ν _B -diameter are the maximum of all of the μ _B -eccentricities and ν _B -eccentricities, respectively i.e., d _{μ B}  (G) = max {e _{μ B}  (y) : y ∈ V} , d _{ν B}  (G) = max {e _{ν B}  (y) : y ∈ V} . The diameter of G is d (G) = (d _{μ B}  (G) , d _{ν B}  (G)).

Definition 3.4. A vertex y at a μ _B -sum distance e _{μ B}  (x) from x is called μ _B -eccentric vertex of x. Similarly, a vertex z at a ν _B -sum distance e _{ν B}  (x) from x is called a ν _B -eccentric vertex of x. If a vertex y is at a sum distance e (x) from x then y is known as an eccentric vertex. A vertex x is said to be μ _B -central vertex if e _{μ B}  (x) = r _{μ B}  (G) and ν _B -central vertex if e _{ν B}  (x) = r _{ν B}  (G). If x is both μ _B -central and ν _B -central then it is called a central vertex of the intuitionistic fuzzy graph. The subgraphs induced by all μ _B -central vertices and ν _B -central vertices are called μ _B -center and ν _B -center of the intuitionistic fuzzy graph. Similarly, the subgraph induced by the central vertices is known as center of the intuitionistic fuzzy graph. If the center of G is G itself then G is called a self-centered intuitionistic fuzzy graph. A vertex y is called a μ _B -peripheral vertex if e _{μ B}  (y) = d _{μ B}  (G), and ν _B -peripheral vertex if e _{ν B}  (y) = d _{ν B}  (G). If y is both μ _B -peripheral and ν _B -peripheral then it is called a peripheral vertex.

Example 3.5. Consider a connected intuitionistic fuzzy graph as shown in Fig. 1. The sum distances between the vertices are d ( a , b ) = ( 0.5 , 1.2 ) , d ( a , c ) = ( 0.8 , 1.5 ) d ( a , d ) = ( 0.7 , 1.2 ) , d ( a , e ) = ( 0.9 , 1.3 ) d ( b , c ) = ( 0.3 , 0.9 ) , d ( b , d ) = ( 0.2 , 1.0 ) d ( b , e ) = ( 0.9 , 1.1 ) , d ( c , d ) = ( 0.5 , 1.3 ) d ( c , e ) = ( 1.2 , 1.4 ) , d ( d , e ) = ( 0.7 , 1.3 ) . e ( a ) = ( 0.9 , 1.5 ) , e ( b ) = ( 0.9 , 1.2 ) e ( c ) = ( 1.2 , 1.5 ) , e ( d ) = ( 0.7 , 1.3 ) e ( e ) = ( 1.2 , 1.4 ) , r ( G ) = ( 0.7 , 1.2 ) d ( G ) = ( 1.2 , 1.5 ) . a is μ _B -central vertex and b ia ν _B -central vertex. c is a peripheral vertex. The μ _B -center of G is • d and ν _B -center is • b

Since there is no vertex of the graph whose eccentricity is equal to radius, therefore, G has no center.

The algorithm for computing the μ _B -sum distance matrix, eccentricities of all the vertices, radius and diameter of any intuitionistic fuzzy graph is shown by the flow chart in Fig. 2 and its running time complexity is calculated in Algorithm 1.

The running time complexity of lines 1 - 4, 13 and 14 is n and lines 5 - 12 is n × n. Therefore, the time complexity of Algorithm 1 is O (n²), where n is the number of vertices of the intuitionistic fuzzy graph. The pseudocode of the algorithm in Fortran language is given as:

Pseudocode in Fortran language

program membership

implicit none

integer:: j, i, k, u(v), m, n, v, l

real:: a(v,v), dist, e(v,v), eccen(v), rad, diam

print*, “Enter the number of vertices”

read*, v

print*, “Enter the enteries of adjacency matrix of membership values row-wise”

read*, ((a(i, j), j=1, v), i=1, v)

print*, “The μ _B -sum distance matrix is”

do m= 1, v

120 u(1)= m

e(m, m)= 0.0

dist= 1000.0

do i= 1, v

if(a(u(1), i) > 0.0)then

e(m, i)= a(u(1), i)

dist= min(dist,e(m, i))

if(e(m, i)== dist)then

u(2)= i

end if

else

e(m, i)= 1000.0

end if

end do

n= 2

u(n)= u(2)

do i= 3, v

u(i)= 0.0

end do

do j= 1, v-2

dist= 100.0

do k= 1, v

do l= 1, v-1

if(k== u(l))then

goto 150

end if

end do

if(a(u(n), k) > 0.0)then

if(e(m, u(n))+ a(u(n), k) < e(m, k))then

e(m, k)= e(m, u(n))+ a(u(n), k)

else

e(m, k)= e(m, k)

end if

end if

150 end do

do k= 1, v

do l =1, v-1

if(k== u(l))then

goto 250

end if

end do

dist= min(dist,e(m, k))

if(e(m, k)== dist)then

u(n+1)= k

end if

250 end do

n= n+1

end do

do j= 1, v

if(m== j)then

e(m, j)= 0.0

end if

end do

do i= 1, v

eccen(m)= max(eccen(m), e(m, i))

end do

print 90, (e(m, j), j= 1, v)

90 format(2x, 10f5.2)

end do

print*, “The μ _B -eccentricities are”

print 200, (eccen(m), m= 1, v)

200 format(3x, 10f4.2)

rad= 1000.0

do m= 1, v

rad= min(rad, eccen(m))

diam= max(diam, eccen(m))

end do

print 20, rad

20 format(“the μ _B -radius is”, f5.2)

print 10, diam

10 format(“the μ _B -diameter is”, f5.2)

end program membership

Example 3.6. Using the above pseudocode, the output results for the membership values of intuitionistic fuzzy graph in Fig. 1 are

Enter the number of vertices 5 Enter the enteries of adjacency matrix of membership values row-wise.

0.0 0.5 0.0 0.0 0.9

0.5 0.0 0.3 0.2 0.0

0.0 0.3 0.0 0.6 0.0

0.0 0.2 0.6 0.0 0.7

0.9 0.0 0.0 0.7 0.0

The μ _B -sum distance matrix is

0.0 0.5 0.8 0.7 0.9

0.5 0.0 0.3 0.2 0.9

0.8 0.3 0.0 0.5 1.2

0.7 0.2 0.5 0.0 0.7

0.9 0.9 1.2 0.7 0.0

The μ _B -eccentricities are

0.9 0.9 1.2 0.7 1.2

the μ _B -radius is 0.7

the μ _B -diameter is 1.2

The algorithm for calculating the ν _B -sum distance matrix, the ν _B -eccentricities of all the vertices, ν _B -radius and ν _B -diameter is explained by the flow chart in Fig. 3 and its running time complexity is discussed in the Algorithm 2.

The running time complexity of lines 1 - 4, 13 and 14 is n and lines 5 - 12 is n × n. Therefore, the time complexity of Algorithm 2 is O (n²), where n is the number of vertices of the intuitionistic fuzzy graph. The pseudocode of the algorithm in Fortran language is given as:

Pseudocode in Fortran language

program Nonmembership

implicit none

integer:: i, j, k, u(v), m, p, n, v, l real:: a(v, v), dist,

e(v, v), value(v, v), eccen(v), rad=-100.0,

diam=100.0

print*, “Enter the number of vertices”

read*, v

print*, “Enter the enteries of adjacency matrix row-wise”

read*, ((a(i, j), j= 1, v), i= 1, v)

do i= 1, v

do j= 1, v

a(i, j)= -a(i, j)

end do

end do

print*, “The ν _B -sum distance matrix is”

do m= 1, v

eccen(m)= 100.0

u(1)= m

do k= 1, v

if(abs(a(u(1), k))> 0.0)then

e(m, k)= a(m, k)

u(2)= k

do p= 1, v

if(p/= k.and. p/= m)then

e(m, p)= 100.0

end if

end do

n= 2

u(n)= u(2)

do i= 3, v

u(i)= 0.0

end do

do j= 1, v-2

dist= 100.0

do i= 1, v

do l= 1, v-1

if(i== u(l))then

goto 400

end if

end do

if(abs(a(u(n), i))> 0.0)then

if(e(m, u(n))+ a(u(n), i)< e(m, i))then

e(m, i)= e(m, u(n))+ a(u(n), i)

end if

end if

400 end do

do i= 1, v

do l= 1, v-1

if(i== u(l))then

goto 300

end if

end do

dist= min(dist, e(m, i))

if(e(m, i)==dist)then

u(n+1)= i

end if

300 end do

n= n+1

end do

end if

do i= 1, v

if(i/= m)then

value(m, i)= min(value(m, i), e(m, i))

end if

end do

end do

do i= 1, v

eccen(m)= min(eccen(m), value(m, i))

end do

print 90, (abs(value(m, j)), j= 1, v)

90 format(2x, 10f5.2)

end do

print*, “The  ν _B -eccentricities are”

print 200, (abs(eccen(m)), m= 1, v)

200 format(3x, 10f4.2)

do m= 1, v

rad= max(rad, eccen(m))

diam= min(diam, eccen(m))

end do

print 20, abs(rad)

20 format(“the ν _B -radius is”, f5.2)

print 10, abs(diam)

10 format(“the ν _B -diameter is”, f5.2)

end program

Example 3.7. Using the above pseudocode, the output results for the non-membership values of the intuitionistic fuzzy graph in Fig. 1 are

Enter the number of vertices

5

Enter the enteries of adjacency matrix of non-membership values row-wise.

0.0 0.2 0.0 0.0 0.1

0.2 0.0 0.7 0.6 0.0

0.0 0.7 0.0 0.3 0.0

0.0 0.6 0.3 0.0 0.1

0.1 0.0 0.0 0.1 0.0

The ν _B -sum distance matrix is

0.0 0.8 1.5 1.2 1.3

1.2 0.0 0.9 1.0 1.1

1.5 0.9 0.0 1.3 1.4

1.2 1.0 1.3 0.0 1.3

0.9 0.7 1.4 1.3 0.0

The ν _B -eccentricities are

1.5 1.2 1.5 1.3 1.4

the ν _B -radius is 1.2

the ν _B -diameter is 1.5

Theorem 3.8. For every intuitionistic fuzzy graph G, radius and diameter satisfy the following inequality r (G) ≤ d (G) ≤2r (G).

Proof. By Definition 3.3, we have r μ B ( G ) ≤ d μ B ( G ) and r ν B ( G ) ≤ d ν B ( G ) ⇒ ( r μ B ( G , r ν B ( G ) ) ≤ ( d μ B ( G ) , d ν B ( G ) ) ⇒ r ( G ) ≤ d ( G ) .

Let u, v be μ _B -central and ν _B -central vertices of G respectively, then e μ B ( u ) = r μ B ( G ) and e ν B ( v ) = r ν B ( G ) .

Let x, y be the μ _B -peripheral and ν _B -peripheral vertices, respectively. Since d defines a metric, for some vertices z, w ∈ V, d ( G ) = ( d μ B ( G ) , d ν B ( G ) ) = ( d μ B ( x , z ) , ( d ν B ( y , w ) ) ≤ ( d μ B ( x , u ) + d μ B ( u , z ) , d ν B ( y , v ) + d ν B ( v , w ) for some u , v ≤ ( r μ B ( G ) + r μ B ( G ) , r ν B ( G ) + r ν B ( G ) ) = ( 2 r μ B ( G ) , 2 r ν B ( G ) ) = 2 r ( G ) . ∥

It completes the proof. □

The following theorem gives an absolute difference between the eccentricities of any two adjacent vertices.

Theorem 3.9. For any two adjacent vertices x and y in a connected intuitionistic fuzzy graph G, |e (x) - e (y) | ≤ (1, d _{ν B}  (x, y)).

Proof. Without loss of generality assume that e (x) ≤ e (y). For any vertices z and w

e ( y ) = ( e μ B ( y ) , e ν B ( y ) ) = ( d μ B ( y , z ) , d ν B ( y , w ) ) ≤ ( d μ B ( y , x ) + d μ B ( x , z ) , d ν B ( y , x ) + d ν B ( x , w ) ) ≤ ( 1 + e μ B ( x ) , d ν B ( y , x ) + e ν B ( x ) ) = ( 1 , d ν B ( x , y ) ) + ( e μ B ( x ) , e ν B ( x ) ) = ( 1 , d ν B ( x , y ) ) + e ( x ) ⇒ - ( e ( x ) - e ( y ) ) ≤ ( 1 , d ν B ( x , y ) ) .(1) For any two vertices u and v, e ( x ) = ( e μ B ( x ) , e ν B ( x ) ) = ( d μ B ( x , u ) , d ν B ( x , v ) ) ≤ ( d μ B ( x , y ) + d μ B ( y , u ) , d ν B ( x , y ) + d ν B ( y , v ) ) ≤ ( 1 + e μ B ( y ) , d ν B ( y , x ) + e ν B ( y ) ) = ( 1 , d ν B ( x , y ) ) + ( e μ B ( y ) , e ν B ( y ) ) = ( 1 , d ν B ( x , y ) ) + e ( y )

⇒ e ( x ) - e ( y ) ≤ ( 1 , d ν B ( x , y ) ) .(2)

Combining the inequalities (1) and (2), we obtain the required result. □

If we consider any two arbitrary vertices x and y in Theorem 3.9, we have the following result.

Theorem 3.10. For any two vertices x and y in a connected intuitionistic fuzzy graph G, |e (x) - e (y) | ≤ (d _{μ B}  (x, y) , d _{ν B}  (x, y)).

Theorem 3.11. For any two adjacent vertices x and y in a connected intuitionistic fuzzy graph G, | d ( x , z ) - d ( y , z ) | ≤ ( 1 , d ν B ( x , y ) ) for every vertex z in G.

Proof. Since d defines a metric on V, d ( x , z ) = ( d μ B ( x , z ) , d ( x , z ) ) ≤ ( d μ B ( x , y ) + d μ B ( y , z ) , d ν B ( x , y ) + d ν B ( y , z ) ) = ( 1 + d μ B ( y , z ) , d ν B ( x , y ) + d ν B ( y , z ) = ( 1 , d ν B ( x , y ) ) + ( d μ B ( y , z ) , d ν B ( y , z ) ) = ( 1 , d ν B ( x , y ) ) + d ( y , z )

d ( x , z ) - d ( y , z ) ≤ ( 1 , d ν B ( x , y ) ) .(3)

For the vertex y, d ( y , z ) = ( d μ B ( y , z ) , d ( y , z ) ) ≤ ( d μ B ( y , x ) + d μ B ( x , z ) , d ν B ( y , x ) + d ν B ( x , z ) ) = ( 1 + d μ B ( x , z ) , d ν B ( x , y ) + d ν B ( x , z ) = ( 1 , d ν B ( x , y ) ) + ( d μ B ( x , z ) , d ν B ( x , z ) ) = ( 1 , d ν B ( x , y ) ) + d ( x , z )

- ( d ( x , z ) - d ( y , z ) ) ≤ ( 1 , d ν B ( x , y ) ) .(4)

From inequalities (3) and (4), we obtain the required result. □ We now generalize Theorem 3.11 for any two vertices x and y in the following theorem.

Theorem 3.12. For any two vertices x and y in a connected intuitionistic fuzzy graph G, |d (x, z) - d (y, z) | ≤ (d _{μ B}  (x, y) , d _{ν B}  (x, y)).

Theorem 3.13. If G is a self centered intuitioniatic fuzzy graph, then each vertex of G is an eccentric vertex.

Proof. Let y be an eccentric vertex of x then y is μ _B -eccentric and ν _B -eccentric vertex of x, that is, e ( x ) = ( e μ B ( x ) , e ν B ( x ) ) = ( d μ B ( x , y ) , d ν B ( x , y ) .

Since G is self centered, e ( x ) = e ( y ) = r ( G ) ⇒ e ( y ) = ( d μ B ( x , y ) , d ν B ( x , y ) ) .

It shows that x is an eccentric vertex of y. Since x was taken to be arbitrary, theorem is true for all vertices of G. □

Remark. Theorem 3.13 also holds when we consider only μ _B -self centered graphs and ν _B -self centered graphs.

Theorem 3.14. Let G be a μ _B -self centered graph then each vertex of G is μ _B -eccentric vertex.

Proof. Since G is a μ _B -self centered graph therefore each vertex of G is a μ _B -central vertex, i.e., e μ B ( x ) = r μ B ( G ) for all x ∈ G .

Let x ba any vertex and y be an eccentric vertex of x then e μ B ( x ) = d μ B ( x , y ) .

As G is self centered, e μ B ( y ) = e μ B ( x ) = d μ B ( x , y ) .

It implies that x in a μ _B -eccentric vertex of y. Hence the proof. □

Theorem 3.15. If G is a ν _B -self centered graph, then each vertex of G is a ν _B -eccentric vertex.

Remark. The converse of Theorems 3.13, 3.14 and 3.15 is not true as it can be seen in Example 3.16.

Example 3.16. Consider a connected intuitionistic fuzzy graph as shown in Fig. 4. d ( x , y ) = ( 0.9 , 1.2 ) d ( x , z ) = ( 0.4 , 0.7 ) d ( x , w ) = ( 0.7 , 0.8 ) d ( y , z ) = ( 0.5 , 0.7 ) d ( y , w ) = ( 0.6 , 0.2 ) d ( z , w ) = ( 0.8 , 1.1 ) e ( x ) = ( 0.9 , 1.2 ) e ( y ) = ( 0.9 , 1.2 ) e ( z ) = ( 0.8 , 1.1 ) e ( w ) = ( 0.8 , 1.1 ) r ( G ) = ( 0.5 , 0.9 ) d ( G ) = ( 0.9 , 1.2 )

All vertices of G are eccentric but G is not self-centered intuitionistic fuzzy graph because the center of G is:

Theorem 3.17. Let G be a self centered intuitionistic fuzzy graph. For every two vertices x, y ∈ G if x ∈ Y^*, then y ∈ X^*, where X^* is the set of all eccentric vertices of x any Y^* is the set of all eccentric vertices of y.

Proof. Since x is a eccentric vertex of y, e (x) = (d _{μ B}  (x, y) , d _{ν B}  (x, y)) ⇒ x ∈ Y^*. It is given that G is a self centered intuitionistic fuzzy graph, so e (y) = e (x) = (d _{μ B}  (y, x) , d _{ν B}  (y, x)), i.e., y is an eccentric vertex of x. Hence y ∈ X^*. □

Remark. The converse of Theorem 3.17 is not true in general. From example 3.16, x ∈ Y^*, y ∈ X^*, z ∈ W^* and w ∈ Z^* but G is not self-centered.

From Theorem 3.17, we can induce the following two lemmas.

Lemma 3.18. Let G be a ν _B -self centered intuitionistic fuzzy graph. For every two vertices x, y ∈ G if x ∈ Y ν B * then y ∈ X ν B *, where X ν B * is the set of all ν _B -eccentric vertices of x any Y ν B * is the set of all ν _B -eccentric vertices of y.

Proof. This theorem can be proved on the same lines as above. Since x is a ν _B -eccentric vertex of y, e _{ν B}  (x) = d _{ν B}  (x, y) ⇒ x ∈ Y ν B *. It is given that G is ν _B -self centered intuitionistic fuzzy graph, so e _{ν B}  (y) = e _{ν B}  (x) = d _{ν B}  (y, x). Hence y ∈ X ν B *. □

Lemma 3.19. Let G be a μ _B -self centered intuitionistic fuzzy graph. For every pair of vertices x, y ∈ G if x ∈ Y μ B * then y ∈ X μ B *, where X μ B * is the set of all μ _B -eccentric vertices of x any Y μ B * is the set of all μ _B -eccentric vertices of y.

Proof. Since x is a μ _B -eccentric vertex of y, e _{μ B}  (x) = d _{μ B}  (x, y) ⇒ x ∈ Y μ B *. It is given that G is a μ _B -self centered intuitionistic fuzzy graph, so e _{μ B}  (x) = e _{μ B}  (y) = d _{μ B}  (y, x). Hence y ∈ X μ B *. □

Remark. The converse of Lemmas 3.18 and 3.19 are not true in general.

Theorem 3.20. Let G be an intuitionistic fuzzy graph then all peripheral vertices are eccentric vertices.

Proof. Let x be a peripheral vertex and y be its eccentric vertex then,

d ( G ) = e ( x ) = ( d μ B ( x , y ) , d ν B ( x , y ) ) = ( d μ B ( y , x ) , d ν B ( y , x ) ) .(5)

It is only possible if e (y) = d (G) = d (y, x). It shows that x is an eccentric vertex of y. Hence the proof. □

Remark. The condition in Theorem 3.20 is not sufficient. In Example 3.16, z and w are eccentric vertices of each other but these are not peripheral vertices.

Lemma 3.21. Let G be an intuitionistic fuzzy graph then all μ _B -peripheral vertices are μ _B -eccentric vertices.

Proof. Let x be a μ _B -peripheral vertex then e _{μ B}  (x) = d _{μ B}  (G). There exists a vertex y such that d μ B ( G ) = e μ B ( x ) = d μ B ( x , y ) ⇒ d μ B ( G ) = d μ B ( y , x ) ⇒ e μ B ( y ) = d μ B ( G ) = d μ B ( y , x ) .

Clearly, x is a μ _B -eccentric vertex of y and the proof is complete. □

Lemma 3.22. Let G be an intuitionistic fuzzy graph then all ν _B -peripheral vertices are ν _B -eccentric vertices.

Remark. The conditions in Lemmas 3.21 and 3.22 are also not sufficient. In Example 3.16, z and w are are μ _B -eccentric and ν _B -eccentric vertices but these are not μ _B -peripheral and ν _B -peripheral vertices, respectively.

Theorem 3.23. Let G be a complete intuitionistic fuzzy graph such that μ _A  (x₁) ≤ μ _A  (x₂) ≤ μ _A  (x₃) ≤ . . . ≤ μ _A  (x _n ) and ν (x₁) ≥ ν _A  (x₂) ≥ ν _A  (x₃) ≥ . . . ≥ ν _A  (x _n ) then the sum distance between any twovertices x _i and x _j is (min {μ _B  (x _i , x _j ) , 2μ _A  (x₁)} , max {ν _B  (x _i , x _j ) , 2ν _A  (x₁)}).

Proof. Let x _i and x _j be any two vertices of G then for some x _k , x _l  ∈ V, the sum distance between x _i and x _j can be defined as:

d ( x i , x j ) = ( d μ B ( x i , x j ) , d ν B ( x i , x j ) ) = ( min { μ B ( x i , x j ) , μ B ( x i , x k ) + μ B ( x k , x j ) } , max { ν B ( x i , x j ) , ν B ( x i , x l ) + ν B ( x l , x j ) } ) .(6)

Since G is a complete intuitionistic fuzzy graph, we have ( μ B ( x i , x k ) , ν B ( x i , x l ) ) = ( min { μ A ( x i ) , μ A ( x k ) } , max { ν A ( x i ) , ν A ( x l ) } )

Since μ _A  (x₁) ≤ μ _A  (x₂) ≤ μ _A  (x₃) ≤ . . . μ _A  (x _n ) and ν _A  (x₁) ≥ ν _A  (x₂) ≥ ν _A  (x₃) ≥ . . . ≥ ν _A  (x _n ), therefore for k = l = 1, ( μ B ( x i , x k ) , ν B ( x i , x l ) ) = ( μ A ( x 1 ) , ν A ( x 1 ) )

Similarly, ( μ B ( x k , x j ) , ν B ( x l , x j ) ) = ( μ A ( x 1 ) , ν A ( x 1 ) ) .

Equation (6) takes the form as d ( x i , x j ) = ( min { μ B ( x i , x j ) , 2 μ A ( x 1 ) } , max { ν B ( x i , x j ) , 2 ν A ( x 1 ) } ) . □

Fuzzy decision making is a most important scientific, social and economic endeavour, there exist several major approaches within the theories of fuzzy decision making. Consider an example, a doctor wants to build a hospital at a place where patient can easily visit him in minimum time. He also wants to choose a standard area of decent reputation, clean with better hygienic conditions and spend minimum time in commuting from one place to another. He has six options of places, namely, P₁, P₂, P₃, P₄, P₅, and P₆. We construct here an intuitionistic fuzzy graph. In this intuitionistic fuzzy graph, membership values of vertices represent the degree of reputation and cleanliness situation of the area, and the non-membership values represent the degree of bad reputation and unhealthy hygienic conditions of the location, respectively. The edges between the vertices represent the number of patients who can travel from one place to another to visit the doctor. The intuitionistic fuzzy graph is shown in Fig. 5.

The degree of membership of P₁ represents that P₁ is 50% suitable with respect to reputation and cleanliness conditions and 40% not well reputed and clean. The edge between P₁ and P₂ represents that 50% of the people can visit the doctor easily from P₁ to P₂. The best location can be obtained by finding the center of intuitionistic fuzzy graph. By routine calculations and using the Fortran algorithms, we have e ( P 1 ) = ( 0.9 , 1.6 ) , e ( P 2 ) = ( 1.0 , 1.8 ) e ( P 3 ) = ( 1.0 , 1.9 ) , e ( P 4 ) = ( 0.9 , 1.7 ) e ( P 5 ) = ( 0.9 , 1.9 ) , e ( P 6 ) = ( 0.8 , 1.6 ) r ( G ) = ( 0.8 , 1.6 ) , d ( G ) = ( 1.0 , 1.9 ) . P₅ is the center of the graph, therefore, it is the best location for the hospital.

5 Conclusion and future work

Fuzzy graph theory has numerous applications in modern sciences and technology, especially in the fields of operations research, neural networks, artificial intelligence and decision making. An intuitionistic fuzzy set is a generalization of the notion of a fuzzy set. The intuitionistic fuzzy models give more precision, flexibility and compatibility to the system as compared to the classical and fuzzy models. We have introduced the concepts of certain metrics in intuitionistic fuzzy graphs in this paper. An algorithm for computing the sum distance matrix, eccentricity of the vertices, radius and diameter in intuitionistic fuzzy graphs is presented and running time complexity of algorithms is calculated. At the end, an application of intuitionistic fuzzy graphs is discussed. We plan to extend our research of intuitionistic fuzzification to (1) Certain metrics in bipolar fuzzy hypergraphs, (2) Certain metrics in intuitionistic fuzzy hypergraphs, and (3) Roughness in hypergraphs.