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Dombi Fuzzy Graphs

Dombi Fuzzy Graphs FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 1, 58–79 https://doi.org/10.1080/16168658.2018.1509520 a b c S. Ashraf ,S.Naz and E. E. Kerre a b Quality Enhancement Cell, Lahore College for Women University, Lahore, Pakistan; Department of Mathematics, University of the Punjab, Lahore, Pakistan; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium ABSTRACT ARTICLE HISTORY Received 3 January 2018 A fuzzy graph is useful in representing structures of relationships Revised 12 February 2018 between items where the existence of a concrete item (vertex) and Accepted 16 February 2018 relationship between two items (edge) are matters of degree. In this paper, the novel concept of Dombi fuzzy graph is introduced. We KEYWORDS shall use graph terminology and introduce fuzzy analogs of several Fuzzy set; t-norm; t-conorm; basic graph-theoretical concepts using Dombi operator. Moreover, t-operators; Dombi fuzzy we consider these results on Dombi fuzzy graphs preserving strong graph property. 1. Introduction Menger presented triangular norms (t-norms) and triangular conorms (t-conorms) in [1] in the framework of probabilistic metric spaces which were later defined and discussed by Schweizer and Sklar [2]. Alsina et al. [3]provedthat t-norms and t-conorms are stan- dard models for intersecting and unifying fuzzy sets, respectively. Since then, many other researchers have presented various types of T-operators for the same purpose [4, 5]. Zadeh’s conventional T-operators, min and max, have been used in almost every applica- tion of fuzzy logic particularly in decision-making processes and fuzzy graph theory. It is a well known fact that from theoretical and experimental aspects other T-operators may work better in some situations, especially in the context of decision-making processes. For example, the product operator may be preferred to the min operator [6]. For the selection of appropriate T-operators for a given application, one has to consider the properties they possess, their suitability to the model, their simplicity, their software and hardware imple- mentation, etc. As the study on these operators has widened, multiple options are available for selecting T-operators that may be better suited for given research. Fuzzy graphs are designed to represent structures of relationships (in the form of edges) between concrete objects (vertices) as a matter of degree. Applications of fuzzy graphs cover an extensive range such as cluster analysis, database theory, decision-making and optimization of networks. Rosenfeld [7] considered fuzzy relations on fuzzy sets and devel- oped the structure of fuzzy graphs, using max and min operations, obtaining analogs of sev- eral graph-theoretical concepts. Some remarks on fuzzy graphs were given by Bhattacharya [8]. Mordeson and Peng [9] defined some operations on fuzzy graphs and introduced the CONTACT E. E. Kerre etienne.kerre@ugent.be © 2018 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 59 concept of strong fuzzy graphs. Later on, Bhutani and Battou [10] considered operations on fuzzy graphs preserving M-strong property. The complement of a fuzzy graph was pro- posed by Mordeson and Peng [9] and then modified by Sunitha and Vijayakumar [11]. Most wide spread T-operators, max and min, have been used to introduce the structure of fuzzy graphs [7–9, 11, 12] since their inception in literature and very little effort is done to make use of new operators. The main purpose of this paper is to stress that the max and min operations are not the only candidates for the generalization of the classical graphs to fuzzy graphs. We propose to incorporate the advancement proposed by Alsina, Klement and other researchers in the area of fuzzy logic in the area of fuzzy graph theory. The paper is reserved to demonstrate the use of a particular T-operator, namely the Dombi operator in the area of fuzzy graph theory. This paper is organized as follows: In Section 2, we have provided the reader an overview of the basic definitions and notations involved in this study. In Section 3, we propose the concepts of Dombi fuzzy graphs and discuss their properties while in Section 4 particular attention to the strong Dombi fuzzy graphs will be paid. We have used standard definitions and terminologies, in this paper. For more details and background, the readers are referred to [13–20]. 2. Preliminaries Throughout this paper, V represents a crisp universe of generic elements, G stands for the crisp graph and G is the Dombi fuzzy graph. Agraph G = (V, E) is a mathematical structure consisting of a set of vertices V = V(G) and a set of edges E = E(G), where each edge is an unordered pair of distinct vertices. Two vertices x and y of a graph G are adjacent if xy ∈ E(G). A vertex joined by an edge to a ver- tex x is called a neighbor of x. The number of edges joined with a vertex x of a graph G is called the degree of x in G denoted by deg (x) or deg(x). A graph with no loops and multi- ple edges is called simple graph. Throughout this paper, we will consider only undirected, simple graphs. The complement G of a graph G, is a graph having vertex set same as in G,in which two vertices are adjacent if and only if they are not adjacent in G. If there exists a one- one correspondence between the vertices of two graphs G = (V , E ) = (V(G ), E(G )) 1 1 1 1 1 and G = (V , E ) = (V(G ), E(G )) which preserves adjacency, then the graphs G and G 2 2 2 2 2 1 2 are called isomorphic. The standard products of graphs: the direct product (tensor product), the Carte- sian product, the semi-strong product, the strong product (symmetric composition) and the lexicographic product (composition) of two graphs G = (V , E ) and G = (V , E ) 1 1 1 2 2 2 will be denoted by G × G , G  G , G • G , G  G and G [G ], respectively. Let 1 2 1 2 1 2 1 2 1 2 (x , x ), (y , y ) ∈ V × V , where V × V is the vertex set of each product graph, then 1 2 1 2 1 2 1 2 E(G × G ) ={(x , x )(y , y ) | x y ∈ E and x y ∈ E }, 1 2 1 2 1 2 1 1 1 2 2 2 E(G  G ) ={(x , x )(y , y ) | x = y and x y ∈ E ,or x y ∈ E and x = y }, 1 2 1 2 1 2 1 1 2 2 2 1 1 1 2 2 E(G • G ) ={(x , x )(y , y ) | x = y and x y ∈ E ,or x y ∈ E and x y ∈ E }, 1 2 1 2 1 2 1 1 2 2 2 1 1 1 2 2 2 E(G  G ) = E(G  G ) ∪ E(G × G ), 1 2 1 2 1 2 E(G [G ]) ={(x , x )(y , y ) | x y ∈ E ,or x = y and x y ∈ E }. 1 2 1 2 1 2 1 1 1 1 1 2 2 2 60 S. ASHRAF ET AL. Definition 2.1 ([21]): A binary operation T : [0, 1] → [0, 1] is a triangular norm ( t-norm) if for all x, y, z ∈ [0, 1] it satisfies the following: (1) T(1, x) = x. (boundary condition) (2) T(x, y) = T(y, x). (commutativity) (3) T(x, T(y, z)) = T(T(x, y), z). (associativity) (4) T(x, y) ≤ T(x, z) if y ≤ z. (monotonicity) Definition 2.2 ([21]): A binary operation S : [0, 1] → [0, 1] is a triangular conorm ( t- conorm) if and only if there exists a t-norm T such that for all (x, y) ∈ [0, 1] S(x, y) = 1 − T(1 − x,1 − y). Popular choices for t-norms are: • The minimum operator M : M(x, y) = min(x, y). • The product operator P : P(x, y) = xy. • The Lukasiewicz’s t-norm W : W(x, y) = max(x + y − 1, 0). Popular choices for corresponding dual t-conorms are: ∗ ∗ • The maximum operator M : M (x, y) = max(x, y). ∗ ∗ • The probabilistic sum P : P (x, y) = x + y − xy. ∗ ∗ • The bounded sum W : W (x, y) = min(x + y,1). The Dombi family t-norm : λ> 0, 1−x 1−y λ λ 1/λ 1 + [( ) + ( ) ] x y t-conorm : λ> 0, 1−y 1−x −λ −λ 1/−λ 1 + [( ) + ( ) ] x y negation 1 − x. The Hamacher family xy t-norm , λ + (1 − λ)(x + y − xy) x + y + (λ − 2)xy t-conorm , 1 + (λ − 1)xy negation 1 − x. Another set of T-operators is xy T(x, y) = , x + y − xy FUZZY INFORMATION AND ENGINEERING 61 x + y − 2xy S(x, y) = , 1 − xy which is obtained by taking λ = 0, in Hamacher family and λ = 1 in Dombi fam- ily of t-norms and t-conorms. Also P(x, y) ≤ xy/(x + y − xy) ≤ M(x, y) and M (x, y) ≤ (x + y − 2xy)/(1 − xy) ≤ P (x, y). Definition 2.3 ([22]): A fuzzy subset η of a set V is a function η : V → [0, 1]. A fuzzy relation [7]on V is a mapping ζ : V × V → [0, 1]. 3. Dombi Fuzzy Graphs Definition 3.1: A Dombi fuzzy graph with a finite set V as the underlying set is a pair G = (η, ζ), where η : V → [0, 1] is a fuzzy subset in V and ζ : V × V → [0, 1] is a symmetric fuzzy relation on η such that η(x)η(y) ζ(xy) ≤ for all x, y ∈ V. η(x) + η(y) − η(x)η(y) We call η the Dombi fuzzy vertex set of G and ζ the Dombi fuzzy edge set of G (Figure 1). Example 3.1: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jk kl jl km η = , , , , ζ = , , , . 0.4 0.5 0.9 0.7 0.1 0.4 0.3 0.2 Definition 3.2: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the direct product G × G = (η × η , ζ × ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η × η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 ζ (x y )ζ (x y ) 1 1 1 2 2 2 (ζ × ζ )((x , x )(y , y )) = for all x y ∈ E , 1 2 1 2 1 2 1 1 1 ζ (x y ) + ζ (x y ) − ζ (x y )ζ (x y ) 1 1 1 2 2 2 1 1 1 2 2 2 for all x y ∈ E . 2 2 2 Figure 1. Dombi fuzzy graph. 62 S. ASHRAF ET AL. Figure 2. Dombi fuzzy graph G × G . 1 2 Example 3.2: Consider two Dombi fuzzy graphs G and G , where η ={j/0.9, k/0.3}, ζ = 1 2 1 1 {jk/0.2}, η ={l/0.2, m/0.7, n/0.4} and ζ ={lm/0.1, mn/0.3}. Then we have 2 2 (ζ × ζ )((j, l)(k, m)) = 0.07, (ζ × ζ )((j, m)(k, l)) = 0.07, 1 2 1 2 (ζ × ζ )((j, m)(k, n)) = 0.14, (ζ × ζ )((j, n)(k, m)) = 0.14. 1 2 1 2 It is easy to check that G × G is the Dombi fuzzy graph of G × G (Figure 2). 1 2 1 2 Proposition 3.1: Let G and G be the Dombi fuzzy graphs of the graphs G and 1 2 1 G , respectively. The direct product G × G of G and G is the Dombi fuzzy graph of 2 1 2 1 2 G × G . 1 2 Proof: Consider x y ∈ E , x y ∈ E . Then 1 1 1 2 2 2 (ζ × ζ )((x , x )(y , y )) 1 2 1 2 1 2 = T (ζ (x y ), ζ (x y )) 1 1 1 2 2 2 η (x )η (y ) η (x )η (y ) 1 1 1 1 2 2 2 2 ≤ T , . η (x ) + η (y ) − η (x )η (y ) η (x ) + η (y ) − η (x )η (y ) 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 Putting a = η (x ), b = η (y ), c = η (x ), d = η (y ), 1 1 1 1 2 2 2 2 ab cd (ζ × ζ )((x , x )(y , y )) ≤ T , 1 2 1 2 1 2 a + b − ab c + d − cd abcd (a+b−ab)(c+d−cd) ab cd abcd + − a+b−ab c+d−cd (a+b−ab)(c+d−cd) FUZZY INFORMATION AND ENGINEERING 63 abcd (a+c−ac)(b+d−bd) ac bd abcd + − a+c−ac b+d−bd (a+c−ac)(b+d−bd) (η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 = . (η × η )((x , x )) + (η × η )((y , y ))−(η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Hence proved. Definition 3.3: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the Cartesian product G  G = (η  η , ζ  ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η  η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ  ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 η (z)ζ (x y ) 2 1 1 1 (ζ  ζ )((x , z)(y , z)) = for all z ∈ V , 1 2 1 1 2 η (z) + ζ (x y ) − η (z)ζ (x y ) 2 1 1 1 2 1 1 1 for all x y ∈ E . 1 1 1 Remark 3.1: The Cartesian product of two Dombi fuzzy graphs is not necessarily a Dombi fuzzy graph. Example 3.3: Consider two Dombi fuzzy graphs as in Example 3.2. Then we have (ζ  ζ )((j, l)(j, m)) = 0.1, (ζ  ζ )((j, m)(j, n)) = 0.29, 1 2 1 2 (ζ  ζ )((k, l)(k, m)) = 0.08, (ζ  ζ )((k, m)(k, n)) = 0.18, 1 2 1 2 (ζ  ζ )((j, l)(k, l)) = 0.11, (ζ  ζ )((j, m)(k, m)) = 0.18, 1 2 1 2 (ζ  ζ )((j, n)(k, n)) = 0.15. 1 2 Clearly, (η  η )((k, m))(η  η )((k, n)) 1 2 1 2 (ζ  ζ )((k, m)(k, n)) = 0.18  0.13 = , 1 2 (η  η )((k, m)) + (η  η )((k, n))− 1 2 1 2 (η  η )((k, m))(η  η )((k, n)) 1 2 1 2 (ζ  ζ )((j, l)(k, l)) = 0.11  0.09 1 2 (η  η )((j, l))(η  η )((k, l)) 1 2 1 2 = . (η  η )((j, l)) + (η  η )((k, l))− 1 2 1 2 (η  η )((j, l))(η  η )((k, l)) 1 2 1 2 Therefore G  G is not a Dombi fuzzy graph (Figure 3). 1 2 Definition 3.4: If a fuzzy membership degree is attached from [0, 1] to each edge of the Dombi fuzzy graph G of a graph G and each vertex is crisply in G, then G is called the Dombi fuzzy edge graph. 64 S. ASHRAF ET AL. Figure 3. Not Dombi fuzzy graph G  G . 1 2 Proposition 3.2: Let G and G be the Dombi fuzzy edge graphs of the graphs G and G , 1 2 1 2 respectively. The Cartesian product G  G of G and G is the Dombi fuzzy edge graph of 1 2 1 2 G  G . 1 2 Proof: Consider x ∈ V , x y ∈ E . Then 1 2 2 2 (ζ  ζ )((x, x )(x, y )) 1 2 2 2 = T (η (x), ζ (x y )) = T (1, ζ (x y )) 1 2 2 2 2 2 2 η (x )η (y ) 2 2 2 2 = ζ (x y ) ≤ 2 2 2 η (x ) + η (y ) − η (x )η (y ) 2 2 2 2 2 2 2 2 (η  η )((x, x ))(η  η )((x, y )) 1 2 2 1 2 2 = . (η  η )((x, x )) + (η  η )((x, y )) − (η  η )((x, x ))(η  η )((x, y )) 1 2 2 1 2 2 1 2 2 1 2 2 Consider z ∈ V , x y ∈ E . Then 2 1 1 1 (ζ  ζ )((x , z)(y , z)) 1 2 1 1 = T (ζ (x y ), η (z)) = T (ζ (x y ),1) 1 1 1 2 1 1 1 η (x )η (y ) 1 1 1 1 = ζ (x y ) ≤ 1 1 1 η (x ) + η (y ) − η (x )η (y ) 1 1 1 1 1 1 1 1 (η  η )((x , z))(η  η )((y , z)) 1 2 1 1 2 1 = . (η  η )((x , z)) + (η  η )((y , z)) − (η  η )((x , z))(η  η )((y , z)) 1 2 1 1 2 1 1 2 1 1 2 1 Hence proved. Example 3.4: Consider two Dombi fuzzy graphs G and G , where η (x) = 1 for all x ∈ V , 1 2 1 1 ζ ={jk/0.7}, η (y) = 1 for all y ∈ V ,and ζ ={lm/0.3, mn/0.9}. Then we have 1 2 2 2 (ζ  ζ )((j, l)(j, m)) = 0.3, (ζ  ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ  ζ )((k, l)(k, m)) = 0.3, (ζ  ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 FUZZY INFORMATION AND ENGINEERING 65 Figure 4. Dombi fuzzy edge graph G  G . 1 2 (ζ  ζ )((j, l)(k, l)) = 0.7, (ζ  ζ )((j, m)(k, m)) = 0.7, 1 2 1 2 (ζ  ζ )((j, n)(k, n)) = 0.7. 1 2 It is easy to check that G  G is the Dombi fuzzy edge graph of G  G (Figure 4). 1 2 1 2 Definition 3.5: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. i i i i Define the semi-strong product G • G = (η • η , ζ • ζ ) of the Dombi fuzzy graphs G = 1 2 1 2 1 2 1 (η , ζ ) and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 1 1 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η • η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ • ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 ζ (x y )ζ (x y ) 1 1 1 2 2 2 (ζ • ζ )((x , x )(y , y )) = for all x y ∈ E , 1 2 1 2 1 2 1 1 1 ζ (x y ) + ζ (x y ) − ζ (x y )ζ (x y ) 1 1 1 2 2 2 1 1 1 2 2 2 for all x y ∈ E . 2 2 2 Proposition 3.3: Let G and G be the Dombi fuzzy edge graphs of the graphs G and G , 1 2 1 2 respectively. The semi-strong product G • G of G and G is the Dombi fuzzy edge graph of 1 2 1 2 G • G . 1 2 Proof: The proof follows at once from the proof of Propositions 3.1 and 3.2. Example 3.5: Consider two Dombi fuzzy graphs G and G as in Example 3.4. Then we have 1 2 (ζ • ζ )((j, l)(j, m)) = 0.3, (ζ • ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ • ζ )((k, l)(k, m)) = 0.3, (ζ • ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 (ζ • ζ )((j, l)(k, m)) = 0.27, (ζ • ζ )((j, m)(k, l)) = 0.27, 1 2 1 2 66 S. ASHRAF ET AL. Figure 5. Dombi fuzzy edge graph G • G . 1 2 (ζ • ζ )((j, m)(k, n)) = 0.65, (ζ • ζ )((j, n)(k, m)) = 0.65. 1 2 1 2 It is easy to check that G • G is the Dombi fuzzy edge graph of G • G (Figure 5). 1 2 1 2 Definition 3.6: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the strong product G  G = (η  η , ζ  ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η  η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ  ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 η (z)ζ (x y ) 2 1 1 1 (ζ  ζ )((x , z)(y , z)) = for all z ∈ V , 1 2 1 1 2 η (z) + ζ (x y ) − η (z)ζ (x y ) 2 1 1 1 2 1 1 1 for all x y ∈ E , 1 1 1 ζ (x y )ζ (x y ) 1 1 1 2 2 2 (ζ  ζ )((x , x )(y , y )) = for all x y ∈ E , 1 2 1 2 1 2 1 1 1 ζ (x y ) + ζ (x y ) − ζ (x y )ζ (x y ) 1 1 1 2 2 2 1 1 1 2 2 2 for all x y ∈ E . 2 2 2 Proposition 3.4: Let G and G be the Dombi fuzzy edge graphs of the graphs G and G , 1 2 1 2 respectively. The strong productG  G ofG andG is the Dombi fuzzy edge graph of G  G . 1 2 1 2 1 2 Proof: The proof follows at once from the proof of Propositions 3.1 and 3.2. Example 3.6: Consider two Dombi fuzzy graphs G and G as in Example 3.4. Then we have 1 2 (ζ  ζ )((j, l)(j, m)) = 0.3, (ζ  ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ  ζ )((k, l)(k, m)) = 0.3, (ζ  ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 (ζ  ζ )((j, l)(k, l)) = 0.7, (ζ  ζ )((j, m)(k, m)) = 0.7, 1 2 1 2 (ζ  ζ )((j, n)(k, n)) = 0.7, (ζ  ζ )((j, l)(k, m)) = 0.27, 1 2 1 2 FUZZY INFORMATION AND ENGINEERING 67 Figure 6. Dombi fuzzy edge graph G  G . 1 2 (ζ  ζ )((j, m)(k, l)) = 0.27, (ζ  ζ )((j, m)(k, n)) = 0.65, 1 2 1 2 (ζ  ζ )((j, n)(k, m)) = 0.65. 1 2 It is easy to check that G  G is the Dombi fuzzy edge graph of G  G (Figure 6). 1 2 1 2 Definition 3.7: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the lexicographic product G [G ] = (η ◦ η , ζ ◦ ζ ) of the Dombi fuzzy graphs G and G 1 2 1 2 1 2 1 2 of G = (V , E ) and G = (V , E ), respectively, as follows: 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η ◦ η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ ◦ ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 η (z)ζ (x y ) 2 1 1 1 (ζ ◦ ζ )((x , z)(y , z)) = for all z ∈ V , 1 2 1 1 2 η (z) + ζ (x y ) − η (z)ζ (x y ) 2 1 1 1 2 1 1 1 for all x y ∈ E , 1 1 1 η (x )η (y )ζ (x y ) 2 2 2 2 1 1 1 (ζ ◦ ζ )((x , x )(y , y )) = 1 2 1 2 1 2 η (x )η (y ) + η (y )ζ (x y )+ 2 2 2 2 2 2 1 1 1 η (x )ζ (x y ) − 2η (x )η (y )ζ (x y ) 2 2 1 1 1 2 2 2 2 1 1 1 for all x y ∈ E , x = y . 1 1 1 2 2 Proposition 3.5: The lexicographic product G [G ] of two Dombi fuzzy edge graphs of G and 1 2 1 G is the Dombi fuzzy edge graph of G [G ]. 2 1 2 Proof: From the proof of Proposition 3.2, it follows that (ζ ◦ ζ )((x, x )(x, y )) 1 2 2 2 (η ◦ η )((x, x ))(η ◦ η )((x, y )) 1 2 2 1 2 2 (η ◦ η )((x, x )) + (η ◦ η )((x, y )) − (η ◦ η )((x, x ))(η ◦ η )((x, y )) 1 2 2 1 2 2 1 2 2 1 2 2 68 S. ASHRAF ET AL. for all x ∈ V , x y ∈ E , 1 2 2 2 (ζ ◦ ζ )((x , z)(y , z)) 1 2 1 1 (η ◦ η )((x , z))(η ◦ η )((y , z)) 1 2 1 1 2 1 (η ◦ η )((x , z)) + (η ◦ η )((y , z)) − (η ◦ η )((x , z))(η ◦ η )((y , z)) 1 2 1 1 2 1 1 2 1 1 2 1 for all z ∈ V , x y ∈ E . 2 1 1 1 Now consider x y ∈ E , x = y . Then 1 1 1 2 2 (ζ ◦ ζ )((x , x )(y , y )) 1 2 1 2 1 2 = T (T (η (x ), η (y )) , ζ (x y )) = T (T (1, 1) , ζ (x y )) 2 2 2 2 1 1 1 1 1 1 η (x )η (y ) 1 1 1 1 = T 1, ζ (x y ) = ζ (x y ) ≤ ( ) 1 1 1 1 1 1 η (x ) + η (y ) − η (x )η (y ) 1 1 1 1 1 1 1 1 (η ◦ η )((x , x ))(η ◦ η )((y , y )) 1 2 1 2 1 2 1 2 = . (η ◦ η )((x , x )) + (η ◦ η )((y , y )) − (η ◦ η )((x , x ))(η ◦ η )((y , y )) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Hence proved. Example 3.7: Consider two Dombi fuzzy graphs G and G as in Example 3.4. Then we have 1 2 (ζ ◦ ζ )((j, l)(j, m)) = 0.3, (ζ ◦ ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ ◦ ζ )((k, l)(k, m)) = 0.3, (ζ ◦ ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 (ζ ◦ ζ )((j, l)(k, l)) = 0.7, (ζ ◦ ζ )((j, m)(k, m)) = 0.7, 1 2 1 2 (ζ ◦ ζ )((j, n)(k, n)) = 0.7, (ζ ◦ ζ )((j, l)(k, m)) = 0.27, 1 2 1 2 (ζ ◦ ζ )((j, m)(k, l)) = 0.27, (ζ ◦ ζ )((j, m)(k, n)) = 0.65, 1 2 1 2 (ζ ◦ ζ )((j, n)(k, m)) = 0.65, (ζ ◦ ζ )((j, l)(k, n)) = 0.7, 1 2 1 2 (ζ ◦ ζ )((j, n)(k, l)) = 0.7. 1 2 It is easy to see that G [G ] is the Dombi fuzzy edge graph of G [G ] (Figure 7). 1 2 1 2 Remark 3.2: In general the semi-strong product, strong product and lexicographic prod- uct of two Dombi fuzzy graphs are not Dombi fuzzy graphs. Figure 7. Dombi fuzzy edge graph G [G ]. 1 2 FUZZY INFORMATION AND ENGINEERING 69 Example 3.8: Consider two Dombi fuzzy graphs as in Example 3.2. Then we have (ζ • ζ )((j, l)(j, m)) = 0.1, (ζ • ζ )((j, m)(j, n)) = 0.29, 1 2 1 2 (ζ • ζ )((k, l)(k, m)) = 0.08, (ζ • ζ )((k, m)(k, n)) = 0.18, 1 2 1 2 (ζ • ζ )((j, l)(k, m)) = 0.07, (ζ • ζ )((j, m)(k, l)) = 0.07, 1 2 1 2 (ζ • ζ )((j, m)(k, n)) = 0.14, (ζ • ζ )((j, n)(k, m)) = 0.14. 1 2 1 2 Clearly, (ζ • ζ )((k, m)(k, n)) = 0.18  0.13 = T((η • η )((k, m)), (η • η )((k, n))). 1 2 1 2 1 2 Therefore, G • G is not a Dombi fuzzy graph. Analogously, we can show that G  G and 1 2 1 2 G [G ] are not Dombi fuzzy graphs (Figure 8). 1 2 Definition 3.8: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the union G ∪ G = (η ∪ η , ζ ∪ ζ ) of the Dombi fuzzy graphs G = (η , ζ ) and G = 1 2 1 2 1 2 1 1 1 2 (η , ζ ) as follows: 2 2 η (x) if x ∈ V \ V , ⎪ 1 1 2 η (x) if x ∈ V \ V , 2 2 1 (η ∪ η )(x) = 1 2 η (x) + η (x) − 2η (x)η (x) ⎪ 1 2 1 2 ⎩ if x ∈ V ∩ V . 1 2 1 − η (x)η (x) 1 2 ζ (xy) if xy ∈ E \ E , ⎪ 1 1 2 ζ (xy) if xy ∈ E \ E , 2 2 1 (ζ ∪ ζ )(xy) = 1 2 ζ (xy) + ζ (xy) − 2ζ (xy)ζ (xy) ⎪ 1 2 1 2 ⎩ if xy ∈ E ∩ E . 1 2 1 − ζ (xy)ζ (xy) 1 2 Example 3.9: We consider two Dombi fuzzy graphs G and G , where η ={j/0.5, k/0.8, 1 2 1 l/0.4}, ζ ={jk/0.4, kl/0.3, jl/0.1}, η ={j/0.6, k/0.5, m/0.3} and ζ ={jk/0.2, km/0.1} 1 2 2 (Figure 9). Then we have j k l m jk kl jl km η ∪ η = , , , , ζ ∪ ζ = , , , . 1 2 1 2 0.71 0.83 0.4 0.3 0.48 0.3 0.18 0.1 Figure 8. Not Dombi fuzzy graph G • G . 1 2 70 S. ASHRAF ET AL. Figure 9. Dombi fuzzy graph G ∪ G . 1 2 Theorem 3.1: The union G ∪ G of G and G is the Dombi fuzzy graph of G ∪ G if and only 1 2 1 2 1 2 if G and G are the Dombi fuzzy graphs of G and G , respectively, where η , η , ζ and ζ are 1 2 1 2 1 2 1 2 fuzzy subsets of V , V , E and E , respectively, and V ∩ V =∅. 1 2 1 2 1 2 Proof: Suppose that G ∪ G is the Dombi fuzzy graph. Let xy ∈ E , then xy ∈ / E and x, y ∈ 1 2 1 2 V \ V . Then 1 2 (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 ζ (xy) = (ζ ∪ ζ )(xy) ≤ 1 1 2 (η ∪ η )(x) + (η ∪ η )(y) − (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 1 2 1 2 η (x)η (y) 1 1 = . η (x) + η (y) − η (x)η (y) 1 1 1 1 G is the Dombi fuzzy graph of G . Similarly, it is easy to show that G is the Dombi Thus 1 1 2 fuzzy graph of G . Conversely, suppose that G and G are the Dombi fuzzy graphs of G and G ,respec- 1 2 1 2 tively. Consider xy ∈ E \ E . Then 1 2 (ζ ∪ ζ )(xy) = ζ (xy)(by Definition 3.8) 1 2 1 ≤ T(η (x), η (y)) (by definition of Dombi fuzzy graph) 1 1 = T((η ∪ η )(x), (η ∪ η )(y)). 1 2 1 2 Similarly, we find for xy ∈ E \ E 2 1 (ζ ∪ ζ )(xy) ≤ T((η ∪ η )(x), (η ∪ η )(y)) 1 2 1 2 1 2 (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 = . (η ∪ η )(x) + (η ∪ η )(y) − (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 1 2 1 2 Hence proved. Definition 3.9: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. i i i i Define the ring sum G ⊕ G = (η ⊕ η , ζ ⊕ ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) as follows: 2 2 2 (η ⊕ η )(x) = (η ∪ η )(x) if x ∈ V ∪ V , 1 2 1 2 1 2 FUZZY INFORMATION AND ENGINEERING 71 ⎪ ζ (xy) if xy ∈ E \ E , 1 1 2 (ζ ⊕ ζ )(xy) = ζ (xy) if xy ∈ E \ E , 1 2 2 2 1 0if xy ∈ E ∩ E . 1 2 Proposition 3.6: The ring sum G ⊕ G of two Dombi fuzzy graphs G and G of G and G is 1 2 1 2 1 2 the Dombi fuzzy graph of G ⊕ G . 1 2 Proof: Consider xy ∈ E \ E . Then there are three possibilities, (i) x, y ∈ V \ V , (ii) x ∈ V \ 1 2 1 2 1 V , y ∈ V ∩ V and (iii) x, y ∈ V ∩ V . 2 1 2 1 2 (i) Suppose x, y ∈ V \ V 1 2 (ζ ⊕ ζ )(xy) = ζ (xy)(by Definition 3.9) 1 2 1 ≤ T(η (x), η (y)) (by definition of Dombi fuzzy graph) 1 1 = T((η ∪ η )(x), (η ∪ η )(y)) 1 2 1 2 = T((η ⊕ η )(x), (η ⊕ η )(y)). 1 2 1 2 (ii) Suppose x ∈ V \ V , y ∈ V ∩ V . Then we obtain: 1 2 1 2 (ζ ⊕ ζ )(xy) = ζ (xy) 1 2 1 ≤ T(η (x), η (y)) 1 1 = T((η ∪ η )(x), η (y)) 1 2 1 = T((η ⊕ η )(x), η (y)).(∗) 1 2 1 Clearly, η (y) + η (y) − 2η (y)η (y) 1 2 1 2 η (y) ≤ . 1 − η (y)η (y) 1 2 As putting η (y) = f, η (y) = h.Weget 1 2 f + h − 2fh f ≤ , 1 − fh f − f h ≤ f + h − 2fh, −f ≤ 1 − 2f, 0 ≤ f − 2f + 1, 0 ≤ (f − 1) . Since T is increasing, so from (∗)weget: (ζ ⊕ ζ )(xy) ≤ T((η ⊕ η )(x), (η ⊕ η )(y)). 1 2 1 2 1 2 (iii) Suppose x ∈ V ∩ V , y ∈ V ∩ V . Then we get: 1 2 1 2 (ζ ⊕ ζ )(xy) = ζ (xy) 1 2 1 ≤ T(η (x), η (y)) 1 1 72 S. ASHRAF ET AL. ≤ T((η ∪ η )(x), (η ∪ η )(y)) 1 2 1 2 = T((η ⊕ η )(x), (η ⊕ η )(y)), 1 2 1 2 since as in (ii): η (x) + η (x) − 2η (x)η (x) η (y) + η (y) − 2η (y)η (y) 1 2 1 2 1 2 1 2 η (x) ≤ and η (y) ≤ . 1 1 1 − η (y)η (y) 1 − η (y)η (y) 1 2 1 2 Similarly, because of symmetry, we find for xy ∈ E \ E in the three possible cases: 2 1 (ζ ⊕ ζ )(xy) ≤ T((η ⊕ η )(x), (η ⊕ η )(y)) 1 2 1 2 1 2 (η ⊕ η )(x)(η ⊕ η )(y) 1 2 1 2 = . (η ⊕ η )(x) + (η ⊕ η )(y) − (η ⊕ η )(x)(η ⊕ η )(y) 1 2 1 2 1 2 1 2 Hence proved. Definition 3.10: Consider the join G + G = (V ∪ V , E ∪ E ∪ E ) of two graphs G = 1 2 1 2 1 2 1 (V , E ) and G = (V , E ), where E is the set of all edges joining the vertices of V and V , 1 1 2 2 2 1 2 V ∩ V =∅.Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define the 1 2 i i i i join G + G = (η + η , ζ + ζ ) of the Dombi fuzzy graphs G = (η , ζ ) and G = (η , ζ ) 1 2 1 2 1 2 1 1 1 2 2 2 as follows: (η + η )(x) = (η ∪ η )(x),if x ∈ V ∪ V , 1 2 1 2 1 2 (ζ + ζ )(xy) = (ζ ∪ ζ )(xy),if xy ∈ E ∪ E , 1 2 1 2 1 2 η (x)η (y) 1 2 (ζ + ζ )(xy) = ,if xy ∈ E . 1 2 η (x) + η (y) − η (x)η (y) 1 2 1 2 Theorem 3.2: The join G + G of G and G is the Dombi fuzzy graph of G + G if and only 1 2 1 2 1 2 if G and G are Dombi fuzzy graphs of G and G , respectively, where η , η , ζ and ζ are fuzzy 1 2 1 2 1 2 1 2 subsets of V , V , E and E , respectively, and V ∩ V =∅. 1 2 1 2 1 2 Proof: Suppose thatG + G is the Dombi fuzzy graph. Then from the proof of Theorem 3.1, 1 2 G and G are Dombi fuzzy graphs. 1 2 Conversely, suppose that G and G are Dombi fuzzy graphs of G and G , respectively. 1 2 1 2 Consider xy ∈ E ∪ E . Then the required result follows from Theorem 3.1. Let xy ∈ E . Then 1 2 η (x)η (y) 1 2 (ζ + ζ )(xy) = 1 2 η (x) + η (y) − η (x)η (y) 1 2 1 2 (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 (η ∪ η )(x) + (η ∪ η )(y) − (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 1 2 1 2 (by definition 3.8 with V ∩ V =∅) 1 2 (η + η )(x)(η + η )(y) 1 2 1 2 = . (η + η )(x) + (η + η )(y) − (η + η )(x)(η + η )(y) 1 2 1 2 1 2 1 2 Hence proved.  FUZZY INFORMATION AND ENGINEERING 73 Definition 3.11: The complement of a Dombi fuzzy graph G = (η, ζ) of G = (V, E) is a ¯ ¯ ¯ Dombi fuzzy graph G = (η ¯, ζ), where η( ¯ x) = η(x) for all x ∈ V and ζ is defined as follows: η(x)η(y) ,if ζ(xy) = 0, η(x) + η(y) − η(x)η(y) ζ(xy) = η(x)η(y) − ζ(xy),if0 <ζ(xy) ≤ 1. η(x) + η(y) − η(x)η(y) Example 3.10: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jm km lm η = , , , , ζ = , , . 0.3 0.6 0.4 0.9 0.29 0.56 0.2 We have η(x) = η(x) = η(x) for all x ∈ V, and η(x)η(y) ζ(xy) = − ζ(xy) η(x) + η(y) − η(x)η(y) η(x)η(y) η(x)η(y) = − − ζ(xy) η(x) + η(y) − η(x)η(y) η(x) + η(y) − η(x)η(y) = ζ(xy) for all x, y ∈ V. Hence G = G (Figure 10). Definition 3.12: A homomorphism ψ : G → G of two Dombi fuzzy graphs G = (η , ζ ) 1 2 1 1 1 and G = (η , ζ ) is a mapping ψ : V → V satisfying the following conditions: 2 2 2 1 2 (a) η (x) ≤ η (ψ (x)) for all x ∈ V , 1 2 1 (b) ζ (xy) ≤ ζ (ψ (x)ψ (y)) for all x, y ∈ V . 1 2 1 Definition 3.13: An isomorphism ψ : G → G of two Dombi fuzzy graphs G = (η , ζ ) 1 2 1 1 1 and G = (η , ζ ) (denoted as G G ) is a bijective mapping ψ : V → V satisfying the 2 2 2 1 2 1 2 following conditions: Figure 10. Dombi fuzzy graph G. 74 S. ASHRAF ET AL. (c) η (x) = η (ψ (x)) for all x ∈ V , 1 2 1 (d) ζ (xy) = ζ (ψ (x)ψ (y)) for all x, y ∈ V . 1 2 1 A weak isomorphism ψ : G → G is a bijective homomorphism with the condition (c) 1 2 above and a co-weak isomorphism ψ : G → G is a bijective homomorphism with the 1 2 condition (d) above. Definition 3.14: A Dombi fuzzy graph G = (η, ζ) is said to be self-complementary if G = ∼ ¯ ¯ (η, ζ) G = (η ¯, ζ). Proposition 3.7: Let G = (η, ζ) be a self-complementary Dombi fuzzy graph, then 1 η(x)η(y) ζ(xy) = . 2 η(x) + η(y) − η(x)η(y) x=y x=y Proof: Let G be a self-complementary Dombi fuzzy graph. Then there exists an isomor- phism ψ : V → V such that η( ¯ ψ (x)) = η(x) for all x ∈ V and ζ(ψ(x)ψ (y)) = ζ(xy) for all xy ∈ E. By definition of G,wehave η( ¯ ψ (x))η( ¯ ψ (y)) ζ(ψ(x)ψ (y)) = − ζ(ψ(x)ψ (y)) η( ¯ ψ (x)) +¯ η(ψ (y)) −¯ η(ψ (x))η( ¯ ψ (y)) η(x)η(y) ζ(xy) = − ζ(ψ(x)ψ (y)) η(x) + η(y) − η(x)η(y) η(x)η(y) ζ(xy) + ζ(ψ(x)ψ (y)) = η(x) + η(y) − η(x)η(y) x=y x=y x=y η(x)η(y) 2 ζ(xy) = η(x) + η(y) − η(x)η(y) x=y x=y 1 η(x)η(y) ζ(xy) = . 2 η(x) + η(y) − η(x)η(y) x=y x=y Hence proved. Proposition 3.8: Let G = (η, ζ) be the Dombi fuzzy graph of G. If ζ(xy) = (η(x)η(y)/ (η(x) + η(y) − η(x)η(y))) for all x, y ∈ V, then G is self-complementary. Proof: Let G be the Dombi fuzzy graph satisfying ζ(xy) = (η(x)η(y)/(η(x) + η(y) − η(x) η(y))) for all x, y ∈ V. Then the identity mapping I : V → V is an isomorphism from G to G. Clearly, I satisfies the condition (c) of Definition 3.13. Since ζ(xy) = (η(x)η(y)/(η(x) + η(y) − η(x)η(y))) for all x, y ∈ V,wehave ¯ ¯ ζ(I(x)I(y)) = ζ(xy) η(x)η(y) = − ζ(xy) η(x) + η(y) − η(x)η(y) FUZZY INFORMATION AND ENGINEERING 75 η(x)η(y) 1 η(x)η(y) = − η(x) + η(y) − η(x)η(y) 2 η(x) + η(y) − η(x)η(y) 1 η(x)η(y) 2 η(x) + η(y) − η(x)η(y) = ζ(xy). Thus the condition (d) of Definition 3.13 is also satisfied by I. Therefore G is self- complementary. Proposition 3.9: The complements of two isomorphic Dombi fuzzy graphs are isomorphic and conversely. Proof: Suppose G and G are two isomorphic Dombi fuzzy graphs. Then there exists a 1 2 bijective mapping ψ : V → V satisfying 1 2 η (x) = η (ψ (x)) for all x ∈ V , 1 2 1 ζ (xy) = ζ (ψ (x)ψ (y)) for all xy ∈ E . 1 2 1 Using the definition of complement, we have η (x)η (y) 1 1 ζ (xy) = − ζ (xy) 1 1 η (x) + η (y) − η (x)η (y) 1 1 1 1 η (ψ (x))η (ψ (y)) 2 2 = − ζ (ψ (x)ψ (y)) = ζ (ψ (x)ψ (y)). 2 2 η (ψ (x)) + η (ψ (y)) − η (ψ (x))η (ψ (y)) 2 2 2 2 Hence G = G . Similarly, we can prove the converse part. 1 2 Proposition 3.10: The complements of two weak-isomorphic Dombi fuzzy graphs are weak isomorphic. Proof: Suppose G and G are two weak-isomorphic Dombi fuzzy graphs. Then there exists 1 2 a bijective mapping ψ : V → V satisfying 1 2 η (x) = η (ψ (x)) for all x ∈ V , 1 2 1 ζ (xy) ≤ ζ (ψ (x)ψ (y)) for all xy ∈ E . 1 2 1 Consider ζ (xy) ≤ ζ (ψ (x)ψ (y)) 1 2 (xy) ≥−ζ (ψ (x)ψ (y)) ⇒−ζ 1 2 ⇒ T(η (x), η (y)) − ζ (xy) ≥ T(η (x), η (y)) − ζ (ψ (x)ψ (y)) 1 1 1 1 1 2 ⇒ T(η (x), η (y)) − ζ (xy) ≥ T(η (ψ (x)), η (ψ (y))) − ζ (ψ (x)ψ (y)) 1 1 1 2 2 2 ⇒ ζ (xy) ≥ ζ (ψ (x)ψ (y)). (by definition of complement) 1 2 Hence G and G are weak isomorphic. 1 2 76 S. ASHRAF ET AL. We state the following proposition without proof. Proposition 3.11: The complements of two co-weak-isomorphic Dombi fuzzy graphs are homomorphic. 4. Strong Dombi Fuzzy Graphs Definition 4.1: A Dombi fuzzy graph G = (η, ζ) is called a strong Dombi fuzzy graph of G = (V, E) if ζ(xy) = η(x)η(y)/(η(x) + η(y) − η(x)η(y)) for all xy ∈ E. Example 4.1: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jk jm ml η = , , , , ζ = , , . 0.5 0.7 0.8 0.3 0.41 0.23 0.28 Proposition 4.1: If G and G are strong Dombi fuzzy graphs, then G × G and G + G are 1 2 1 2 1 2 also strong Dombi fuzzy graphs (Figure 11). Proof: The proof follows at once from the proof of Proposition 3.1 and Theorem 3.2. Proposition 4.2: If G and G are strong Dombi fuzzy edge graphs, then G  G , G • G , 1 2 1 2 1 2 G  G and G [G ] are also strong Dombi fuzzy edge graphs. 1 2 1 2 Proof: The proof follows at once from the proof of Propositions 3.2–3.5. Remark 4.1: In general the Cartesian product, semi-strong product, strong product and lexicographic product of two strong Dombi fuzzy graphs are not Dombi fuzzy graphs. Remark 4.2: The union of two strong Dombi fuzzy graphs need not be a strong Dombi fuzzy graph. Here is a counterexample. Example 4.2: We consider two strong Dombi fuzzy graphs G and G , where η = 1 2 1 {j/0.1, k/0.4, l/0.3, m/0.5}, ζ ={jk/0.09, kl/0.21, jl/0.08, lm/0.23}, η ={j/0.3, k/0.7, l/ 1 2 0.9, n/0.2} and ζ ={jk/0.27, nl/0.2, nj/0.14}. Then we have j k l n m η ∪ η = , , , , , 1 2 0.35 0.75 0.9 0.2 0.5 Figure 11. Strong Dombi fuzzy graph. FUZZY INFORMATION AND ENGINEERING 77 Figure 12. Not strong Dombi fuzzy graph G ∪ G . 1 2 jk kl ln nj jl ml ζ ∪ ζ = , , , , , . 1 2 0.32 0.21 0.2 0.14 0.08 0.23 It is easy to check that G ∪ G is not the strong Dombi fuzzy graph of G ∪ G (Figure 12). 1 2 1 2 Proposition 4.3: IfG × G is strong Dombi fuzzy graph, then at leastG orG must be strong. 1 2 1 2 Proof: Assume that G and G are not strong Dombi fuzzy graphs. Then there exists x y ∈ 1 2 1 1 E and x y ∈ E such that 1 2 2 2 η (x )η (y ) ab 1 1 1 1 ζ (x y )< = , 1 1 1 η (x ) + η (y ) − η (x )η (y ) a + b − ab 1 1 1 1 1 1 1 1 η (x )η (y ) cd 2 2 2 2 ζ (x y )< = . 2 2 2 η (x ) + η (y ) − η (x )η (y ) c + d − cd 2 2 2 2 2 2 2 2 Suppose that ab ζ (x y ) ≤ ζ (x y )< ≤ a. 2 2 2 1 1 1 a + b − ab Let (x , x )(y , y ) ∈ E(G × G ). Then we have 1 2 1 2 1 2 (ζ × ζ )((x , x )(y , y )) 1 2 1 2 1 2 = T (ζ (x y ), ζ (x y )) 1 1 1 2 2 2 ab cd < T , a + b − ab c + d − cd abcd (a+b−ab)(c+d−cd) ab cd abcd + − a+b−ab c+d−cd (a+b−ab)(c+d−cd) abcd (a+c−ac)(b+d−bd) ac bd abcd + − a+c−ac b+d−bd (a+c−ac)(b+d−bd) (η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 = . (η × η )((x , x )) + (η × η )((y , y )) − (η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 That is, G × G is not a strong Dombi fuzzy graph, a contradiction. 1 2 78 S. ASHRAF ET AL. Figure 13. Complete Dombi fuzzy graph. Proposition 4.4: If G + G is a strong Dombi fuzzy graph, then G and G are both strong. 1 2 1 2 Proof: Obvious. Definition 4.2: The complement of a strong Dombi fuzzy graph G = (η, ζ) of G = (V, E) is ¯ ¯ ¯ ¯ ¯ ¯ a strong Dombi fuzzy graph G = (η ¯, ζ) of G = (V, E), where η( ¯ x) = η(x) for all x ∈ V and ζ is defined as follows: η(x)η(y) if ζ(xy) = 0, η(x) + η(y) − η(x)η(y) ζ(xy) = 0if0 <ζ(xy) ≤ 1. Definition 4.3: A Dombi fuzzy graph G = (η, ζ) is said to be complete if ζ(xy) = η(x)η(y)/(η(x) + η(y) − η(x)η(y)) for all x, y ∈ V. Example 4.3: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jk kl lm mj jl km η = , , , , ζ = , , , , , . 0.3 0.6 0.2 0.8 0.25 0.18 0.19 0.28 0.14 0.52 Remark 4.3: Every complete Dombi fuzzy graph is strong (Figure 13). 5. Conclusion In this paper, the new concept of Dombi fuzzy graph is introduced. It has been proved that the direct product, join and ring sum of two Dombi fuzzy graphs are the Dombi fuzzy graphs. But in general, the Cartesian product, the semi-strong product, the strong prod- uct and the lexicographic product of two Dombi fuzzy graphs are not Dombi fuzzy graphs. However, these graphs may be Dombi fuzzy if they have crisp vertices. Moreover, these results on Dombi fuzzy graphs are considered preserving strong property. The Dombi fuzzy graph can well describe the uncertainty of all kinds of networks. Acknowledgments Thanks for the platform provided by Fuzzy Information and Engineering magazine for Memorizing Prof. Zadeh. FUZZY INFORMATION AND ENGINEERING 79 Disclosure statement No potential conflict of interest was reported by the authors. References [1] Menger K. Statistical metrics. Proc Natl Acad Sci. 1942;28(12):535–537. [2] Schweizer B, Sklar A. Probabilistic metric spaces. Amsterdam: North-Holland; 1983. [3] Alsina C, Trillas E, Valverde L. On some logical connectives for fuzzy sets theory. J Math Anal Appl. 1983;93(1):15–26. [4] Dombi J. A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst. 1982;8:149–163. [5] Hamacher H. Uber logische Verknupfungen unscharfer Aussagen und deren Augehörige Bewer- tungsfunctionen. In: Klir G, Ricciardi I, editors. Progress in cybernetics and systems research. Vol. 3. Washington, D.C.: Hemisphere; 1978. p. 276–288. [6] Dubois D, Ostasiewicz W, Prade H. Fuzzy sets: history and basic notions. In: Dubois D, Prade H, edi- tors. Handbook of fuzzy sets and possibility theory. Boston: Springer; 2000. p. 121–123. Chapter 1.2. [7] Rosenfeld A. Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M, editors. Fuzzy sets and their applica- tions. New York: Academic Press; 1975. p. 77–95. [8] Bhattacharya P. Some remarks on fuzzy graphs. Pattern Recognit Lett. 1987;6(5):297–302. [9] Mordeson JN, Peng CS. Operations on fuzzy graphs. Inf Sci (NY). 1994;79(3):159–170. [10] Bhutani KR, Battou A. On M-strong fuzzy graphs. Inf Sci. 2003;155:103–109. [11] Sunitha MS, Vijayakumar A. Complement of a fuzzy graph. Indian J Pure Appl Math. 2002;33(9):1451–1464. [12] Bhutani KR. On automorphisms of fuzzy graphs. Pattern Recognit Lett. 1989;9(3):159–162. [13] Harary F. Graph theory. 3rd ed. Reading, MA: Addison-Wesley; 1972. [14] Mordeson JN, Nair PS. Fuzzy graphs and fuzzy hypergraphs. Heidelberg: Physica-Verlag; 2000. [15] Nguyen HT, Walker EA. A first course in fuzzy logic. 3rd ed. London: CRC Press; 2006. [16] Sunil M, Mordeson JN. Fuzzy influence graphs. New Math Nat Comput. 2017;13(3):311–325. [17] Naz S, Ashraf S, Akram M. A novel approach to decision-making with Pythagorean fuzzy infor- mation. Mathematics. 2018;6:1–28. [18] Sarwar M, Akram M. An algorithm for computing certain metrics in intuitionistic fuzzy graphs. J Intell Fuzzy Syst. 2016;30(4):2405–2416. [19] Naz S, Rashmanlou H, Malik MA. Operations on single valued neutrosophic graphs with applica- tion. J Intell Fuzzy Syst. 2017;32(3):2137–2151. [20] Ashraf S, Naz S, Rashmanlou H, et al. Regularity of graphs in single valued neutrosophic environ- ment. J Intell Fuzzy Syst. 2017;33(1):529–542. [21] Klement EP, Mesiar R, Pap E. Triangular norms. Kluwer Academic: Dordrecht; 2000. (Trends in Logic; 8). [22] Zadeh LA. Fuzzy sets. Inf Control. 1965;8(3):338–353. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Dombi Fuzzy Graphs

Fuzzy Information and Engineering , Volume 10 (1): 22 – Jan 2, 2018

Dombi Fuzzy Graphs

Abstract

A fuzzy graph is useful in representing structures of relationships between items where the existence of a concrete item (vertex) and relationship between two items (edge) are matters of degree. In this paper, the novel concept of Dombi fuzzy graph is introduced. We shall use graph terminology and introduce fuzzy analogs of several basic graph-theoretical concepts using Dombi operator. Moreover, we consider these results on Dombi fuzzy graphs preserving strong property.
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© 2018 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province.
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10.1080/16168658.2018.1509520
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FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 1, 58–79 https://doi.org/10.1080/16168658.2018.1509520 a b c S. Ashraf ,S.Naz and E. E. Kerre a b Quality Enhancement Cell, Lahore College for Women University, Lahore, Pakistan; Department of Mathematics, University of the Punjab, Lahore, Pakistan; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium ABSTRACT ARTICLE HISTORY Received 3 January 2018 A fuzzy graph is useful in representing structures of relationships Revised 12 February 2018 between items where the existence of a concrete item (vertex) and Accepted 16 February 2018 relationship between two items (edge) are matters of degree. In this paper, the novel concept of Dombi fuzzy graph is introduced. We KEYWORDS shall use graph terminology and introduce fuzzy analogs of several Fuzzy set; t-norm; t-conorm; basic graph-theoretical concepts using Dombi operator. Moreover, t-operators; Dombi fuzzy we consider these results on Dombi fuzzy graphs preserving strong graph property. 1. Introduction Menger presented triangular norms (t-norms) and triangular conorms (t-conorms) in [1] in the framework of probabilistic metric spaces which were later defined and discussed by Schweizer and Sklar [2]. Alsina et al. [3]provedthat t-norms and t-conorms are stan- dard models for intersecting and unifying fuzzy sets, respectively. Since then, many other researchers have presented various types of T-operators for the same purpose [4, 5]. Zadeh’s conventional T-operators, min and max, have been used in almost every applica- tion of fuzzy logic particularly in decision-making processes and fuzzy graph theory. It is a well known fact that from theoretical and experimental aspects other T-operators may work better in some situations, especially in the context of decision-making processes. For example, the product operator may be preferred to the min operator [6]. For the selection of appropriate T-operators for a given application, one has to consider the properties they possess, their suitability to the model, their simplicity, their software and hardware imple- mentation, etc. As the study on these operators has widened, multiple options are available for selecting T-operators that may be better suited for given research. Fuzzy graphs are designed to represent structures of relationships (in the form of edges) between concrete objects (vertices) as a matter of degree. Applications of fuzzy graphs cover an extensive range such as cluster analysis, database theory, decision-making and optimization of networks. Rosenfeld [7] considered fuzzy relations on fuzzy sets and devel- oped the structure of fuzzy graphs, using max and min operations, obtaining analogs of sev- eral graph-theoretical concepts. Some remarks on fuzzy graphs were given by Bhattacharya [8]. Mordeson and Peng [9] defined some operations on fuzzy graphs and introduced the CONTACT E. E. Kerre etienne.kerre@ugent.be © 2018 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 59 concept of strong fuzzy graphs. Later on, Bhutani and Battou [10] considered operations on fuzzy graphs preserving M-strong property. The complement of a fuzzy graph was pro- posed by Mordeson and Peng [9] and then modified by Sunitha and Vijayakumar [11]. Most wide spread T-operators, max and min, have been used to introduce the structure of fuzzy graphs [7–9, 11, 12] since their inception in literature and very little effort is done to make use of new operators. The main purpose of this paper is to stress that the max and min operations are not the only candidates for the generalization of the classical graphs to fuzzy graphs. We propose to incorporate the advancement proposed by Alsina, Klement and other researchers in the area of fuzzy logic in the area of fuzzy graph theory. The paper is reserved to demonstrate the use of a particular T-operator, namely the Dombi operator in the area of fuzzy graph theory. This paper is organized as follows: In Section 2, we have provided the reader an overview of the basic definitions and notations involved in this study. In Section 3, we propose the concepts of Dombi fuzzy graphs and discuss their properties while in Section 4 particular attention to the strong Dombi fuzzy graphs will be paid. We have used standard definitions and terminologies, in this paper. For more details and background, the readers are referred to [13–20]. 2. Preliminaries Throughout this paper, V represents a crisp universe of generic elements, G stands for the crisp graph and G is the Dombi fuzzy graph. Agraph G = (V, E) is a mathematical structure consisting of a set of vertices V = V(G) and a set of edges E = E(G), where each edge is an unordered pair of distinct vertices. Two vertices x and y of a graph G are adjacent if xy ∈ E(G). A vertex joined by an edge to a ver- tex x is called a neighbor of x. The number of edges joined with a vertex x of a graph G is called the degree of x in G denoted by deg (x) or deg(x). A graph with no loops and multi- ple edges is called simple graph. Throughout this paper, we will consider only undirected, simple graphs. The complement G of a graph G, is a graph having vertex set same as in G,in which two vertices are adjacent if and only if they are not adjacent in G. If there exists a one- one correspondence between the vertices of two graphs G = (V , E ) = (V(G ), E(G )) 1 1 1 1 1 and G = (V , E ) = (V(G ), E(G )) which preserves adjacency, then the graphs G and G 2 2 2 2 2 1 2 are called isomorphic. The standard products of graphs: the direct product (tensor product), the Carte- sian product, the semi-strong product, the strong product (symmetric composition) and the lexicographic product (composition) of two graphs G = (V , E ) and G = (V , E ) 1 1 1 2 2 2 will be denoted by G × G , G  G , G • G , G  G and G [G ], respectively. Let 1 2 1 2 1 2 1 2 1 2 (x , x ), (y , y ) ∈ V × V , where V × V is the vertex set of each product graph, then 1 2 1 2 1 2 1 2 E(G × G ) ={(x , x )(y , y ) | x y ∈ E and x y ∈ E }, 1 2 1 2 1 2 1 1 1 2 2 2 E(G  G ) ={(x , x )(y , y ) | x = y and x y ∈ E ,or x y ∈ E and x = y }, 1 2 1 2 1 2 1 1 2 2 2 1 1 1 2 2 E(G • G ) ={(x , x )(y , y ) | x = y and x y ∈ E ,or x y ∈ E and x y ∈ E }, 1 2 1 2 1 2 1 1 2 2 2 1 1 1 2 2 2 E(G  G ) = E(G  G ) ∪ E(G × G ), 1 2 1 2 1 2 E(G [G ]) ={(x , x )(y , y ) | x y ∈ E ,or x = y and x y ∈ E }. 1 2 1 2 1 2 1 1 1 1 1 2 2 2 60 S. ASHRAF ET AL. Definition 2.1 ([21]): A binary operation T : [0, 1] → [0, 1] is a triangular norm ( t-norm) if for all x, y, z ∈ [0, 1] it satisfies the following: (1) T(1, x) = x. (boundary condition) (2) T(x, y) = T(y, x). (commutativity) (3) T(x, T(y, z)) = T(T(x, y), z). (associativity) (4) T(x, y) ≤ T(x, z) if y ≤ z. (monotonicity) Definition 2.2 ([21]): A binary operation S : [0, 1] → [0, 1] is a triangular conorm ( t- conorm) if and only if there exists a t-norm T such that for all (x, y) ∈ [0, 1] S(x, y) = 1 − T(1 − x,1 − y). Popular choices for t-norms are: • The minimum operator M : M(x, y) = min(x, y). • The product operator P : P(x, y) = xy. • The Lukasiewicz’s t-norm W : W(x, y) = max(x + y − 1, 0). Popular choices for corresponding dual t-conorms are: ∗ ∗ • The maximum operator M : M (x, y) = max(x, y). ∗ ∗ • The probabilistic sum P : P (x, y) = x + y − xy. ∗ ∗ • The bounded sum W : W (x, y) = min(x + y,1). The Dombi family t-norm : λ> 0, 1−x 1−y λ λ 1/λ 1 + [( ) + ( ) ] x y t-conorm : λ> 0, 1−y 1−x −λ −λ 1/−λ 1 + [( ) + ( ) ] x y negation 1 − x. The Hamacher family xy t-norm , λ + (1 − λ)(x + y − xy) x + y + (λ − 2)xy t-conorm , 1 + (λ − 1)xy negation 1 − x. Another set of T-operators is xy T(x, y) = , x + y − xy FUZZY INFORMATION AND ENGINEERING 61 x + y − 2xy S(x, y) = , 1 − xy which is obtained by taking λ = 0, in Hamacher family and λ = 1 in Dombi fam- ily of t-norms and t-conorms. Also P(x, y) ≤ xy/(x + y − xy) ≤ M(x, y) and M (x, y) ≤ (x + y − 2xy)/(1 − xy) ≤ P (x, y). Definition 2.3 ([22]): A fuzzy subset η of a set V is a function η : V → [0, 1]. A fuzzy relation [7]on V is a mapping ζ : V × V → [0, 1]. 3. Dombi Fuzzy Graphs Definition 3.1: A Dombi fuzzy graph with a finite set V as the underlying set is a pair G = (η, ζ), where η : V → [0, 1] is a fuzzy subset in V and ζ : V × V → [0, 1] is a symmetric fuzzy relation on η such that η(x)η(y) ζ(xy) ≤ for all x, y ∈ V. η(x) + η(y) − η(x)η(y) We call η the Dombi fuzzy vertex set of G and ζ the Dombi fuzzy edge set of G (Figure 1). Example 3.1: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jk kl jl km η = , , , , ζ = , , , . 0.4 0.5 0.9 0.7 0.1 0.4 0.3 0.2 Definition 3.2: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the direct product G × G = (η × η , ζ × ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η × η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 ζ (x y )ζ (x y ) 1 1 1 2 2 2 (ζ × ζ )((x , x )(y , y )) = for all x y ∈ E , 1 2 1 2 1 2 1 1 1 ζ (x y ) + ζ (x y ) − ζ (x y )ζ (x y ) 1 1 1 2 2 2 1 1 1 2 2 2 for all x y ∈ E . 2 2 2 Figure 1. Dombi fuzzy graph. 62 S. ASHRAF ET AL. Figure 2. Dombi fuzzy graph G × G . 1 2 Example 3.2: Consider two Dombi fuzzy graphs G and G , where η ={j/0.9, k/0.3}, ζ = 1 2 1 1 {jk/0.2}, η ={l/0.2, m/0.7, n/0.4} and ζ ={lm/0.1, mn/0.3}. Then we have 2 2 (ζ × ζ )((j, l)(k, m)) = 0.07, (ζ × ζ )((j, m)(k, l)) = 0.07, 1 2 1 2 (ζ × ζ )((j, m)(k, n)) = 0.14, (ζ × ζ )((j, n)(k, m)) = 0.14. 1 2 1 2 It is easy to check that G × G is the Dombi fuzzy graph of G × G (Figure 2). 1 2 1 2 Proposition 3.1: Let G and G be the Dombi fuzzy graphs of the graphs G and 1 2 1 G , respectively. The direct product G × G of G and G is the Dombi fuzzy graph of 2 1 2 1 2 G × G . 1 2 Proof: Consider x y ∈ E , x y ∈ E . Then 1 1 1 2 2 2 (ζ × ζ )((x , x )(y , y )) 1 2 1 2 1 2 = T (ζ (x y ), ζ (x y )) 1 1 1 2 2 2 η (x )η (y ) η (x )η (y ) 1 1 1 1 2 2 2 2 ≤ T , . η (x ) + η (y ) − η (x )η (y ) η (x ) + η (y ) − η (x )η (y ) 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 Putting a = η (x ), b = η (y ), c = η (x ), d = η (y ), 1 1 1 1 2 2 2 2 ab cd (ζ × ζ )((x , x )(y , y )) ≤ T , 1 2 1 2 1 2 a + b − ab c + d − cd abcd (a+b−ab)(c+d−cd) ab cd abcd + − a+b−ab c+d−cd (a+b−ab)(c+d−cd) FUZZY INFORMATION AND ENGINEERING 63 abcd (a+c−ac)(b+d−bd) ac bd abcd + − a+c−ac b+d−bd (a+c−ac)(b+d−bd) (η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 = . (η × η )((x , x )) + (η × η )((y , y ))−(η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Hence proved. Definition 3.3: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the Cartesian product G  G = (η  η , ζ  ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η  η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ  ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 η (z)ζ (x y ) 2 1 1 1 (ζ  ζ )((x , z)(y , z)) = for all z ∈ V , 1 2 1 1 2 η (z) + ζ (x y ) − η (z)ζ (x y ) 2 1 1 1 2 1 1 1 for all x y ∈ E . 1 1 1 Remark 3.1: The Cartesian product of two Dombi fuzzy graphs is not necessarily a Dombi fuzzy graph. Example 3.3: Consider two Dombi fuzzy graphs as in Example 3.2. Then we have (ζ  ζ )((j, l)(j, m)) = 0.1, (ζ  ζ )((j, m)(j, n)) = 0.29, 1 2 1 2 (ζ  ζ )((k, l)(k, m)) = 0.08, (ζ  ζ )((k, m)(k, n)) = 0.18, 1 2 1 2 (ζ  ζ )((j, l)(k, l)) = 0.11, (ζ  ζ )((j, m)(k, m)) = 0.18, 1 2 1 2 (ζ  ζ )((j, n)(k, n)) = 0.15. 1 2 Clearly, (η  η )((k, m))(η  η )((k, n)) 1 2 1 2 (ζ  ζ )((k, m)(k, n)) = 0.18  0.13 = , 1 2 (η  η )((k, m)) + (η  η )((k, n))− 1 2 1 2 (η  η )((k, m))(η  η )((k, n)) 1 2 1 2 (ζ  ζ )((j, l)(k, l)) = 0.11  0.09 1 2 (η  η )((j, l))(η  η )((k, l)) 1 2 1 2 = . (η  η )((j, l)) + (η  η )((k, l))− 1 2 1 2 (η  η )((j, l))(η  η )((k, l)) 1 2 1 2 Therefore G  G is not a Dombi fuzzy graph (Figure 3). 1 2 Definition 3.4: If a fuzzy membership degree is attached from [0, 1] to each edge of the Dombi fuzzy graph G of a graph G and each vertex is crisply in G, then G is called the Dombi fuzzy edge graph. 64 S. ASHRAF ET AL. Figure 3. Not Dombi fuzzy graph G  G . 1 2 Proposition 3.2: Let G and G be the Dombi fuzzy edge graphs of the graphs G and G , 1 2 1 2 respectively. The Cartesian product G  G of G and G is the Dombi fuzzy edge graph of 1 2 1 2 G  G . 1 2 Proof: Consider x ∈ V , x y ∈ E . Then 1 2 2 2 (ζ  ζ )((x, x )(x, y )) 1 2 2 2 = T (η (x), ζ (x y )) = T (1, ζ (x y )) 1 2 2 2 2 2 2 η (x )η (y ) 2 2 2 2 = ζ (x y ) ≤ 2 2 2 η (x ) + η (y ) − η (x )η (y ) 2 2 2 2 2 2 2 2 (η  η )((x, x ))(η  η )((x, y )) 1 2 2 1 2 2 = . (η  η )((x, x )) + (η  η )((x, y )) − (η  η )((x, x ))(η  η )((x, y )) 1 2 2 1 2 2 1 2 2 1 2 2 Consider z ∈ V , x y ∈ E . Then 2 1 1 1 (ζ  ζ )((x , z)(y , z)) 1 2 1 1 = T (ζ (x y ), η (z)) = T (ζ (x y ),1) 1 1 1 2 1 1 1 η (x )η (y ) 1 1 1 1 = ζ (x y ) ≤ 1 1 1 η (x ) + η (y ) − η (x )η (y ) 1 1 1 1 1 1 1 1 (η  η )((x , z))(η  η )((y , z)) 1 2 1 1 2 1 = . (η  η )((x , z)) + (η  η )((y , z)) − (η  η )((x , z))(η  η )((y , z)) 1 2 1 1 2 1 1 2 1 1 2 1 Hence proved. Example 3.4: Consider two Dombi fuzzy graphs G and G , where η (x) = 1 for all x ∈ V , 1 2 1 1 ζ ={jk/0.7}, η (y) = 1 for all y ∈ V ,and ζ ={lm/0.3, mn/0.9}. Then we have 1 2 2 2 (ζ  ζ )((j, l)(j, m)) = 0.3, (ζ  ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ  ζ )((k, l)(k, m)) = 0.3, (ζ  ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 FUZZY INFORMATION AND ENGINEERING 65 Figure 4. Dombi fuzzy edge graph G  G . 1 2 (ζ  ζ )((j, l)(k, l)) = 0.7, (ζ  ζ )((j, m)(k, m)) = 0.7, 1 2 1 2 (ζ  ζ )((j, n)(k, n)) = 0.7. 1 2 It is easy to check that G  G is the Dombi fuzzy edge graph of G  G (Figure 4). 1 2 1 2 Definition 3.5: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. i i i i Define the semi-strong product G • G = (η • η , ζ • ζ ) of the Dombi fuzzy graphs G = 1 2 1 2 1 2 1 (η , ζ ) and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 1 1 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η • η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ • ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 ζ (x y )ζ (x y ) 1 1 1 2 2 2 (ζ • ζ )((x , x )(y , y )) = for all x y ∈ E , 1 2 1 2 1 2 1 1 1 ζ (x y ) + ζ (x y ) − ζ (x y )ζ (x y ) 1 1 1 2 2 2 1 1 1 2 2 2 for all x y ∈ E . 2 2 2 Proposition 3.3: Let G and G be the Dombi fuzzy edge graphs of the graphs G and G , 1 2 1 2 respectively. The semi-strong product G • G of G and G is the Dombi fuzzy edge graph of 1 2 1 2 G • G . 1 2 Proof: The proof follows at once from the proof of Propositions 3.1 and 3.2. Example 3.5: Consider two Dombi fuzzy graphs G and G as in Example 3.4. Then we have 1 2 (ζ • ζ )((j, l)(j, m)) = 0.3, (ζ • ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ • ζ )((k, l)(k, m)) = 0.3, (ζ • ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 (ζ • ζ )((j, l)(k, m)) = 0.27, (ζ • ζ )((j, m)(k, l)) = 0.27, 1 2 1 2 66 S. ASHRAF ET AL. Figure 5. Dombi fuzzy edge graph G • G . 1 2 (ζ • ζ )((j, m)(k, n)) = 0.65, (ζ • ζ )((j, n)(k, m)) = 0.65. 1 2 1 2 It is easy to check that G • G is the Dombi fuzzy edge graph of G • G (Figure 5). 1 2 1 2 Definition 3.6: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the strong product G  G = (η  η , ζ  ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) of G = (V , E ) and G = (V , E ), respectively, as follows: 2 2 2 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η  η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ  ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 η (z)ζ (x y ) 2 1 1 1 (ζ  ζ )((x , z)(y , z)) = for all z ∈ V , 1 2 1 1 2 η (z) + ζ (x y ) − η (z)ζ (x y ) 2 1 1 1 2 1 1 1 for all x y ∈ E , 1 1 1 ζ (x y )ζ (x y ) 1 1 1 2 2 2 (ζ  ζ )((x , x )(y , y )) = for all x y ∈ E , 1 2 1 2 1 2 1 1 1 ζ (x y ) + ζ (x y ) − ζ (x y )ζ (x y ) 1 1 1 2 2 2 1 1 1 2 2 2 for all x y ∈ E . 2 2 2 Proposition 3.4: Let G and G be the Dombi fuzzy edge graphs of the graphs G and G , 1 2 1 2 respectively. The strong productG  G ofG andG is the Dombi fuzzy edge graph of G  G . 1 2 1 2 1 2 Proof: The proof follows at once from the proof of Propositions 3.1 and 3.2. Example 3.6: Consider two Dombi fuzzy graphs G and G as in Example 3.4. Then we have 1 2 (ζ  ζ )((j, l)(j, m)) = 0.3, (ζ  ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ  ζ )((k, l)(k, m)) = 0.3, (ζ  ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 (ζ  ζ )((j, l)(k, l)) = 0.7, (ζ  ζ )((j, m)(k, m)) = 0.7, 1 2 1 2 (ζ  ζ )((j, n)(k, n)) = 0.7, (ζ  ζ )((j, l)(k, m)) = 0.27, 1 2 1 2 FUZZY INFORMATION AND ENGINEERING 67 Figure 6. Dombi fuzzy edge graph G  G . 1 2 (ζ  ζ )((j, m)(k, l)) = 0.27, (ζ  ζ )((j, m)(k, n)) = 0.65, 1 2 1 2 (ζ  ζ )((j, n)(k, m)) = 0.65. 1 2 It is easy to check that G  G is the Dombi fuzzy edge graph of G  G (Figure 6). 1 2 1 2 Definition 3.7: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the lexicographic product G [G ] = (η ◦ η , ζ ◦ ζ ) of the Dombi fuzzy graphs G and G 1 2 1 2 1 2 1 2 of G = (V , E ) and G = (V , E ), respectively, as follows: 1 1 1 2 2 2 η (x )η (x ) 1 1 2 2 (η ◦ η )(x , x ) = for all (x , x ) ∈ V × V , 1 2 1 2 1 2 1 2 η (x ) + η (x ) − η (x )η (x ) 1 1 2 2 1 1 2 2 η (x)ζ (x y ) 1 2 2 2 (ζ ◦ ζ )((x, x )(x, y )) = for all x ∈ V , 1 2 2 2 1 η (x) + ζ (x y ) − η (x)ζ (x y ) 1 2 2 2 1 2 2 2 for all x y ∈ E , 2 2 2 η (z)ζ (x y ) 2 1 1 1 (ζ ◦ ζ )((x , z)(y , z)) = for all z ∈ V , 1 2 1 1 2 η (z) + ζ (x y ) − η (z)ζ (x y ) 2 1 1 1 2 1 1 1 for all x y ∈ E , 1 1 1 η (x )η (y )ζ (x y ) 2 2 2 2 1 1 1 (ζ ◦ ζ )((x , x )(y , y )) = 1 2 1 2 1 2 η (x )η (y ) + η (y )ζ (x y )+ 2 2 2 2 2 2 1 1 1 η (x )ζ (x y ) − 2η (x )η (y )ζ (x y ) 2 2 1 1 1 2 2 2 2 1 1 1 for all x y ∈ E , x = y . 1 1 1 2 2 Proposition 3.5: The lexicographic product G [G ] of two Dombi fuzzy edge graphs of G and 1 2 1 G is the Dombi fuzzy edge graph of G [G ]. 2 1 2 Proof: From the proof of Proposition 3.2, it follows that (ζ ◦ ζ )((x, x )(x, y )) 1 2 2 2 (η ◦ η )((x, x ))(η ◦ η )((x, y )) 1 2 2 1 2 2 (η ◦ η )((x, x )) + (η ◦ η )((x, y )) − (η ◦ η )((x, x ))(η ◦ η )((x, y )) 1 2 2 1 2 2 1 2 2 1 2 2 68 S. ASHRAF ET AL. for all x ∈ V , x y ∈ E , 1 2 2 2 (ζ ◦ ζ )((x , z)(y , z)) 1 2 1 1 (η ◦ η )((x , z))(η ◦ η )((y , z)) 1 2 1 1 2 1 (η ◦ η )((x , z)) + (η ◦ η )((y , z)) − (η ◦ η )((x , z))(η ◦ η )((y , z)) 1 2 1 1 2 1 1 2 1 1 2 1 for all z ∈ V , x y ∈ E . 2 1 1 1 Now consider x y ∈ E , x = y . Then 1 1 1 2 2 (ζ ◦ ζ )((x , x )(y , y )) 1 2 1 2 1 2 = T (T (η (x ), η (y )) , ζ (x y )) = T (T (1, 1) , ζ (x y )) 2 2 2 2 1 1 1 1 1 1 η (x )η (y ) 1 1 1 1 = T 1, ζ (x y ) = ζ (x y ) ≤ ( ) 1 1 1 1 1 1 η (x ) + η (y ) − η (x )η (y ) 1 1 1 1 1 1 1 1 (η ◦ η )((x , x ))(η ◦ η )((y , y )) 1 2 1 2 1 2 1 2 = . (η ◦ η )((x , x )) + (η ◦ η )((y , y )) − (η ◦ η )((x , x ))(η ◦ η )((y , y )) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Hence proved. Example 3.7: Consider two Dombi fuzzy graphs G and G as in Example 3.4. Then we have 1 2 (ζ ◦ ζ )((j, l)(j, m)) = 0.3, (ζ ◦ ζ )((j, m)(j, n)) = 0.9, 1 2 1 2 (ζ ◦ ζ )((k, l)(k, m)) = 0.3, (ζ ◦ ζ )((k, m)(k, n)) = 0.9, 1 2 1 2 (ζ ◦ ζ )((j, l)(k, l)) = 0.7, (ζ ◦ ζ )((j, m)(k, m)) = 0.7, 1 2 1 2 (ζ ◦ ζ )((j, n)(k, n)) = 0.7, (ζ ◦ ζ )((j, l)(k, m)) = 0.27, 1 2 1 2 (ζ ◦ ζ )((j, m)(k, l)) = 0.27, (ζ ◦ ζ )((j, m)(k, n)) = 0.65, 1 2 1 2 (ζ ◦ ζ )((j, n)(k, m)) = 0.65, (ζ ◦ ζ )((j, l)(k, n)) = 0.7, 1 2 1 2 (ζ ◦ ζ )((j, n)(k, l)) = 0.7. 1 2 It is easy to see that G [G ] is the Dombi fuzzy edge graph of G [G ] (Figure 7). 1 2 1 2 Remark 3.2: In general the semi-strong product, strong product and lexicographic prod- uct of two Dombi fuzzy graphs are not Dombi fuzzy graphs. Figure 7. Dombi fuzzy edge graph G [G ]. 1 2 FUZZY INFORMATION AND ENGINEERING 69 Example 3.8: Consider two Dombi fuzzy graphs as in Example 3.2. Then we have (ζ • ζ )((j, l)(j, m)) = 0.1, (ζ • ζ )((j, m)(j, n)) = 0.29, 1 2 1 2 (ζ • ζ )((k, l)(k, m)) = 0.08, (ζ • ζ )((k, m)(k, n)) = 0.18, 1 2 1 2 (ζ • ζ )((j, l)(k, m)) = 0.07, (ζ • ζ )((j, m)(k, l)) = 0.07, 1 2 1 2 (ζ • ζ )((j, m)(k, n)) = 0.14, (ζ • ζ )((j, n)(k, m)) = 0.14. 1 2 1 2 Clearly, (ζ • ζ )((k, m)(k, n)) = 0.18  0.13 = T((η • η )((k, m)), (η • η )((k, n))). 1 2 1 2 1 2 Therefore, G • G is not a Dombi fuzzy graph. Analogously, we can show that G  G and 1 2 1 2 G [G ] are not Dombi fuzzy graphs (Figure 8). 1 2 Definition 3.8: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define i i i i the union G ∪ G = (η ∪ η , ζ ∪ ζ ) of the Dombi fuzzy graphs G = (η , ζ ) and G = 1 2 1 2 1 2 1 1 1 2 (η , ζ ) as follows: 2 2 η (x) if x ∈ V \ V , ⎪ 1 1 2 η (x) if x ∈ V \ V , 2 2 1 (η ∪ η )(x) = 1 2 η (x) + η (x) − 2η (x)η (x) ⎪ 1 2 1 2 ⎩ if x ∈ V ∩ V . 1 2 1 − η (x)η (x) 1 2 ζ (xy) if xy ∈ E \ E , ⎪ 1 1 2 ζ (xy) if xy ∈ E \ E , 2 2 1 (ζ ∪ ζ )(xy) = 1 2 ζ (xy) + ζ (xy) − 2ζ (xy)ζ (xy) ⎪ 1 2 1 2 ⎩ if xy ∈ E ∩ E . 1 2 1 − ζ (xy)ζ (xy) 1 2 Example 3.9: We consider two Dombi fuzzy graphs G and G , where η ={j/0.5, k/0.8, 1 2 1 l/0.4}, ζ ={jk/0.4, kl/0.3, jl/0.1}, η ={j/0.6, k/0.5, m/0.3} and ζ ={jk/0.2, km/0.1} 1 2 2 (Figure 9). Then we have j k l m jk kl jl km η ∪ η = , , , , ζ ∪ ζ = , , , . 1 2 1 2 0.71 0.83 0.4 0.3 0.48 0.3 0.18 0.1 Figure 8. Not Dombi fuzzy graph G • G . 1 2 70 S. ASHRAF ET AL. Figure 9. Dombi fuzzy graph G ∪ G . 1 2 Theorem 3.1: The union G ∪ G of G and G is the Dombi fuzzy graph of G ∪ G if and only 1 2 1 2 1 2 if G and G are the Dombi fuzzy graphs of G and G , respectively, where η , η , ζ and ζ are 1 2 1 2 1 2 1 2 fuzzy subsets of V , V , E and E , respectively, and V ∩ V =∅. 1 2 1 2 1 2 Proof: Suppose that G ∪ G is the Dombi fuzzy graph. Let xy ∈ E , then xy ∈ / E and x, y ∈ 1 2 1 2 V \ V . Then 1 2 (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 ζ (xy) = (ζ ∪ ζ )(xy) ≤ 1 1 2 (η ∪ η )(x) + (η ∪ η )(y) − (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 1 2 1 2 η (x)η (y) 1 1 = . η (x) + η (y) − η (x)η (y) 1 1 1 1 G is the Dombi fuzzy graph of G . Similarly, it is easy to show that G is the Dombi Thus 1 1 2 fuzzy graph of G . Conversely, suppose that G and G are the Dombi fuzzy graphs of G and G ,respec- 1 2 1 2 tively. Consider xy ∈ E \ E . Then 1 2 (ζ ∪ ζ )(xy) = ζ (xy)(by Definition 3.8) 1 2 1 ≤ T(η (x), η (y)) (by definition of Dombi fuzzy graph) 1 1 = T((η ∪ η )(x), (η ∪ η )(y)). 1 2 1 2 Similarly, we find for xy ∈ E \ E 2 1 (ζ ∪ ζ )(xy) ≤ T((η ∪ η )(x), (η ∪ η )(y)) 1 2 1 2 1 2 (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 = . (η ∪ η )(x) + (η ∪ η )(y) − (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 1 2 1 2 Hence proved. Definition 3.9: Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. i i i i Define the ring sum G ⊕ G = (η ⊕ η , ζ ⊕ ζ ) of the Dombi fuzzy graphs G = (η , ζ ) 1 2 1 2 1 2 1 1 1 and G = (η , ζ ) as follows: 2 2 2 (η ⊕ η )(x) = (η ∪ η )(x) if x ∈ V ∪ V , 1 2 1 2 1 2 FUZZY INFORMATION AND ENGINEERING 71 ⎪ ζ (xy) if xy ∈ E \ E , 1 1 2 (ζ ⊕ ζ )(xy) = ζ (xy) if xy ∈ E \ E , 1 2 2 2 1 0if xy ∈ E ∩ E . 1 2 Proposition 3.6: The ring sum G ⊕ G of two Dombi fuzzy graphs G and G of G and G is 1 2 1 2 1 2 the Dombi fuzzy graph of G ⊕ G . 1 2 Proof: Consider xy ∈ E \ E . Then there are three possibilities, (i) x, y ∈ V \ V , (ii) x ∈ V \ 1 2 1 2 1 V , y ∈ V ∩ V and (iii) x, y ∈ V ∩ V . 2 1 2 1 2 (i) Suppose x, y ∈ V \ V 1 2 (ζ ⊕ ζ )(xy) = ζ (xy)(by Definition 3.9) 1 2 1 ≤ T(η (x), η (y)) (by definition of Dombi fuzzy graph) 1 1 = T((η ∪ η )(x), (η ∪ η )(y)) 1 2 1 2 = T((η ⊕ η )(x), (η ⊕ η )(y)). 1 2 1 2 (ii) Suppose x ∈ V \ V , y ∈ V ∩ V . Then we obtain: 1 2 1 2 (ζ ⊕ ζ )(xy) = ζ (xy) 1 2 1 ≤ T(η (x), η (y)) 1 1 = T((η ∪ η )(x), η (y)) 1 2 1 = T((η ⊕ η )(x), η (y)).(∗) 1 2 1 Clearly, η (y) + η (y) − 2η (y)η (y) 1 2 1 2 η (y) ≤ . 1 − η (y)η (y) 1 2 As putting η (y) = f, η (y) = h.Weget 1 2 f + h − 2fh f ≤ , 1 − fh f − f h ≤ f + h − 2fh, −f ≤ 1 − 2f, 0 ≤ f − 2f + 1, 0 ≤ (f − 1) . Since T is increasing, so from (∗)weget: (ζ ⊕ ζ )(xy) ≤ T((η ⊕ η )(x), (η ⊕ η )(y)). 1 2 1 2 1 2 (iii) Suppose x ∈ V ∩ V , y ∈ V ∩ V . Then we get: 1 2 1 2 (ζ ⊕ ζ )(xy) = ζ (xy) 1 2 1 ≤ T(η (x), η (y)) 1 1 72 S. ASHRAF ET AL. ≤ T((η ∪ η )(x), (η ∪ η )(y)) 1 2 1 2 = T((η ⊕ η )(x), (η ⊕ η )(y)), 1 2 1 2 since as in (ii): η (x) + η (x) − 2η (x)η (x) η (y) + η (y) − 2η (y)η (y) 1 2 1 2 1 2 1 2 η (x) ≤ and η (y) ≤ . 1 1 1 − η (y)η (y) 1 − η (y)η (y) 1 2 1 2 Similarly, because of symmetry, we find for xy ∈ E \ E in the three possible cases: 2 1 (ζ ⊕ ζ )(xy) ≤ T((η ⊕ η )(x), (η ⊕ η )(y)) 1 2 1 2 1 2 (η ⊕ η )(x)(η ⊕ η )(y) 1 2 1 2 = . (η ⊕ η )(x) + (η ⊕ η )(y) − (η ⊕ η )(x)(η ⊕ η )(y) 1 2 1 2 1 2 1 2 Hence proved. Definition 3.10: Consider the join G + G = (V ∪ V , E ∪ E ∪ E ) of two graphs G = 1 2 1 2 1 2 1 (V , E ) and G = (V , E ), where E is the set of all edges joining the vertices of V and V , 1 1 2 2 2 1 2 V ∩ V =∅.Let η be a fuzzy subset of V and let ζ be a fuzzy subset of E , i = 1, 2. Define the 1 2 i i i i join G + G = (η + η , ζ + ζ ) of the Dombi fuzzy graphs G = (η , ζ ) and G = (η , ζ ) 1 2 1 2 1 2 1 1 1 2 2 2 as follows: (η + η )(x) = (η ∪ η )(x),if x ∈ V ∪ V , 1 2 1 2 1 2 (ζ + ζ )(xy) = (ζ ∪ ζ )(xy),if xy ∈ E ∪ E , 1 2 1 2 1 2 η (x)η (y) 1 2 (ζ + ζ )(xy) = ,if xy ∈ E . 1 2 η (x) + η (y) − η (x)η (y) 1 2 1 2 Theorem 3.2: The join G + G of G and G is the Dombi fuzzy graph of G + G if and only 1 2 1 2 1 2 if G and G are Dombi fuzzy graphs of G and G , respectively, where η , η , ζ and ζ are fuzzy 1 2 1 2 1 2 1 2 subsets of V , V , E and E , respectively, and V ∩ V =∅. 1 2 1 2 1 2 Proof: Suppose thatG + G is the Dombi fuzzy graph. Then from the proof of Theorem 3.1, 1 2 G and G are Dombi fuzzy graphs. 1 2 Conversely, suppose that G and G are Dombi fuzzy graphs of G and G , respectively. 1 2 1 2 Consider xy ∈ E ∪ E . Then the required result follows from Theorem 3.1. Let xy ∈ E . Then 1 2 η (x)η (y) 1 2 (ζ + ζ )(xy) = 1 2 η (x) + η (y) − η (x)η (y) 1 2 1 2 (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 (η ∪ η )(x) + (η ∪ η )(y) − (η ∪ η )(x)(η ∪ η )(y) 1 2 1 2 1 2 1 2 (by definition 3.8 with V ∩ V =∅) 1 2 (η + η )(x)(η + η )(y) 1 2 1 2 = . (η + η )(x) + (η + η )(y) − (η + η )(x)(η + η )(y) 1 2 1 2 1 2 1 2 Hence proved.  FUZZY INFORMATION AND ENGINEERING 73 Definition 3.11: The complement of a Dombi fuzzy graph G = (η, ζ) of G = (V, E) is a ¯ ¯ ¯ Dombi fuzzy graph G = (η ¯, ζ), where η( ¯ x) = η(x) for all x ∈ V and ζ is defined as follows: η(x)η(y) ,if ζ(xy) = 0, η(x) + η(y) − η(x)η(y) ζ(xy) = η(x)η(y) − ζ(xy),if0 <ζ(xy) ≤ 1. η(x) + η(y) − η(x)η(y) Example 3.10: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jm km lm η = , , , , ζ = , , . 0.3 0.6 0.4 0.9 0.29 0.56 0.2 We have η(x) = η(x) = η(x) for all x ∈ V, and η(x)η(y) ζ(xy) = − ζ(xy) η(x) + η(y) − η(x)η(y) η(x)η(y) η(x)η(y) = − − ζ(xy) η(x) + η(y) − η(x)η(y) η(x) + η(y) − η(x)η(y) = ζ(xy) for all x, y ∈ V. Hence G = G (Figure 10). Definition 3.12: A homomorphism ψ : G → G of two Dombi fuzzy graphs G = (η , ζ ) 1 2 1 1 1 and G = (η , ζ ) is a mapping ψ : V → V satisfying the following conditions: 2 2 2 1 2 (a) η (x) ≤ η (ψ (x)) for all x ∈ V , 1 2 1 (b) ζ (xy) ≤ ζ (ψ (x)ψ (y)) for all x, y ∈ V . 1 2 1 Definition 3.13: An isomorphism ψ : G → G of two Dombi fuzzy graphs G = (η , ζ ) 1 2 1 1 1 and G = (η , ζ ) (denoted as G G ) is a bijective mapping ψ : V → V satisfying the 2 2 2 1 2 1 2 following conditions: Figure 10. Dombi fuzzy graph G. 74 S. ASHRAF ET AL. (c) η (x) = η (ψ (x)) for all x ∈ V , 1 2 1 (d) ζ (xy) = ζ (ψ (x)ψ (y)) for all x, y ∈ V . 1 2 1 A weak isomorphism ψ : G → G is a bijective homomorphism with the condition (c) 1 2 above and a co-weak isomorphism ψ : G → G is a bijective homomorphism with the 1 2 condition (d) above. Definition 3.14: A Dombi fuzzy graph G = (η, ζ) is said to be self-complementary if G = ∼ ¯ ¯ (η, ζ) G = (η ¯, ζ). Proposition 3.7: Let G = (η, ζ) be a self-complementary Dombi fuzzy graph, then 1 η(x)η(y) ζ(xy) = . 2 η(x) + η(y) − η(x)η(y) x=y x=y Proof: Let G be a self-complementary Dombi fuzzy graph. Then there exists an isomor- phism ψ : V → V such that η( ¯ ψ (x)) = η(x) for all x ∈ V and ζ(ψ(x)ψ (y)) = ζ(xy) for all xy ∈ E. By definition of G,wehave η( ¯ ψ (x))η( ¯ ψ (y)) ζ(ψ(x)ψ (y)) = − ζ(ψ(x)ψ (y)) η( ¯ ψ (x)) +¯ η(ψ (y)) −¯ η(ψ (x))η( ¯ ψ (y)) η(x)η(y) ζ(xy) = − ζ(ψ(x)ψ (y)) η(x) + η(y) − η(x)η(y) η(x)η(y) ζ(xy) + ζ(ψ(x)ψ (y)) = η(x) + η(y) − η(x)η(y) x=y x=y x=y η(x)η(y) 2 ζ(xy) = η(x) + η(y) − η(x)η(y) x=y x=y 1 η(x)η(y) ζ(xy) = . 2 η(x) + η(y) − η(x)η(y) x=y x=y Hence proved. Proposition 3.8: Let G = (η, ζ) be the Dombi fuzzy graph of G. If ζ(xy) = (η(x)η(y)/ (η(x) + η(y) − η(x)η(y))) for all x, y ∈ V, then G is self-complementary. Proof: Let G be the Dombi fuzzy graph satisfying ζ(xy) = (η(x)η(y)/(η(x) + η(y) − η(x) η(y))) for all x, y ∈ V. Then the identity mapping I : V → V is an isomorphism from G to G. Clearly, I satisfies the condition (c) of Definition 3.13. Since ζ(xy) = (η(x)η(y)/(η(x) + η(y) − η(x)η(y))) for all x, y ∈ V,wehave ¯ ¯ ζ(I(x)I(y)) = ζ(xy) η(x)η(y) = − ζ(xy) η(x) + η(y) − η(x)η(y) FUZZY INFORMATION AND ENGINEERING 75 η(x)η(y) 1 η(x)η(y) = − η(x) + η(y) − η(x)η(y) 2 η(x) + η(y) − η(x)η(y) 1 η(x)η(y) 2 η(x) + η(y) − η(x)η(y) = ζ(xy). Thus the condition (d) of Definition 3.13 is also satisfied by I. Therefore G is self- complementary. Proposition 3.9: The complements of two isomorphic Dombi fuzzy graphs are isomorphic and conversely. Proof: Suppose G and G are two isomorphic Dombi fuzzy graphs. Then there exists a 1 2 bijective mapping ψ : V → V satisfying 1 2 η (x) = η (ψ (x)) for all x ∈ V , 1 2 1 ζ (xy) = ζ (ψ (x)ψ (y)) for all xy ∈ E . 1 2 1 Using the definition of complement, we have η (x)η (y) 1 1 ζ (xy) = − ζ (xy) 1 1 η (x) + η (y) − η (x)η (y) 1 1 1 1 η (ψ (x))η (ψ (y)) 2 2 = − ζ (ψ (x)ψ (y)) = ζ (ψ (x)ψ (y)). 2 2 η (ψ (x)) + η (ψ (y)) − η (ψ (x))η (ψ (y)) 2 2 2 2 Hence G = G . Similarly, we can prove the converse part. 1 2 Proposition 3.10: The complements of two weak-isomorphic Dombi fuzzy graphs are weak isomorphic. Proof: Suppose G and G are two weak-isomorphic Dombi fuzzy graphs. Then there exists 1 2 a bijective mapping ψ : V → V satisfying 1 2 η (x) = η (ψ (x)) for all x ∈ V , 1 2 1 ζ (xy) ≤ ζ (ψ (x)ψ (y)) for all xy ∈ E . 1 2 1 Consider ζ (xy) ≤ ζ (ψ (x)ψ (y)) 1 2 (xy) ≥−ζ (ψ (x)ψ (y)) ⇒−ζ 1 2 ⇒ T(η (x), η (y)) − ζ (xy) ≥ T(η (x), η (y)) − ζ (ψ (x)ψ (y)) 1 1 1 1 1 2 ⇒ T(η (x), η (y)) − ζ (xy) ≥ T(η (ψ (x)), η (ψ (y))) − ζ (ψ (x)ψ (y)) 1 1 1 2 2 2 ⇒ ζ (xy) ≥ ζ (ψ (x)ψ (y)). (by definition of complement) 1 2 Hence G and G are weak isomorphic. 1 2 76 S. ASHRAF ET AL. We state the following proposition without proof. Proposition 3.11: The complements of two co-weak-isomorphic Dombi fuzzy graphs are homomorphic. 4. Strong Dombi Fuzzy Graphs Definition 4.1: A Dombi fuzzy graph G = (η, ζ) is called a strong Dombi fuzzy graph of G = (V, E) if ζ(xy) = η(x)η(y)/(η(x) + η(y) − η(x)η(y)) for all xy ∈ E. Example 4.1: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jk jm ml η = , , , , ζ = , , . 0.5 0.7 0.8 0.3 0.41 0.23 0.28 Proposition 4.1: If G and G are strong Dombi fuzzy graphs, then G × G and G + G are 1 2 1 2 1 2 also strong Dombi fuzzy graphs (Figure 11). Proof: The proof follows at once from the proof of Proposition 3.1 and Theorem 3.2. Proposition 4.2: If G and G are strong Dombi fuzzy edge graphs, then G  G , G • G , 1 2 1 2 1 2 G  G and G [G ] are also strong Dombi fuzzy edge graphs. 1 2 1 2 Proof: The proof follows at once from the proof of Propositions 3.2–3.5. Remark 4.1: In general the Cartesian product, semi-strong product, strong product and lexicographic product of two strong Dombi fuzzy graphs are not Dombi fuzzy graphs. Remark 4.2: The union of two strong Dombi fuzzy graphs need not be a strong Dombi fuzzy graph. Here is a counterexample. Example 4.2: We consider two strong Dombi fuzzy graphs G and G , where η = 1 2 1 {j/0.1, k/0.4, l/0.3, m/0.5}, ζ ={jk/0.09, kl/0.21, jl/0.08, lm/0.23}, η ={j/0.3, k/0.7, l/ 1 2 0.9, n/0.2} and ζ ={jk/0.27, nl/0.2, nj/0.14}. Then we have j k l n m η ∪ η = , , , , , 1 2 0.35 0.75 0.9 0.2 0.5 Figure 11. Strong Dombi fuzzy graph. FUZZY INFORMATION AND ENGINEERING 77 Figure 12. Not strong Dombi fuzzy graph G ∪ G . 1 2 jk kl ln nj jl ml ζ ∪ ζ = , , , , , . 1 2 0.32 0.21 0.2 0.14 0.08 0.23 It is easy to check that G ∪ G is not the strong Dombi fuzzy graph of G ∪ G (Figure 12). 1 2 1 2 Proposition 4.3: IfG × G is strong Dombi fuzzy graph, then at leastG orG must be strong. 1 2 1 2 Proof: Assume that G and G are not strong Dombi fuzzy graphs. Then there exists x y ∈ 1 2 1 1 E and x y ∈ E such that 1 2 2 2 η (x )η (y ) ab 1 1 1 1 ζ (x y )< = , 1 1 1 η (x ) + η (y ) − η (x )η (y ) a + b − ab 1 1 1 1 1 1 1 1 η (x )η (y ) cd 2 2 2 2 ζ (x y )< = . 2 2 2 η (x ) + η (y ) − η (x )η (y ) c + d − cd 2 2 2 2 2 2 2 2 Suppose that ab ζ (x y ) ≤ ζ (x y )< ≤ a. 2 2 2 1 1 1 a + b − ab Let (x , x )(y , y ) ∈ E(G × G ). Then we have 1 2 1 2 1 2 (ζ × ζ )((x , x )(y , y )) 1 2 1 2 1 2 = T (ζ (x y ), ζ (x y )) 1 1 1 2 2 2 ab cd < T , a + b − ab c + d − cd abcd (a+b−ab)(c+d−cd) ab cd abcd + − a+b−ab c+d−cd (a+b−ab)(c+d−cd) abcd (a+c−ac)(b+d−bd) ac bd abcd + − a+c−ac b+d−bd (a+c−ac)(b+d−bd) (η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 = . (η × η )((x , x )) + (η × η )((y , y )) − (η × η )((x , x ))(η × η )((y , y )) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 That is, G × G is not a strong Dombi fuzzy graph, a contradiction. 1 2 78 S. ASHRAF ET AL. Figure 13. Complete Dombi fuzzy graph. Proposition 4.4: If G + G is a strong Dombi fuzzy graph, then G and G are both strong. 1 2 1 2 Proof: Obvious. Definition 4.2: The complement of a strong Dombi fuzzy graph G = (η, ζ) of G = (V, E) is ¯ ¯ ¯ ¯ ¯ ¯ a strong Dombi fuzzy graph G = (η ¯, ζ) of G = (V, E), where η( ¯ x) = η(x) for all x ∈ V and ζ is defined as follows: η(x)η(y) if ζ(xy) = 0, η(x) + η(y) − η(x)η(y) ζ(xy) = 0if0 <ζ(xy) ≤ 1. Definition 4.3: A Dombi fuzzy graph G = (η, ζ) is said to be complete if ζ(xy) = η(x)η(y)/(η(x) + η(y) − η(x)η(y)) for all x, y ∈ V. Example 4.3: Consider a Dombi fuzzy graph over V ={j, k, l, m} defined by j k l m jk kl lm mj jl km η = , , , , ζ = , , , , , . 0.3 0.6 0.2 0.8 0.25 0.18 0.19 0.28 0.14 0.52 Remark 4.3: Every complete Dombi fuzzy graph is strong (Figure 13). 5. Conclusion In this paper, the new concept of Dombi fuzzy graph is introduced. It has been proved that the direct product, join and ring sum of two Dombi fuzzy graphs are the Dombi fuzzy graphs. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Jan 2, 2018

Keywords: Fuzzy set; t -norm; t -conorm; t -operators; Dombi fuzzy graph

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