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Interval-valued fuzzy graphs
Computers and Mathematics with Applications 61 (2011) 289–299
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Interval-valued fuzzy graphs Muhammad Akram a,∗ , Wieslaw A. Dudek b a
Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore-54000, Pakistan
b
Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370, Wroclaw, Poland
article
info
Article history: Received 25 August 2010 Received in revised form 1 November 2010 Accepted 1 November 2010 Keywords: Interval-valued fuzzy graph Self-complementary Interval-valued fuzzy complete graph
abstract We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of intervalvalued fuzzy complete graphs and present some properties of self-complementary and selfweak complementary interval-valued fuzzy complete graphs. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction In 1975, Zadeh [1] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy sets [2] in which the values of the membership degrees are intervals of numbers instead of the numbers. Interval-valued fuzzy sets provide a more adequate description of uncertainty 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 [3]. Since interval-valued fuzzy sets are widely studied and used, we describe briefly the work of Gorzalczany on approximate reasoning [4,5], Roy and Biswas on medical diagnosis [6], Turksen on multivalued logic [7] and Mendel on intelligent control [3]. The fuzzy graph theory as a generalization of Euler’s graph theory was first introduced by Rosenfeld [8] in 1975. The fuzzy relations between fuzzy sets were first considered by Rosenfeld and he developed the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts. Later, Bhattacharya [9] gave some remarks on fuzzy graphs, and some operations on fuzzy graphs were introduced by Mordeson and Peng [10]. The complement of a fuzzy graph was defined by Mordeson [11] and further studied by Sunitha and Vijayakumar [12]. Bhutani and Rosenfeld introduced the concept of M-strong fuzzy graphs in [13] and studied some properties. The concept of strong arcs in fuzzy graphs was discussed in [14]. Hongmei and Lianhua gave the definition of interval-valued graph in [15]. In this paper, we define the operations of Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some properties. We study isomorphism (resp. weak isomorphism) between interval-valued fuzzy graphs in an equivalence relation (resp. partial order). We introduce the notion of interval-valued fuzzy complete graphs and present some properties of self-complementary and self-weak complementary interval-valued fuzzy complete graphs.
∗
Corresponding author. Tel.: +92 3334510258; fax: +92 99212505. E-mail addresses: [email protected], [email protected] (M. Akram), [email protected] (W.A. Dudek).
0898-1221/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.11.004
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M. Akram, W.A. Dudek / Computers and Mathematics with Applications 61 (2011) 289–299
The definitions and terminologies that we used in this paper are standard. For other notations, terminologies and applications, the readers are referred to [16–28].
2. Preliminaries A graph is an ordered pair G∗ = (V , E ), where V is the set of vertices of G∗ and E is the set of edges of G∗ . Two vertices x and y in a graph G∗ are said to be adjacent in G∗ if {x, y} is in an edge of G∗ . (For simplicity an edge {x, y} will be denoted by xy.) A simple graph is a graph without loops and multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. The complete graph on n vertices has n vertices and n(n − 1)/2 edges. We will consider only graphs with the finite number of vertices and edges. By a complementary graph G∗ of a simple graph G∗ we mean a graph having the same vertices as G∗ and such that two vertices are adjacent in G∗ if and only if they are not adjacent in G∗ . An isomorphism of graphs G∗1 and G∗2 is a bijection between the vertex sets of G∗1 and G∗2 such that any two vertices v1 and v2 of G∗1 are adjacent in G∗1 if and only if f (v1 ) and f (v2 ) are adjacent in G∗2 . Isomorphic graphs are denoted by G∗1 ≃ G∗2 . Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be two simple graphs, we can construct several new graphs. The first construction called the Cartesian product of G∗1 and G∗2 gives a graph G∗1 × G∗2 = (V , E ) with V = V1 × V2 and E = {(x, x2 )(x, y2 ) | x ∈ V1 , x2 y2 ∈ E2 } ∪ {(x1 , z )(y1 , z ) | x1 y1 ∈ E1 , z ∈ V2 } . The composition of graphs G∗1 and G∗2 is the graph G∗1 [G∗2 ] = (V1 × V2 , E 0 ), where E 0 = E ∪ {(x1 , x2 )(y1 , y2 ) | x1 y1 ∈ E1 , x2 ̸= y2 } and E is defined as in G∗1 × G∗2 . Note that G∗1 [G∗2 ] ̸= G∗2 [G∗1 ]. The union of graphs G∗1 and G∗2 is defined as G∗1 ∪ G∗2 = (V1 ∪ V2 , E1 ∪ E2 ). The join of G∗1 and G∗2 is the simple graph G∗1 + G∗2 = (V1 ∪ V2 , E1 ∪ E2 ∪ E ′ ), where E ′ is the set of all edges joining the nodes of V1 and V2 . In this construction it is assumed that V1 ∩ V2 ̸= ∅. By a fuzzy subset µ on a set X is mean a map µ : X → [0, 1]. A map ν : X × X → [0, 1] is called a fuzzy relation on X if ν(x, y) ≤ min(µ(x), µ(y)) for all x, y ∈ X . A fuzzy relation ν is symmetric if ν(x, y) = ν(y, x) for all x, y ∈ X . An interval number D is an interval [a− , a+ ] with 0 ≤ a− ≤ a+ ≤ 1. The interval [a, a] is identified with the number a ∈ [0, 1]. D[0, 1] denotes the set of all interval numbers. + − + For interval numbers D1 = [a− 1 , b1 ] and D2 = [a2 , b2 ], we define
• • • • • • •
+ − + − − + + rmin(D1 , D2 ) = rmin([a− 1 , b1 ], [a2 , b2 ]) = [min{a1 , a2 }, min{b1 , b2 }], + − + − − + + rmax(D1 , D2 ) = rmax([a− 1 , b1 ], [a2 , b2 ]) = [max{a1 , a2 }, max{b1 , b2 }], − − − + + + + D1 + D2 = [a− 1 + a2 − a1 · a2 , b1 + b2 − b1 · b2 ], − + + D1 ≤ D2 ⇐⇒ a− 1 ≤ a2 and b1 ≤ b2 , − + + D1 = D2 ⇐⇒ a− 1 = a2 and b1 = b2 , D1 < D2 ⇐⇒ D1 ≤ D2 and D1 ̸= D2 , + − + kD = k[a− 1 , b1 ] = [ka1 , kb1 ], where 0 ≤ k ≤ 1.
Then, (D[0, 1], ≤, ∨, ∧) is a complete lattice with [0, 0] as the least element and [1, 1] as the greatest. The interval-valued fuzzy set A in V is defined by A=
− x, µA (x), µ+ :x∈V , A ( x)
+ − + where µ− A (x) and µA (x) are fuzzy subsets of V such that µA (x) ≤ µA (x) for all x ∈ V . For any two interval-valued sets − + − + A = [µA (x), µA (x)] and B = [µB (x), µB (x)] in V we define:
− + + • A B = {(x, max(µ− A (x), µB (x)), max(µA (x), µB (x))) : x ∈ V }, − + + • A B = {(x, min(µ− A (x), µB (x)), min(µA (x), µB (x))) : x ∈ V }. If G∗ = (V , E ) is a graph, then by an interval-valued fuzzy relation B on a set E we mean an interval-valued fuzzy set such that − − µ− B (xy) ≤ min(µA (x), µA (y)), + + µ+ B (xy) ≤ min(µA (x), µA (y))
for all xy ∈ E.
M. Akram, W.A. Dudek / Computers and Mathematics with Applications 61 (2011) 289–299
291
3. Operations on interval-valued fuzzy graphs Throughout this paper, G∗ is a crisp graph, and G is an interval-valued fuzzy graph. + Definition 3.1. By an interval-valued fuzzy graph of a graph G∗ = (V , E ) we mean a pair G = (A, B), where A = [µ− A , µA ] is − + an interval-valued fuzzy set on V and B = [µB , µB ] is an interval-valued fuzzy relation on E.
Example 3.2. Consider a graph G∗ = (V , E ) such that V = {x, y, z }, E = {xy, yz , zx}. Let A be an interval-valued fuzzy set of V and let B be an interval-valued fuzzy set of E ⊆ V × V defined by
x y z x y z , , , , , , 0.2 0.3 0.4 0.4 0.5 0.5 xy yz zx xy yz zx B= , , , , , . 0.1 0.2 0.1 0.3 0.4 0.4 A=
By routine computations, it is easy to see that G = (A, B) is an interval-valued fuzzy graph of G∗ . Definition 3.3. The Cartesian product G1 × G2 of two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ) of the graphs G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) is defined as a pair (A1 × A1 , B1 × B2 ) such that
(i) (ii) (iii)
(µ− × µ− )(x1 , x2 ) = min(µ− (x1 ), µ− (x2 )) A A A A 1
2
1
1
2
1
2
(µ+ × µ+ )(x1 , x2 ) = min(µ+ (x1 ), µ+ (x2 )) A A A A
for all (x1 , x2 ) ∈ V ,
2
−
−
−
−
1
2
1
2
1
2
1
2
−
−
−
1
2
1
2
1
2
1
2
(µB × µB )((x, x2 )(x, y2 )) = min(µA (x), µB (x2 y2 )) (µ+ × µ+ )((x, x2 )(x, y2 )) = min(µ+ (x), µ+ (x2 y2 )) B B A B
for all x ∈ V1 and x2 y2 ∈ E2 ,
−
(µB × µB )((x1 , z )(y1 , z )) = min(µB (x1 y1 ), µA (z )) (µ+ × µ+ )((x1 , z )(y1 , z )) = min(µ+ (x1 y1 ), µ+ (z )) B B B A
for all z ∈ V2 and x1 y1 ∈ E1 .
Example 3.4. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be graphs such that V1 = {a, b}, V2 = {c , d}, E1 = {ab} and E2 = {cd}. Consider two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ), where
a b , , , 0.2 0.3 0.4 0.5 c d c d , , , , A2 = 0.1 0.2 0.4 0.6 A1 =
a
,
b
B1 =
,
ab
0.1 0.2
B2 =
ab cd
,
cd
0.1 0.3
, .
Then, it is not difficult to verify − (µ− B1 × µB2 )((a, c )(a, d)) = 0.1,
+ (µ+ B1 × µB2 )((a, c )(a, d)) = 0.3,
− (µ− B1 × µB2 )((a, c )(b, c )) = 0.1,
+ (µ+ B1 × µB2 )((a, c )(b, c )) = 0.2,
− (µ− B1 × µB2 )((a, d)(b, d)) = 0.1,
+ (µ+ B1 × µB2 )((a, d)(b, d)) = 0.2,
− (µ− B1 × µB2 )((b, c )(b, d)) = 0.1,
+ (µ+ B1 × µB2 )((b, c )(b, d)) = 0.3.
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M. Akram, W.A. Dudek / Computers and Mathematics with Applications 61 (2011) 289–299
By routine computations, it is easy to see that G1 × G2 is an interval-valued fuzzy graph of G∗1 × G∗2 . Proposition 3.5. The Cartesian product G1 × G2 = (A1 × A2 , B1 × B2 ) of two interval-valued fuzzy graphs of the graphs G∗1 and G∗2 is an interval-valued fuzzy graph of G∗1 × G∗2 . Proof. We verify only conditions for B1 × B2 because conditions for A1 × A2 are obvious. Let x ∈ V1 , x2 y2 ∈ E2 . Then − (µ− B1 × µB2 )((x, x2 )(x, y2 )) = ≤ = =
− min(µ− A1 (x), µB2 (x2 y2 ))
− − min(µ− A1 (x), min(µA2 (x2 ), µA2 (y2 )))
− − − min(min(µ− A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
− − − min((µ− A1 × µA2 )(x, x2 ), (µA1 × µA2 )(x, y2 )), + min(µ+ A1 (x), µB2 (x2 y2 ))
+ (µ+ B1 × µB2 )((x, x2 )(x, y2 )) = ≤ = =
+ + min(µ+ A1 (x), min(µA2 (x2 ), µA2 (y2 )))
+ + + min(min(µ+ A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
+ + + min((µ+ A1 × µA2 )(x, x2 ), (µA1 × µA2 )(x, y2 )).
Similarly for z ∈ V2 and x1 y1 ∈ E1 we have − (µ− B1 × µB2 )((x1 , z )(y1 , z )) = ≤ = =
− min(µ− B1 (x1 y1 ), µA2 (z ))
− − min(min(µ− A1 (x1 ), µA1 (y1 )), µA2 (z ))
− − − min(min(µ− A1 (x), µA2 (z )), min(µA1 (y1 ), µA2 (z )))
− − − min((µ− A1 × µA2 )(x1 , z ), (µA1 × µA2 )(y1 , z )),
+ (µ+ B1 × µB2 )((x1 , z )(y1 , z )) = ≤ = =
This completes the proof.
+ min(µ+ B1 (x1 y1 ), µA2 (z ))
+ + min(min(µ+ A1 (x1 ), µA1 (y1 )), µA2 (z ))
+ + + min(min(µ+ A1 (x), µA2 (z )), min(µA1 (y1 ), µA2 (z )))
+ + + min((µ+ A1 × µA2 )(x1 , z ), (µA1 × µA2 )(y1 , z )).
Definition 3.6. The composition G1 [G2 ] = (A1 ◦ A2 , B1 ◦ B2 ) of two interval-valued fuzzy graphs G1 and G2 of the graphs G∗1 and G∗2 is defined as follows:
(i) (ii) (iii)
(µ− ◦ µ− )(x1 , x2 ) = min(µ− (x1 ), µ− (x2 )) A A A A
1
2
1
2
1
2
1
2
(µ+ ◦ µ+ )(x1 , x2 ) = min(µ+ (x1 ), µ+ (x2 )) A A A A
for all (x1 , x2 ) ∈ V ,
(µ− ◦ µ− )((x, x2 )(x, y2 )) = min(µ− (x), µ− (x2 y2 )) B B A B
1
2
1
2
1
2
1
2
(µ+ ◦ µ+ )((x, x2 )(x, y2 )) = min(µ+ (x), µ+ (x2 y2 )) B B A B (µ− ◦ µ− )((x1 , z )(y1 , z )) = min(µ− (x1 y1 ), µ− (z )) B B B A
for all x ∈ V1 and x2 y2 ∈ E2 ,
1
2
1
2
1
2
1
2
(µ+ ◦ µ+ )((x1 , z )(y1 , z )) = min(µ+ (x1 y1 ), µ+ (z )) B B B A
for all z ∈ V2 and x1 y1 ∈ E1 ,
M. Akram, W.A. Dudek / Computers and Mathematics with Applications 61 (2011) 289–299
(iv)
(µ− ◦ µ− )((x1 , x2 )(y1 , y2 )) = min(µ− (x2 ), µ− (y2 ), µ− (x1 y1 )) B B A A B 1
2
2
2
1
1
2
2
2
1
(µ+ ◦ µ+ )((x1 , x2 )(y1 , y2 )) = min(µ+ (x2 ), µ+ (y2 ), µ+ (x1 y1 )) B B A A B
293
for all (x1 , x2 )(y1 , y2 ) ∈ E 0 − E.
Example 3.7. Let G∗1 and G∗2 be as in the previous example. Consider two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ) defined by
a b , , , , A1 = 0.2 0.3 0.5 0.5 d c d c , , , , A2 = 0.1 0.3 0.4 0.6
a
b
B1 =
,
ab
0.2 0.4
B2 =
ab cd
,
cd
0.1 0.3
, .
Then we have − (µ− B1 ◦ µB2 )((a, c )(a, d)) = 0.2,
+ (µ+ B1 ◦ µB2 )((a, c )(a, d)) = 0.3,
− (µ− B1 ◦ µB2 )((b, c )(b, d)) = 0.1,
+ (µ+ B1 ◦ µB2 )((b, c )(b, d)) = 0.3,
− (µ− B1 ◦ µB2 )((a, c )(b, c )) = 0.1,
+ (µ+ B1 ◦ µB2 )((a, c )(b, c )) = 0.4,
− (µ− B1 ◦ µB2 )((a, d)(b, d)) = 0.2,
+ (µ+ B1 ◦ µB2 )((a, d)(b, d)) = 0.4,
− (µ− B1 ◦ µB2 )((a, c )(b, d)) = 0.1,
+ (µ+ B1 ◦ µB2 )((a, c )(b, d)) = 0.4,
− (µ− B1 ◦ µB2 )((b, c )(a, d)) = 0.1,
+ (µ+ B1 ◦ µB2 )((b, c )(a, d)) = 0.4.
By routine computations, it is easy to see that G1 [G2 ] = (A1 ◦ A2 , B1 ◦ B2 ) is an interval-valued fuzzy graph of G∗1 [G∗2 ]. Proposition 3.8. The composition G1 [G2 ] of interval-valued fuzzy graphs G1 and G2 of G∗1 and G∗2 is an interval-valued fuzzy graph of G∗1 [G∗2 ]. Proof. Similarly as in the previous proof we verify the conditions for B1 ◦ B2 only. In the case x ∈ V1 , x2 y2 ∈ E2 , according to (ii) we obtain − (µ− B1 ◦ µB2 )((x, x2 )(x, y2 )) = ≤ = = + (µ+ B1 ◦ µB2 )((x, x2 )(x, y2 )) = ≤ = =
− min(µ− A1 (x), µB2 (x2 y2 ))
− − min(µ− A1 (x), min(µA2 (x2 ), µA2 (y2 )))
− − − min(min(µ− A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
− − − min((µ− A1 ◦ µA2 )(x, x2 ), (µA1 ◦ µA2 )(x, y2 )), + min(µ+ A1 (x), µB2 (x2 y2 ))
+ + min(µ+ A1 (x), min(µA2 (x2 ), µA2 (y2 )))
+ + + min(min(µ+ A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
+ + + min((µ+ A1 ◦ µA2 )(x, x2 ), (µA1 ◦ µA2 )(x, y2 )).
In the case z ∈ V2 , x1 y1 ∈ E1 the proof is similar.
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In the case (x1 , x2 )(y1 , y2 ) ∈ E 0 − E we have x1 y1 ∈ E1 and x2 ̸= y2 , which according to (iv) implies − (µ− B1 ◦ µB2 )((x1 , x2 )(y1 , y2 )) = ≤ = = + (µ+ B1 ◦ µB2 )((x1 , x2 )(y1 , y2 )) = ≤ = =
This completes the proof.
− − min(µ− A2 (x2 ), µA2 (y2 ), µB1 (x1 y1 ))
− − − min(µ− A2 (x2 ), µA2 (y2 ), min(µA1 (x1 ), µA1 (y1 )))
− − − min(min(µ− A1 (x1 ), µA2 (x2 )), min(µA1 (y1 ), µA2 (y2 )))
− − − min((µ− A1 ◦ µA2 )(x1 , x2 ), (µA1 ◦ µA2 )(y1 , y2 )), + + min(µ+ A2 (x2 ), µA2 (y2 ), µB1 (x1 y1 ))
+ + + min(µ+ A2 (x2 ), µA2 (y2 ), min(µA1 (x1 ), µA1 (y1 )))
+ + + min(min(µ+ A1 (x1 ), µA2 (x2 )), min(µA1 (y1 ), µA2 (y2 )))
+ + + min((µ+ A1 ◦ µA2 )(x1 , x2 ), (µA1 ◦ µA2 )(y1 , y2 )).
Definition 3.9. The union G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) of two interval-valued fuzzy graphs G1 and G2 of the graphs G∗1 and G∗2 is defined as follows:
(A)
(B)
(C)
(D)
− (µA1 ∪ µ−A2 )(x) = µ−A1 (x)
if x ∈ V1 and x ̸∈ V2 ,
(µ− ∪ µ− )(x) = µ− (x) if x ∈ V2 and x ̸∈ V1 , A A A
(µ−1 ∪ µ−2 )(x) = max2 (µ− (x), µ− (x)) if x ∈ V ∩ V , 1 2 A A A1 A2 +1 +2 + (µA1 ∪ µA2 )(x) = µA1 (x) if x ∈ V1 and x ̸∈ V2 , (µ+ ∪ µ+ )(x) = µ+ (x) if x ∈ V2 and x ̸∈ V1 , A A A
(µ+1 ∪ µ+2 )(x) = max2 (µ+ (x), µ+ (x)) if x ∈ V ∩ V , 1 2 A A A1 A2 −1 −2 − (µB1 ∪ µB2 )(xy) = µB1 (xy) if xy ∈ E1 and xy ̸∈ E2 , (µ− ∪ µ− )(xy) = µ− (xy) if xy ∈ E2 and xy ̸∈ E1 , B B B
(µ−1 ∪ µ−2 )(xy) = max2 (µ− (xy), µ− (xy)) if xy ∈ E ∩ E , 1 2 B B B1 B2 +1 +2 + (µB1 ∪ µB2 )(xy) = µB1 (xy) if xy ∈ E1 and xy ̸∈ E2 , (µ+ ∪ µ+ )(xy) = µ+ (xy) if xy ∈ E2 and xy ̸∈ E1 , B B B
(µ+1 ∪ µ+2 )(xy) = max2 (µ+ (xy), µ+ (xy)) B1
B2
B1
B2
if xy ∈ E1 ∩ E2 .
Example 3.10. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be graphs such that V1 = {a, b, c , d, e}, E1 = {ab, bc , be, ce, ad, ed}, V2 = {a, b, c , d, f } and E2 = {ab, bc , cf , bf , bd}. Consider two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ) defined by
A1 =
a
,
b
,
c
,
d
,
e
0.2 0.4 0.3 0.3 0.2
ab
,
ce
,
be
,
ad
,
a
,
b
,
c
,
bc
,
d
,
ce
,
e
,
be
0.4 0.5 0.6 0.7 0.6
de
,
ab
,
ad
, , a b c d f b c d f a A2 = , , , , , , , , , , 0.2 0.2 0.3 0.2 0.4 0.4 0.5 0.6 0.6 0.6 ab bc cf bf bd ab bc cf bf bd B2 = , , , , , , , , , . 0.1 0.2 0.1 0.1 0.2 0.2 0.4 0.5 0.2 0.5 B1 =
,
bc
,
0.1 0.2 0.1 0.2 0.1 0.1
de
0.3 0.4 0.5 0.5 0.3 0.6
,
Then, according to the above definition:
+ (µ+ A1 ∪ µA2 )(c ) = 0.6, − − (µB1 ∪ µB2 )(ab) = 0.1, − (µ− B1 ∪ µB2 )(ad) = 0.1, + (µB1 ∪ µ+ B3 )(ab) = 0.3, + (µ+ ∪ µ B1 B2 )(ad) = 0.3,
− (µ− A1 ∪ µA2 )(a) = 0.2, − (µA1 ∪ µ− A2 )(c ) = 0.3, − (µ− ∪ µ A1 A2 )(e) = 0.2, + + (µA1 ∪ µA2 )(a) = 0.4, + (µ+ A1 ∪ µA2 )(d) = 0.7, − − (µB1 ∪ µB2 )(bc ) = 0.2, − (µ− B1 ∪ µB2 )(de) = 0.1, + + (µB1 ∪ µB2 )(bc ) = 0.4, + (µ+ B1 ∪ µB2 )(de) = 0.6,
− (µ− A1 ∪ µA2 )(b) = 0.4, − − (µA1 ∪ µA2 )(d) = 0.3, − (µ− A1 ∪ µA2 )(f ) = 0.4, + + (µA1 ∪ µA2 )(b) = 0.5, + (µ+ A1 ∪ µA2 )(e) = 0.1, − − (µB1 ∪ µB2 )(ce) = 0.1, − (µ− B1 ∪ µB2 )(bd) = 0.2, + (µB1 ∪ µ+ B2 )(ce) = 0.5, + (µ+ ∪ µ B1 B2 )(bd) = 0.5,
+ (µ+ A1 ∪ µA2 )(f ) = 0.6, − − (µB1 ∪ µB2 )(be) = 0.2, − (µ− B1 ∪ µB2 )(bf ) = 0.1, + (µB1 ∪ µ+ B2 )(be) = 0.5, + (µ+ ∪ µ B1 B2 )(bf ) = 0.2.
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Clearly, G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of the graph G∗1 ∪ G∗2 . Proposition 3.11. The union of two interval-valued fuzzy graphs is an interval-valued fuzzy graph. Proof. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. We prove that G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of the graph G∗1 ∪ G∗2 . Since all conditions for A1 ∪ A2 are automatically satisfied we verify only conditions for B1 ∪ B2 . At first we consider the case when xy ∈ E1 ∩ E2 . Then − (µ− B1 ∪ µB2 )(xy) = ≤ = = + (µ+ B1 ∪ µB2 )(xy) = ≤ = =
− max(µ− B1 (xy), µB2 (xy))
− − − max(min(µ− A1 (x), µA1 (y)), min(µA2 (x), µA2 (y)))
− − − min(max(µ− A1 (x), µA2 (x)), max(µA1 (y), µA2 (y)))
− − − min((µ− A1 ∪ µA2 )(x), (µA−1 ∪ µA2 )(y)), + max(µ+ B1 (xy), µB2 (xy))
+ + + max(min(µ+ A1 (x), µA1 (y)), min(µA2 (x), µA2 (y)))
+ + + min(max(µ+ A1 (x), µA2 (x)), max(µA1 (y), µA2 (y)))
+ + + min((µ+ A1 ∪ µA2 )(x), (µA−1 ∪ µA2 )(y)).
If xy ∈ E1 and xy ̸∈ E2 , then − − − − − (µ− B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)), + + + + + (µ+ B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)).
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If xy ∈ E2 and xy ̸∈ E1 , then − − − − − (µ− B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)), + + + + + (µ+ B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)).
This completes the proof.
Definition 3.12. The join G1 + G2 = (A1 + A2 , B1 + B2 ) of two interval-valued fuzzy graphs G1 and G2 of the graphs G∗1 and G∗2 is defined as follows: (µ− + µ− )(x) = (µ− ∪ µ− )(x) A A A A
(A)
1
2
1
2
1
2
1
2
(µ+ + µ+ )(x) = (µ+ ∪ µ+ )(x) A A A A
if x ∈ V1 ∪ V2 ,
(µ− + µ− )(xy) = (µ− ∪ µ− )(xy) B B B B
(B)
1
2
1
2
1
2
1
2
)(xy) = (µ+ ∪ µ+ )(xy) + µ+ (µ+ B B B B
if xy ∈ E1 ∩ E2 ,
(µ− + µ− )(xy) = min(µ− (x), µ− (y)) B B A A
(C)
1
2
1
2
1
2
1
2
(µ+ + µ+ )(xy) = min(µ+ (x), µ+ (y)) B B A A
if xy ∈ E ′ , where E ′ is the set of all edges joining the nodes of V1 and V2 .
Proposition 3.13. The join of interval-valued fuzzy graphs is an interval-valued fuzzy graph. Proof. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. We prove that G1 + G2 = (A1 + A2 , B1 + B2 ) is an interval-valued fuzzy graph of the graph G∗1 + G∗2 . In view of Proposition 3.11 is sufficient to verify the case when xy ∈ E ′ . In this case we have − − − (µ− B1 + µB2 )(xy) = min(µA1 (x), µA2 (y)) − − − ≤ min((µA1 ∪ µA2 )(x), (µ− A1 ∪ µA2 )(y)) − − − = min((µA1 + µA2 )(x), (µA1 + µ− A2 )(y)), + + + (µ+ B1 + µB2 )(xy) = min(µA1 (x), µA2 (y)) + + + ≤ min((µ+ A1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)) + + + = min((µA1 + µA2 )(x), (µA1 + µ+ A2 )(y)).
This completes the proof.
Proposition 3.14. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be crisp graphs with V1 ∩ V2 = ∅. Let A1 , A2 , B1 and B2 be interval-valued fuzzy subsets of V1 , V2 , E1 and E2 , respectively. Then G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of G∗1 ∪ G∗2 if and only if G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. Proof. Suppose that G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of G∗1 ∪ G∗2 . Let xy ∈ E1 . Then xy ̸∈ E2 and x, y ∈ V1 − V2 . Thus − − µ− B1 (xy) = (µB1 ∪ µB2 )(xy) − − − ≤ min((µA1 ∪ µ− A2 )(x), (µA1 ∪ µA2 )(y)) − − = min(µA1 (x), µA1 (y)), + + µ+ B1 (xy) = (µB1 ∪ µB2 )(xy) + + + ≤ min((µA1 ∪ µ+ A2 )(x), (µA1 ∪ µA2 )(y)) + = min(µ+ A1 (x), µA1 (y)).
This shows that G1 = (A1 , B1 ) is an interval-valued fuzzy graph. Similarly, we can show that G2 = (A2 , B2 ) is an intervalvalued fuzzy graph. The converse statement is given by Proposition 3.11. As a consequence of Propositions 3.13 and 3.14 we obtain Proposition 3.15. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be crisp graphs and let V1 ∩ V2 = ∅. Let A1 , A2 , B1 and B2 be intervalvalued fuzzy subsets of V1 , V2 , E1 and E2 , respectively. Then G1 + G2 = (A1 + A2 , B1 + B2 ) is an interval-valued fuzzy graph of G∗ if and only if G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively.
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4. Isomorphisms of interval-valued fuzzy graphs In this section we characterize various types of (weak) isomorphisms of interval-valued fuzzy graphs. Definition 4.1. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be two interval-valued fuzzy graphs. A homomorphism f : G1 → G2 is a mapping f : V1 → V2 such that − + + (a) µ− A1 (x1 ) ≤ µA2 (f (x1 )), µA1 (x1 ) ≤ µA2 (f (x1 )), − − + (b) µB1 (x1 y1 ) ≤ µB2 (f (x1 )f (y1 )), µB1 (x1 y1 ) ≤ µ+ B2 (f (x1 )f (y1 )) for all x1 ∈ V1 , x1 y1 ∈ E1 .
A bijective homomorphism with the property − + + (c) µ− A1 (x1 ) = µA2 (f (x1 )), µA1 (x1 ) = µA2 (f (x1 )),
is called a weak isomorphism. A weak isomorphism preserves the weights of the nodes but not necessarily the weights of the arcs. A bijective homomorphism preserving the weights of the arcs but not necessarily the weights of the nodes, i.e., a bijective homomorphism f : G1 → G2 such that − + + (d) µ− B1 (x1 y1 ) = µB2 (f (x1 )f (y1 )), µB1 (x1 y1 ) = µB2 (f (x1 )f (y1 )) for all x1 y1 ∈ V1 is called a weak co-isomorphism.
A bijective mapping f : G1 → G2 satisfying (c) and (d) is called an isomorphism. Example 4.2. Consider graphs G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) such that V1 = {a1 , b1 }, V2 = {a2 , b2 }, E1 = {a1 b1 } and E2 = {a2 b2 }. Let A1 , A2 , B1 and B2 be interval-valued fuzzy subsets defined by
a1 b 1 , , , 0.2 0.3 0.5 0.6 a2 b 2 a2 b 2 A2 = , , , , 0.3 0.2 0.6 0.5
A1 =
a1
,
b1
B1 =
a1 b1 a1 b1 0.1
B2 =
,
0.3
a2 b2 a2 b2 0.1
,
0.4
, .
Then, as is easy to see, G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. The map f : V1 → V2 defined by f (a1 ) = b2 and f (b1 ) = a2 is a weak isomorphism but it is not an isomorphism. Example 4.3. Let G∗1 and G∗2 be as in the previous example and let A1 , A2 , B1 and B2 be interval-valued fuzzy subsets defined by
a1 b 1 , , , 0.2 0.3 0.4 0.5 a2 b 2 a2 b 2 , , , , A2 = 0.4 0.3 0.5 0.6 A1 =
a1
,
b1
B1 =
0.1
B2 =
a1 b1 a1 b1
,
0.3
a2 b2 a2 b2 0.1
,
0.3
, .
Then G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. The map f : V1 → V2 defined by f (a1 ) = b2 and f (b1 ) = a2 is a weak co-isomorphism but it is not an isomorphism. Proposition 4.4. An isomorphism between interval-valued fuzzy graphs is an equivalence relation. Problem. Prove or disprove that weak isomorphism (co-isomorphism) between interval-valued fuzzy graphs is a partial ordering relation. 5. Interval-valued fuzzy complete graphs Definition 5.1. An interval-valued fuzzy graph G = (A, B) is called complete if
− − µ− B (xy) = min µA (x), µA (y)
and
+ + µ+ B (xy) = min µA (x), µA (y)
for all xy ∈ E .
Example 5.2. Consider a graph G∗ = (V , E ) such that V = {x, y, z }, E = {xy, yz , zx}. If A and B are interval-valued fuzzy subset defined by
x y z x y z , , , , , , 0.2 0.3 0.4 0.4 0.5 0.5 xy yz zx xy yz zx B= , , , , , , 0.2 0.3 0.2 0.4 0.5 0.4 then G = (A, B) is an interval-valued fuzzy complete graph of G∗ . A=
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As a consequence of Proposition 3.8 we obtain Proposition 5.3. If G = (A, B) be an interval-valued fuzzy complete graph, then also G[G] is an interval-valued fuzzy complete graph. Definition 5.4. The complement of an interval-valued fuzzy complete graph G = (A, B) of G∗ = (V , E ) is an interval-valued + − + fuzzy complete graph G = (A, B) on G∗ = (V , E ), where A = A = [µ− A , µA ] and B = [µ B , µ B ] is defined by
µ− B (xy) = µ+ B (xy) =
0 − min(µ− A (x), µA (y))
if µ− B (xy) > 0, if if µ− B (xy) = 0,
0 + min(µ+ A (x), µA (y))
if µ+ B (xy) > 0, if if µ+ B (xy) = 0.
Definition 5.5. An interval-valued fuzzy complete graph G = (A, B) is called self-complementary if G ∼ = G. Example 5.6. Consider a graph G∗ = (V , E ) such that V = {a, b, c }, E = {ab, bc }. Then an interval-valued fuzzy graph G = (A, B), where
c a b c , , , , , 0.1 0.2 0.3 0.3 0.4 0.5 ab bc ab bc , , , , B= 0.1 0.2 0.3 0.4 A=
a
,
b
is self-complementary. Proposition 5.7. In a self-complementary interval-valued fuzzy complete graph G = (A, B) we have
∑ ∑ − − min(µ− (a) A (x), µA (y)), x̸=y µB (xy) = ∑ ∑x̸=y + + + (b) x̸=y µB (xy) = x̸=y min(µA (x), µA (y)). Proof. Let G = (A, B) be a self-complementary interval-valued fuzzy complete graph. Then there exists an automorphism − + + − − + + f : V → V such that µ− A (f (x)) = µA (x), µA (f (x)) = µA (x), µB (f (x)f (y)) = µB (xy) and µB (f (x)f (y)) = µB (xy) for all x, y ∈ V . Hence, for x, y ∈ V we obtain
− − − − − µ− B (xy) = µ B (f (x)f (y)) = min µA (f (x)), µA (f (y)) = min(µA (x), µA (y)), which implies (a). The proof of (b) is analogous.
− − + Proposition 5.8. Let G = (A, B) be an interval-valued fuzzy complete graph. If µ− B (xy) = min(µA (x), µA (y)) and µB (xy) = + + min(µA (x), µA (y)) for all x, y ∈ V , then G is self-complementary. − − Proof. Let G = (A, B) be an interval-valued fuzzy complete graph such that µ− B (xy) = min(µA (x), µA (y)) and + + + µB (xy) = min(µA (x), µA (y)) for all x, y ∈ V . Then G = G under the identity map I : V → V . Hence G is selfcomplementary.
Proposition 5.9. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be interval-valued fuzzy complete graphs. Then G1 ∼ = G2 if and only if G1 ∼ = G2 . Proof. Assume that G1 and G2 are isomorphic, there exists a bijective map f : V1 → V2 satisfying − µ− A1 (x) = µA2 (f (x)),
+ µ+ A1 (x) = µA2 (f (x)) for all x ∈ V1 ,
− µ− B1 (xy) = µB2 (f (x)f (y)),
+ µ+ B1 (xy) = µB2 (f (x)f (y)) for all xy ∈ E1 .
By the definition of complement, we have − − − − µ− B1 (xy) = min(µ− A1 (x), µA1 (y)) = min(µA2 (f (x)), µA2 (f (y))) = µ B2 (f (x)f (y)), + + + + µ+ B1 (xy) = min(µ+ A1 (x), µA1 (y)) = min(µA2 (f (x)), µA2 (f (y))) = µ B2 (f (x)f (y)) for all xy ∈ E1 .
Hence G1 ∼ = G2 . The proof of the converse part is straightforward.
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6. Conclusions It is well known that interval-valued fuzzy sets constitute a generalization of the notion of fuzzy sets. The interval-valued fuzzy models give more precision, flexibility and compatibility to the system as compared to the classical and fuzzy models. So we have introduced interval-valued fuzzy graphs and have presented several properties in this paper. The further study of interval-valued fuzzy graphs may also be extended with an application of interval-valued fuzzy graphs in database theory, an expert system, neural networks and geographical information system. Acknowledgements The authors are thankful to the referees for their valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
L.A. Zadeh, The concept of a linguistic and application to approximate reasoning I, Inf. Sci. 8 (1975) 199–249. L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. J.M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, Upper Saddle River, New Jersey, 2001. M.B. Gorzalczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets Syst. 21 (1987) 1–17. M.B. Gorzalczany, An interval-valued fuzzy inference method some basic properties, Fuzzy Sets Syst. 31 (1989) 243–251. M.K. Roy, R. Biswas, I-V fuzzy relations and Sanchez’s approach for medical diagnosis, Fuzzy Sets Syst. 47 (1992) 35–38. I.B. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets Syst. 20 (1986) 191–210. A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K.S. Fu, M. Shimura (Eds.), Fuzzy Sets and Their Applications, Academic Press, New York, 1975, pp. 77–95. P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognit. Lett. 6 (1987) 297–302. J.N. Mordeson, C.S. Peng, Operations on fuzzy graphs, Inf. Sci. 79 (1994) 159–170. J.N. Mordeson, P.S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, Physica Verlag, Heidelberg, 1998. Second edition 2001. M.S. Sunitha, A. Vijayakumar, Complement of a fuzzy graph, Indian J. Pure Appl. Math. 33 (2002) 1451–1464. K.R. Bhutani, A. Battou, On M-strong fuzzy graphs, Inf. Sci. 155 (2003) 103–109. K.R. Bhutani, A. Rosenfeld, Strong arcs in fuzzy graphs, Inf. Sci. 152 (2003) 319–322. J. Hongmei, W. Lianhua, Interval-valued fuzzy subsemigroups and subgroups sssociated by intervalvalued suzzy graphs, in: 2009 WRI Global Congress on Intelligent Systems, 2009, pp. 484–487. M. Akram, Fuzzy Lie ideals of lie algebras with interval-valued membership functions, Quasigroups Related Systems 16 (2008) 1–12. M. Akram, K.H. Dar, Generalized Fuzzy K -Algebras, VDM Verlag, ISBN: 978-3-639-27095-2, 2010, pp. 288. A. Alaoui, On fuzzification of some concepts of graphs, Fuzzy Sets Syst. 101 (1999) 363–389. K.T. Atanassov, Intuitionistic fuzzy sets: Theory and applications, Studies in fuzziness and soft computing, Physica-Verl. (1999). F. Harary, Graph Theory, 3rd Edition, Addison-Wesley, Reading, MA, 1972. K.P. Huber, M.R. Berthold, Application of fuzzy graphs for metamodeling, in: Proceedings of the 2002 IEEE Conference, pp. 640-644. S. Mathew, M.S. Sunitha, Node connectivity and arc connectivity of a fuzzy graph, Inf. Sci. 180 (4) (2010) 519–531. J.N. Mordeson, Fuzzy line graphs, Pattern Recognit. Lett. 14 (1993) 381–384. A. Nagoorgani, K. Radha, Isomorphism on fuzzy graphs, Int. J. Comput. Math. Sci. 2 (2008) 190–196. A. Perchant, I. Bloch, Fuzzy morphisms between graphs, Fuzzy Sets Syst. 128 (2002) 149–168. L.A. Zadeh, Similarity relations and fuzzy orderings, Inf. Sci. 3 (1971) 177–200. K.R. Bhutani, On automorphism of fuzzy graphs, Pattern Recognit. Lett. 9 (1989) 159–162. J.M. Mendel, X. Gang, Fast computation of centroids for constant-width interval-valued fuzzy sets, Fuzzy information processing society, NAFIPS (2006) 621–626.
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Interval-valued fuzzy graphs Muhammad Akram a,∗ , Wieslaw A. Dudek b a
Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore-54000, Pakistan
b
Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370, Wroclaw, Poland
article
info
Article history: Received 25 August 2010 Received in revised form 1 November 2010 Accepted 1 November 2010 Keywords: Interval-valued fuzzy graph Self-complementary Interval-valued fuzzy complete graph
abstract We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of intervalvalued fuzzy complete graphs and present some properties of self-complementary and selfweak complementary interval-valued fuzzy complete graphs. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction In 1975, Zadeh [1] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy sets [2] in which the values of the membership degrees are intervals of numbers instead of the numbers. Interval-valued fuzzy sets provide a more adequate description of uncertainty 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 [3]. Since interval-valued fuzzy sets are widely studied and used, we describe briefly the work of Gorzalczany on approximate reasoning [4,5], Roy and Biswas on medical diagnosis [6], Turksen on multivalued logic [7] and Mendel on intelligent control [3]. The fuzzy graph theory as a generalization of Euler’s graph theory was first introduced by Rosenfeld [8] in 1975. The fuzzy relations between fuzzy sets were first considered by Rosenfeld and he developed the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts. Later, Bhattacharya [9] gave some remarks on fuzzy graphs, and some operations on fuzzy graphs were introduced by Mordeson and Peng [10]. The complement of a fuzzy graph was defined by Mordeson [11] and further studied by Sunitha and Vijayakumar [12]. Bhutani and Rosenfeld introduced the concept of M-strong fuzzy graphs in [13] and studied some properties. The concept of strong arcs in fuzzy graphs was discussed in [14]. Hongmei and Lianhua gave the definition of interval-valued graph in [15]. In this paper, we define the operations of Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some properties. We study isomorphism (resp. weak isomorphism) between interval-valued fuzzy graphs in an equivalence relation (resp. partial order). We introduce the notion of interval-valued fuzzy complete graphs and present some properties of self-complementary and self-weak complementary interval-valued fuzzy complete graphs.
∗
Corresponding author. Tel.: +92 3334510258; fax: +92 99212505. E-mail addresses: [email protected], [email protected] (M. Akram), [email protected] (W.A. Dudek).
0898-1221/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.11.004
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The definitions and terminologies that we used in this paper are standard. For other notations, terminologies and applications, the readers are referred to [16–28].
2. Preliminaries A graph is an ordered pair G∗ = (V , E ), where V is the set of vertices of G∗ and E is the set of edges of G∗ . Two vertices x and y in a graph G∗ are said to be adjacent in G∗ if {x, y} is in an edge of G∗ . (For simplicity an edge {x, y} will be denoted by xy.) A simple graph is a graph without loops and multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. The complete graph on n vertices has n vertices and n(n − 1)/2 edges. We will consider only graphs with the finite number of vertices and edges. By a complementary graph G∗ of a simple graph G∗ we mean a graph having the same vertices as G∗ and such that two vertices are adjacent in G∗ if and only if they are not adjacent in G∗ . An isomorphism of graphs G∗1 and G∗2 is a bijection between the vertex sets of G∗1 and G∗2 such that any two vertices v1 and v2 of G∗1 are adjacent in G∗1 if and only if f (v1 ) and f (v2 ) are adjacent in G∗2 . Isomorphic graphs are denoted by G∗1 ≃ G∗2 . Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be two simple graphs, we can construct several new graphs. The first construction called the Cartesian product of G∗1 and G∗2 gives a graph G∗1 × G∗2 = (V , E ) with V = V1 × V2 and E = {(x, x2 )(x, y2 ) | x ∈ V1 , x2 y2 ∈ E2 } ∪ {(x1 , z )(y1 , z ) | x1 y1 ∈ E1 , z ∈ V2 } . The composition of graphs G∗1 and G∗2 is the graph G∗1 [G∗2 ] = (V1 × V2 , E 0 ), where E 0 = E ∪ {(x1 , x2 )(y1 , y2 ) | x1 y1 ∈ E1 , x2 ̸= y2 } and E is defined as in G∗1 × G∗2 . Note that G∗1 [G∗2 ] ̸= G∗2 [G∗1 ]. The union of graphs G∗1 and G∗2 is defined as G∗1 ∪ G∗2 = (V1 ∪ V2 , E1 ∪ E2 ). The join of G∗1 and G∗2 is the simple graph G∗1 + G∗2 = (V1 ∪ V2 , E1 ∪ E2 ∪ E ′ ), where E ′ is the set of all edges joining the nodes of V1 and V2 . In this construction it is assumed that V1 ∩ V2 ̸= ∅. By a fuzzy subset µ on a set X is mean a map µ : X → [0, 1]. A map ν : X × X → [0, 1] is called a fuzzy relation on X if ν(x, y) ≤ min(µ(x), µ(y)) for all x, y ∈ X . A fuzzy relation ν is symmetric if ν(x, y) = ν(y, x) for all x, y ∈ X . An interval number D is an interval [a− , a+ ] with 0 ≤ a− ≤ a+ ≤ 1. The interval [a, a] is identified with the number a ∈ [0, 1]. D[0, 1] denotes the set of all interval numbers. + − + For interval numbers D1 = [a− 1 , b1 ] and D2 = [a2 , b2 ], we define
• • • • • • •
+ − + − − + + rmin(D1 , D2 ) = rmin([a− 1 , b1 ], [a2 , b2 ]) = [min{a1 , a2 }, min{b1 , b2 }], + − + − − + + rmax(D1 , D2 ) = rmax([a− 1 , b1 ], [a2 , b2 ]) = [max{a1 , a2 }, max{b1 , b2 }], − − − + + + + D1 + D2 = [a− 1 + a2 − a1 · a2 , b1 + b2 − b1 · b2 ], − + + D1 ≤ D2 ⇐⇒ a− 1 ≤ a2 and b1 ≤ b2 , − + + D1 = D2 ⇐⇒ a− 1 = a2 and b1 = b2 , D1 < D2 ⇐⇒ D1 ≤ D2 and D1 ̸= D2 , + − + kD = k[a− 1 , b1 ] = [ka1 , kb1 ], where 0 ≤ k ≤ 1.
Then, (D[0, 1], ≤, ∨, ∧) is a complete lattice with [0, 0] as the least element and [1, 1] as the greatest. The interval-valued fuzzy set A in V is defined by A=
− x, µA (x), µ+ :x∈V , A ( x)
+ − + where µ− A (x) and µA (x) are fuzzy subsets of V such that µA (x) ≤ µA (x) for all x ∈ V . For any two interval-valued sets − + − + A = [µA (x), µA (x)] and B = [µB (x), µB (x)] in V we define:
− + + • A B = {(x, max(µ− A (x), µB (x)), max(µA (x), µB (x))) : x ∈ V }, − + + • A B = {(x, min(µ− A (x), µB (x)), min(µA (x), µB (x))) : x ∈ V }. If G∗ = (V , E ) is a graph, then by an interval-valued fuzzy relation B on a set E we mean an interval-valued fuzzy set such that − − µ− B (xy) ≤ min(µA (x), µA (y)), + + µ+ B (xy) ≤ min(µA (x), µA (y))
for all xy ∈ E.
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3. Operations on interval-valued fuzzy graphs Throughout this paper, G∗ is a crisp graph, and G is an interval-valued fuzzy graph. + Definition 3.1. By an interval-valued fuzzy graph of a graph G∗ = (V , E ) we mean a pair G = (A, B), where A = [µ− A , µA ] is − + an interval-valued fuzzy set on V and B = [µB , µB ] is an interval-valued fuzzy relation on E.
Example 3.2. Consider a graph G∗ = (V , E ) such that V = {x, y, z }, E = {xy, yz , zx}. Let A be an interval-valued fuzzy set of V and let B be an interval-valued fuzzy set of E ⊆ V × V defined by
x y z x y z , , , , , , 0.2 0.3 0.4 0.4 0.5 0.5 xy yz zx xy yz zx B= , , , , , . 0.1 0.2 0.1 0.3 0.4 0.4 A=
By routine computations, it is easy to see that G = (A, B) is an interval-valued fuzzy graph of G∗ . Definition 3.3. The Cartesian product G1 × G2 of two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ) of the graphs G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) is defined as a pair (A1 × A1 , B1 × B2 ) such that
(i) (ii) (iii)
(µ− × µ− )(x1 , x2 ) = min(µ− (x1 ), µ− (x2 )) A A A A 1
2
1
1
2
1
2
(µ+ × µ+ )(x1 , x2 ) = min(µ+ (x1 ), µ+ (x2 )) A A A A
for all (x1 , x2 ) ∈ V ,
2
−
−
−
−
1
2
1
2
1
2
1
2
−
−
−
1
2
1
2
1
2
1
2
(µB × µB )((x, x2 )(x, y2 )) = min(µA (x), µB (x2 y2 )) (µ+ × µ+ )((x, x2 )(x, y2 )) = min(µ+ (x), µ+ (x2 y2 )) B B A B
for all x ∈ V1 and x2 y2 ∈ E2 ,
−
(µB × µB )((x1 , z )(y1 , z )) = min(µB (x1 y1 ), µA (z )) (µ+ × µ+ )((x1 , z )(y1 , z )) = min(µ+ (x1 y1 ), µ+ (z )) B B B A
for all z ∈ V2 and x1 y1 ∈ E1 .
Example 3.4. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be graphs such that V1 = {a, b}, V2 = {c , d}, E1 = {ab} and E2 = {cd}. Consider two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ), where
a b , , , 0.2 0.3 0.4 0.5 c d c d , , , , A2 = 0.1 0.2 0.4 0.6 A1 =
a
,
b
B1 =
,
ab
0.1 0.2
B2 =
ab cd
,
cd
0.1 0.3
, .
Then, it is not difficult to verify − (µ− B1 × µB2 )((a, c )(a, d)) = 0.1,
+ (µ+ B1 × µB2 )((a, c )(a, d)) = 0.3,
− (µ− B1 × µB2 )((a, c )(b, c )) = 0.1,
+ (µ+ B1 × µB2 )((a, c )(b, c )) = 0.2,
− (µ− B1 × µB2 )((a, d)(b, d)) = 0.1,
+ (µ+ B1 × µB2 )((a, d)(b, d)) = 0.2,
− (µ− B1 × µB2 )((b, c )(b, d)) = 0.1,
+ (µ+ B1 × µB2 )((b, c )(b, d)) = 0.3.
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By routine computations, it is easy to see that G1 × G2 is an interval-valued fuzzy graph of G∗1 × G∗2 . Proposition 3.5. The Cartesian product G1 × G2 = (A1 × A2 , B1 × B2 ) of two interval-valued fuzzy graphs of the graphs G∗1 and G∗2 is an interval-valued fuzzy graph of G∗1 × G∗2 . Proof. We verify only conditions for B1 × B2 because conditions for A1 × A2 are obvious. Let x ∈ V1 , x2 y2 ∈ E2 . Then − (µ− B1 × µB2 )((x, x2 )(x, y2 )) = ≤ = =
− min(µ− A1 (x), µB2 (x2 y2 ))
− − min(µ− A1 (x), min(µA2 (x2 ), µA2 (y2 )))
− − − min(min(µ− A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
− − − min((µ− A1 × µA2 )(x, x2 ), (µA1 × µA2 )(x, y2 )), + min(µ+ A1 (x), µB2 (x2 y2 ))
+ (µ+ B1 × µB2 )((x, x2 )(x, y2 )) = ≤ = =
+ + min(µ+ A1 (x), min(µA2 (x2 ), µA2 (y2 )))
+ + + min(min(µ+ A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
+ + + min((µ+ A1 × µA2 )(x, x2 ), (µA1 × µA2 )(x, y2 )).
Similarly for z ∈ V2 and x1 y1 ∈ E1 we have − (µ− B1 × µB2 )((x1 , z )(y1 , z )) = ≤ = =
− min(µ− B1 (x1 y1 ), µA2 (z ))
− − min(min(µ− A1 (x1 ), µA1 (y1 )), µA2 (z ))
− − − min(min(µ− A1 (x), µA2 (z )), min(µA1 (y1 ), µA2 (z )))
− − − min((µ− A1 × µA2 )(x1 , z ), (µA1 × µA2 )(y1 , z )),
+ (µ+ B1 × µB2 )((x1 , z )(y1 , z )) = ≤ = =
This completes the proof.
+ min(µ+ B1 (x1 y1 ), µA2 (z ))
+ + min(min(µ+ A1 (x1 ), µA1 (y1 )), µA2 (z ))
+ + + min(min(µ+ A1 (x), µA2 (z )), min(µA1 (y1 ), µA2 (z )))
+ + + min((µ+ A1 × µA2 )(x1 , z ), (µA1 × µA2 )(y1 , z )).
Definition 3.6. The composition G1 [G2 ] = (A1 ◦ A2 , B1 ◦ B2 ) of two interval-valued fuzzy graphs G1 and G2 of the graphs G∗1 and G∗2 is defined as follows:
(i) (ii) (iii)
(µ− ◦ µ− )(x1 , x2 ) = min(µ− (x1 ), µ− (x2 )) A A A A
1
2
1
2
1
2
1
2
(µ+ ◦ µ+ )(x1 , x2 ) = min(µ+ (x1 ), µ+ (x2 )) A A A A
for all (x1 , x2 ) ∈ V ,
(µ− ◦ µ− )((x, x2 )(x, y2 )) = min(µ− (x), µ− (x2 y2 )) B B A B
1
2
1
2
1
2
1
2
(µ+ ◦ µ+ )((x, x2 )(x, y2 )) = min(µ+ (x), µ+ (x2 y2 )) B B A B (µ− ◦ µ− )((x1 , z )(y1 , z )) = min(µ− (x1 y1 ), µ− (z )) B B B A
for all x ∈ V1 and x2 y2 ∈ E2 ,
1
2
1
2
1
2
1
2
(µ+ ◦ µ+ )((x1 , z )(y1 , z )) = min(µ+ (x1 y1 ), µ+ (z )) B B B A
for all z ∈ V2 and x1 y1 ∈ E1 ,
M. Akram, W.A. Dudek / Computers and Mathematics with Applications 61 (2011) 289–299
(iv)
(µ− ◦ µ− )((x1 , x2 )(y1 , y2 )) = min(µ− (x2 ), µ− (y2 ), µ− (x1 y1 )) B B A A B 1
2
2
2
1
1
2
2
2
1
(µ+ ◦ µ+ )((x1 , x2 )(y1 , y2 )) = min(µ+ (x2 ), µ+ (y2 ), µ+ (x1 y1 )) B B A A B
293
for all (x1 , x2 )(y1 , y2 ) ∈ E 0 − E.
Example 3.7. Let G∗1 and G∗2 be as in the previous example. Consider two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ) defined by
a b , , , , A1 = 0.2 0.3 0.5 0.5 d c d c , , , , A2 = 0.1 0.3 0.4 0.6
a
b
B1 =
,
ab
0.2 0.4
B2 =
ab cd
,
cd
0.1 0.3
, .
Then we have − (µ− B1 ◦ µB2 )((a, c )(a, d)) = 0.2,
+ (µ+ B1 ◦ µB2 )((a, c )(a, d)) = 0.3,
− (µ− B1 ◦ µB2 )((b, c )(b, d)) = 0.1,
+ (µ+ B1 ◦ µB2 )((b, c )(b, d)) = 0.3,
− (µ− B1 ◦ µB2 )((a, c )(b, c )) = 0.1,
+ (µ+ B1 ◦ µB2 )((a, c )(b, c )) = 0.4,
− (µ− B1 ◦ µB2 )((a, d)(b, d)) = 0.2,
+ (µ+ B1 ◦ µB2 )((a, d)(b, d)) = 0.4,
− (µ− B1 ◦ µB2 )((a, c )(b, d)) = 0.1,
+ (µ+ B1 ◦ µB2 )((a, c )(b, d)) = 0.4,
− (µ− B1 ◦ µB2 )((b, c )(a, d)) = 0.1,
+ (µ+ B1 ◦ µB2 )((b, c )(a, d)) = 0.4.
By routine computations, it is easy to see that G1 [G2 ] = (A1 ◦ A2 , B1 ◦ B2 ) is an interval-valued fuzzy graph of G∗1 [G∗2 ]. Proposition 3.8. The composition G1 [G2 ] of interval-valued fuzzy graphs G1 and G2 of G∗1 and G∗2 is an interval-valued fuzzy graph of G∗1 [G∗2 ]. Proof. Similarly as in the previous proof we verify the conditions for B1 ◦ B2 only. In the case x ∈ V1 , x2 y2 ∈ E2 , according to (ii) we obtain − (µ− B1 ◦ µB2 )((x, x2 )(x, y2 )) = ≤ = = + (µ+ B1 ◦ µB2 )((x, x2 )(x, y2 )) = ≤ = =
− min(µ− A1 (x), µB2 (x2 y2 ))
− − min(µ− A1 (x), min(µA2 (x2 ), µA2 (y2 )))
− − − min(min(µ− A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
− − − min((µ− A1 ◦ µA2 )(x, x2 ), (µA1 ◦ µA2 )(x, y2 )), + min(µ+ A1 (x), µB2 (x2 y2 ))
+ + min(µ+ A1 (x), min(µA2 (x2 ), µA2 (y2 )))
+ + + min(min(µ+ A1 (x), µA2 (x2 )), min(µA1 (x), µA2 (y2 )))
+ + + min((µ+ A1 ◦ µA2 )(x, x2 ), (µA1 ◦ µA2 )(x, y2 )).
In the case z ∈ V2 , x1 y1 ∈ E1 the proof is similar.
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In the case (x1 , x2 )(y1 , y2 ) ∈ E 0 − E we have x1 y1 ∈ E1 and x2 ̸= y2 , which according to (iv) implies − (µ− B1 ◦ µB2 )((x1 , x2 )(y1 , y2 )) = ≤ = = + (µ+ B1 ◦ µB2 )((x1 , x2 )(y1 , y2 )) = ≤ = =
This completes the proof.
− − min(µ− A2 (x2 ), µA2 (y2 ), µB1 (x1 y1 ))
− − − min(µ− A2 (x2 ), µA2 (y2 ), min(µA1 (x1 ), µA1 (y1 )))
− − − min(min(µ− A1 (x1 ), µA2 (x2 )), min(µA1 (y1 ), µA2 (y2 )))
− − − min((µ− A1 ◦ µA2 )(x1 , x2 ), (µA1 ◦ µA2 )(y1 , y2 )), + + min(µ+ A2 (x2 ), µA2 (y2 ), µB1 (x1 y1 ))
+ + + min(µ+ A2 (x2 ), µA2 (y2 ), min(µA1 (x1 ), µA1 (y1 )))
+ + + min(min(µ+ A1 (x1 ), µA2 (x2 )), min(µA1 (y1 ), µA2 (y2 )))
+ + + min((µ+ A1 ◦ µA2 )(x1 , x2 ), (µA1 ◦ µA2 )(y1 , y2 )).
Definition 3.9. The union G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) of two interval-valued fuzzy graphs G1 and G2 of the graphs G∗1 and G∗2 is defined as follows:
(A)
(B)
(C)
(D)
− (µA1 ∪ µ−A2 )(x) = µ−A1 (x)
if x ∈ V1 and x ̸∈ V2 ,
(µ− ∪ µ− )(x) = µ− (x) if x ∈ V2 and x ̸∈ V1 , A A A
(µ−1 ∪ µ−2 )(x) = max2 (µ− (x), µ− (x)) if x ∈ V ∩ V , 1 2 A A A1 A2 +1 +2 + (µA1 ∪ µA2 )(x) = µA1 (x) if x ∈ V1 and x ̸∈ V2 , (µ+ ∪ µ+ )(x) = µ+ (x) if x ∈ V2 and x ̸∈ V1 , A A A
(µ+1 ∪ µ+2 )(x) = max2 (µ+ (x), µ+ (x)) if x ∈ V ∩ V , 1 2 A A A1 A2 −1 −2 − (µB1 ∪ µB2 )(xy) = µB1 (xy) if xy ∈ E1 and xy ̸∈ E2 , (µ− ∪ µ− )(xy) = µ− (xy) if xy ∈ E2 and xy ̸∈ E1 , B B B
(µ−1 ∪ µ−2 )(xy) = max2 (µ− (xy), µ− (xy)) if xy ∈ E ∩ E , 1 2 B B B1 B2 +1 +2 + (µB1 ∪ µB2 )(xy) = µB1 (xy) if xy ∈ E1 and xy ̸∈ E2 , (µ+ ∪ µ+ )(xy) = µ+ (xy) if xy ∈ E2 and xy ̸∈ E1 , B B B
(µ+1 ∪ µ+2 )(xy) = max2 (µ+ (xy), µ+ (xy)) B1
B2
B1
B2
if xy ∈ E1 ∩ E2 .
Example 3.10. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be graphs such that V1 = {a, b, c , d, e}, E1 = {ab, bc , be, ce, ad, ed}, V2 = {a, b, c , d, f } and E2 = {ab, bc , cf , bf , bd}. Consider two interval-valued fuzzy graphs G1 = (A1 , B1 ) and G2 = (A2 , B2 ) defined by
A1 =
a
,
b
,
c
,
d
,
e
0.2 0.4 0.3 0.3 0.2
ab
,
ce
,
be
,
ad
,
a
,
b
,
c
,
bc
,
d
,
ce
,
e
,
be
0.4 0.5 0.6 0.7 0.6
de
,
ab
,
ad
, , a b c d f b c d f a A2 = , , , , , , , , , , 0.2 0.2 0.3 0.2 0.4 0.4 0.5 0.6 0.6 0.6 ab bc cf bf bd ab bc cf bf bd B2 = , , , , , , , , , . 0.1 0.2 0.1 0.1 0.2 0.2 0.4 0.5 0.2 0.5 B1 =
,
bc
,
0.1 0.2 0.1 0.2 0.1 0.1
de
0.3 0.4 0.5 0.5 0.3 0.6
,
Then, according to the above definition:
+ (µ+ A1 ∪ µA2 )(c ) = 0.6, − − (µB1 ∪ µB2 )(ab) = 0.1, − (µ− B1 ∪ µB2 )(ad) = 0.1, + (µB1 ∪ µ+ B3 )(ab) = 0.3, + (µ+ ∪ µ B1 B2 )(ad) = 0.3,
− (µ− A1 ∪ µA2 )(a) = 0.2, − (µA1 ∪ µ− A2 )(c ) = 0.3, − (µ− ∪ µ A1 A2 )(e) = 0.2, + + (µA1 ∪ µA2 )(a) = 0.4, + (µ+ A1 ∪ µA2 )(d) = 0.7, − − (µB1 ∪ µB2 )(bc ) = 0.2, − (µ− B1 ∪ µB2 )(de) = 0.1, + + (µB1 ∪ µB2 )(bc ) = 0.4, + (µ+ B1 ∪ µB2 )(de) = 0.6,
− (µ− A1 ∪ µA2 )(b) = 0.4, − − (µA1 ∪ µA2 )(d) = 0.3, − (µ− A1 ∪ µA2 )(f ) = 0.4, + + (µA1 ∪ µA2 )(b) = 0.5, + (µ+ A1 ∪ µA2 )(e) = 0.1, − − (µB1 ∪ µB2 )(ce) = 0.1, − (µ− B1 ∪ µB2 )(bd) = 0.2, + (µB1 ∪ µ+ B2 )(ce) = 0.5, + (µ+ ∪ µ B1 B2 )(bd) = 0.5,
+ (µ+ A1 ∪ µA2 )(f ) = 0.6, − − (µB1 ∪ µB2 )(be) = 0.2, − (µ− B1 ∪ µB2 )(bf ) = 0.1, + (µB1 ∪ µ+ B2 )(be) = 0.5, + (µ+ ∪ µ B1 B2 )(bf ) = 0.2.
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Clearly, G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of the graph G∗1 ∪ G∗2 . Proposition 3.11. The union of two interval-valued fuzzy graphs is an interval-valued fuzzy graph. Proof. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. We prove that G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of the graph G∗1 ∪ G∗2 . Since all conditions for A1 ∪ A2 are automatically satisfied we verify only conditions for B1 ∪ B2 . At first we consider the case when xy ∈ E1 ∩ E2 . Then − (µ− B1 ∪ µB2 )(xy) = ≤ = = + (µ+ B1 ∪ µB2 )(xy) = ≤ = =
− max(µ− B1 (xy), µB2 (xy))
− − − max(min(µ− A1 (x), µA1 (y)), min(µA2 (x), µA2 (y)))
− − − min(max(µ− A1 (x), µA2 (x)), max(µA1 (y), µA2 (y)))
− − − min((µ− A1 ∪ µA2 )(x), (µA−1 ∪ µA2 )(y)), + max(µ+ B1 (xy), µB2 (xy))
+ + + max(min(µ+ A1 (x), µA1 (y)), min(µA2 (x), µA2 (y)))
+ + + min(max(µ+ A1 (x), µA2 (x)), max(µA1 (y), µA2 (y)))
+ + + min((µ+ A1 ∪ µA2 )(x), (µA−1 ∪ µA2 )(y)).
If xy ∈ E1 and xy ̸∈ E2 , then − − − − − (µ− B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)), + + + + + (µ+ B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)).
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If xy ∈ E2 and xy ̸∈ E1 , then − − − − − (µ− B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)), + + + + + (µ+ B1 ∪ µB2 )(xy) ≤ min((µA1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)).
This completes the proof.
Definition 3.12. The join G1 + G2 = (A1 + A2 , B1 + B2 ) of two interval-valued fuzzy graphs G1 and G2 of the graphs G∗1 and G∗2 is defined as follows: (µ− + µ− )(x) = (µ− ∪ µ− )(x) A A A A
(A)
1
2
1
2
1
2
1
2
(µ+ + µ+ )(x) = (µ+ ∪ µ+ )(x) A A A A
if x ∈ V1 ∪ V2 ,
(µ− + µ− )(xy) = (µ− ∪ µ− )(xy) B B B B
(B)
1
2
1
2
1
2
1
2
)(xy) = (µ+ ∪ µ+ )(xy) + µ+ (µ+ B B B B
if xy ∈ E1 ∩ E2 ,
(µ− + µ− )(xy) = min(µ− (x), µ− (y)) B B A A
(C)
1
2
1
2
1
2
1
2
(µ+ + µ+ )(xy) = min(µ+ (x), µ+ (y)) B B A A
if xy ∈ E ′ , where E ′ is the set of all edges joining the nodes of V1 and V2 .
Proposition 3.13. The join of interval-valued fuzzy graphs is an interval-valued fuzzy graph. Proof. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. We prove that G1 + G2 = (A1 + A2 , B1 + B2 ) is an interval-valued fuzzy graph of the graph G∗1 + G∗2 . In view of Proposition 3.11 is sufficient to verify the case when xy ∈ E ′ . In this case we have − − − (µ− B1 + µB2 )(xy) = min(µA1 (x), µA2 (y)) − − − ≤ min((µA1 ∪ µA2 )(x), (µ− A1 ∪ µA2 )(y)) − − − = min((µA1 + µA2 )(x), (µA1 + µ− A2 )(y)), + + + (µ+ B1 + µB2 )(xy) = min(µA1 (x), µA2 (y)) + + + ≤ min((µ+ A1 ∪ µA2 )(x), (µA1 ∪ µA2 )(y)) + + + = min((µA1 + µA2 )(x), (µA1 + µ+ A2 )(y)).
This completes the proof.
Proposition 3.14. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be crisp graphs with V1 ∩ V2 = ∅. Let A1 , A2 , B1 and B2 be interval-valued fuzzy subsets of V1 , V2 , E1 and E2 , respectively. Then G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of G∗1 ∪ G∗2 if and only if G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. Proof. Suppose that G1 ∪ G2 = (A1 ∪ A2 , B1 ∪ B2 ) is an interval-valued fuzzy graph of G∗1 ∪ G∗2 . Let xy ∈ E1 . Then xy ̸∈ E2 and x, y ∈ V1 − V2 . Thus − − µ− B1 (xy) = (µB1 ∪ µB2 )(xy) − − − ≤ min((µA1 ∪ µ− A2 )(x), (µA1 ∪ µA2 )(y)) − − = min(µA1 (x), µA1 (y)), + + µ+ B1 (xy) = (µB1 ∪ µB2 )(xy) + + + ≤ min((µA1 ∪ µ+ A2 )(x), (µA1 ∪ µA2 )(y)) + = min(µ+ A1 (x), µA1 (y)).
This shows that G1 = (A1 , B1 ) is an interval-valued fuzzy graph. Similarly, we can show that G2 = (A2 , B2 ) is an intervalvalued fuzzy graph. The converse statement is given by Proposition 3.11. As a consequence of Propositions 3.13 and 3.14 we obtain Proposition 3.15. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be crisp graphs and let V1 ∩ V2 = ∅. Let A1 , A2 , B1 and B2 be intervalvalued fuzzy subsets of V1 , V2 , E1 and E2 , respectively. Then G1 + G2 = (A1 + A2 , B1 + B2 ) is an interval-valued fuzzy graph of G∗ if and only if G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively.
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297
4. Isomorphisms of interval-valued fuzzy graphs In this section we characterize various types of (weak) isomorphisms of interval-valued fuzzy graphs. Definition 4.1. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be two interval-valued fuzzy graphs. A homomorphism f : G1 → G2 is a mapping f : V1 → V2 such that − + + (a) µ− A1 (x1 ) ≤ µA2 (f (x1 )), µA1 (x1 ) ≤ µA2 (f (x1 )), − − + (b) µB1 (x1 y1 ) ≤ µB2 (f (x1 )f (y1 )), µB1 (x1 y1 ) ≤ µ+ B2 (f (x1 )f (y1 )) for all x1 ∈ V1 , x1 y1 ∈ E1 .
A bijective homomorphism with the property − + + (c) µ− A1 (x1 ) = µA2 (f (x1 )), µA1 (x1 ) = µA2 (f (x1 )),
is called a weak isomorphism. A weak isomorphism preserves the weights of the nodes but not necessarily the weights of the arcs. A bijective homomorphism preserving the weights of the arcs but not necessarily the weights of the nodes, i.e., a bijective homomorphism f : G1 → G2 such that − + + (d) µ− B1 (x1 y1 ) = µB2 (f (x1 )f (y1 )), µB1 (x1 y1 ) = µB2 (f (x1 )f (y1 )) for all x1 y1 ∈ V1 is called a weak co-isomorphism.
A bijective mapping f : G1 → G2 satisfying (c) and (d) is called an isomorphism. Example 4.2. Consider graphs G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) such that V1 = {a1 , b1 }, V2 = {a2 , b2 }, E1 = {a1 b1 } and E2 = {a2 b2 }. Let A1 , A2 , B1 and B2 be interval-valued fuzzy subsets defined by
a1 b 1 , , , 0.2 0.3 0.5 0.6 a2 b 2 a2 b 2 A2 = , , , , 0.3 0.2 0.6 0.5
A1 =
a1
,
b1
B1 =
a1 b1 a1 b1 0.1
B2 =
,
0.3
a2 b2 a2 b2 0.1
,
0.4
, .
Then, as is easy to see, G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. The map f : V1 → V2 defined by f (a1 ) = b2 and f (b1 ) = a2 is a weak isomorphism but it is not an isomorphism. Example 4.3. Let G∗1 and G∗2 be as in the previous example and let A1 , A2 , B1 and B2 be interval-valued fuzzy subsets defined by
a1 b 1 , , , 0.2 0.3 0.4 0.5 a2 b 2 a2 b 2 , , , , A2 = 0.4 0.3 0.5 0.6 A1 =
a1
,
b1
B1 =
0.1
B2 =
a1 b1 a1 b1
,
0.3
a2 b2 a2 b2 0.1
,
0.3
, .
Then G1 = (A1 , B1 ) and G2 = (A2 , B2 ) are interval-valued fuzzy graphs of G∗1 and G∗2 , respectively. The map f : V1 → V2 defined by f (a1 ) = b2 and f (b1 ) = a2 is a weak co-isomorphism but it is not an isomorphism. Proposition 4.4. An isomorphism between interval-valued fuzzy graphs is an equivalence relation. Problem. Prove or disprove that weak isomorphism (co-isomorphism) between interval-valued fuzzy graphs is a partial ordering relation. 5. Interval-valued fuzzy complete graphs Definition 5.1. An interval-valued fuzzy graph G = (A, B) is called complete if
− − µ− B (xy) = min µA (x), µA (y)
and
+ + µ+ B (xy) = min µA (x), µA (y)
for all xy ∈ E .
Example 5.2. Consider a graph G∗ = (V , E ) such that V = {x, y, z }, E = {xy, yz , zx}. If A and B are interval-valued fuzzy subset defined by
x y z x y z , , , , , , 0.2 0.3 0.4 0.4 0.5 0.5 xy yz zx xy yz zx B= , , , , , , 0.2 0.3 0.2 0.4 0.5 0.4 then G = (A, B) is an interval-valued fuzzy complete graph of G∗ . A=
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As a consequence of Proposition 3.8 we obtain Proposition 5.3. If G = (A, B) be an interval-valued fuzzy complete graph, then also G[G] is an interval-valued fuzzy complete graph. Definition 5.4. The complement of an interval-valued fuzzy complete graph G = (A, B) of G∗ = (V , E ) is an interval-valued + − + fuzzy complete graph G = (A, B) on G∗ = (V , E ), where A = A = [µ− A , µA ] and B = [µ B , µ B ] is defined by
µ− B (xy) = µ+ B (xy) =
0 − min(µ− A (x), µA (y))
if µ− B (xy) > 0, if if µ− B (xy) = 0,
0 + min(µ+ A (x), µA (y))
if µ+ B (xy) > 0, if if µ+ B (xy) = 0.
Definition 5.5. An interval-valued fuzzy complete graph G = (A, B) is called self-complementary if G ∼ = G. Example 5.6. Consider a graph G∗ = (V , E ) such that V = {a, b, c }, E = {ab, bc }. Then an interval-valued fuzzy graph G = (A, B), where
c a b c , , , , , 0.1 0.2 0.3 0.3 0.4 0.5 ab bc ab bc , , , , B= 0.1 0.2 0.3 0.4 A=
a
,
b
is self-complementary. Proposition 5.7. In a self-complementary interval-valued fuzzy complete graph G = (A, B) we have
∑ ∑ − − min(µ− (a) A (x), µA (y)), x̸=y µB (xy) = ∑ ∑x̸=y + + + (b) x̸=y µB (xy) = x̸=y min(µA (x), µA (y)). Proof. Let G = (A, B) be a self-complementary interval-valued fuzzy complete graph. Then there exists an automorphism − + + − − + + f : V → V such that µ− A (f (x)) = µA (x), µA (f (x)) = µA (x), µB (f (x)f (y)) = µB (xy) and µB (f (x)f (y)) = µB (xy) for all x, y ∈ V . Hence, for x, y ∈ V we obtain
− − − − − µ− B (xy) = µ B (f (x)f (y)) = min µA (f (x)), µA (f (y)) = min(µA (x), µA (y)), which implies (a). The proof of (b) is analogous.
− − + Proposition 5.8. Let G = (A, B) be an interval-valued fuzzy complete graph. If µ− B (xy) = min(µA (x), µA (y)) and µB (xy) = + + min(µA (x), µA (y)) for all x, y ∈ V , then G is self-complementary. − − Proof. Let G = (A, B) be an interval-valued fuzzy complete graph such that µ− B (xy) = min(µA (x), µA (y)) and + + + µB (xy) = min(µA (x), µA (y)) for all x, y ∈ V . Then G = G under the identity map I : V → V . Hence G is selfcomplementary.
Proposition 5.9. Let G1 = (A1 , B1 ) and G2 = (A2 , B2 ) be interval-valued fuzzy complete graphs. Then G1 ∼ = G2 if and only if G1 ∼ = G2 . Proof. Assume that G1 and G2 are isomorphic, there exists a bijective map f : V1 → V2 satisfying − µ− A1 (x) = µA2 (f (x)),
+ µ+ A1 (x) = µA2 (f (x)) for all x ∈ V1 ,
− µ− B1 (xy) = µB2 (f (x)f (y)),
+ µ+ B1 (xy) = µB2 (f (x)f (y)) for all xy ∈ E1 .
By the definition of complement, we have − − − − µ− B1 (xy) = min(µ− A1 (x), µA1 (y)) = min(µA2 (f (x)), µA2 (f (y))) = µ B2 (f (x)f (y)), + + + + µ+ B1 (xy) = min(µ+ A1 (x), µA1 (y)) = min(µA2 (f (x)), µA2 (f (y))) = µ B2 (f (x)f (y)) for all xy ∈ E1 .
Hence G1 ∼ = G2 . The proof of the converse part is straightforward.
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6. Conclusions It is well known that interval-valued fuzzy sets constitute a generalization of the notion of fuzzy sets. The interval-valued fuzzy models give more precision, flexibility and compatibility to the system as compared to the classical and fuzzy models. So we have introduced interval-valued fuzzy graphs and have presented several properties in this paper. The further study of interval-valued fuzzy graphs may also be extended with an application of interval-valued fuzzy graphs in database theory, an expert system, neural networks and geographical information system. Acknowledgements The authors are thankful to the referees for their valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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