Some operations on vague graphs 1 Introduction

Journal of Advanced Research in Pure Mathematics Online ISSN: 1943-2380

Vol. 6, Issue. 1, 2014, pp. 61-77 doi: 10.5373/jarpm.1562.092512

Some operations on vague graphs Ali Asghar Talebi∗ , Narges Mehdipoor, Hossein Rashmanlou Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

Abstract. The concept of vague sets is due to Gau and Buehrer who studied the concept with the aim of interpreting the real life problems in better way than the existing mechanisms such as fuzzy sets. If G = (σ, µ) be a fuzzy graph then Gµ = (σ, µµ ) is its µ-complement. In this paper we define strong product, categorical product and µ-complement on vague graphs and investigate some properties of µ- complement. We also study some definition of self µ-complementary and self µ-weak complementary on vague graphs. Keywords: Vague graph; Strong product; Categorical product; µ-complement; Self µ-complement; Self weak µ-complement. Mathematics Subject Classification 2010: 03E72.

1

Introduction

In 1965, Zadeh [24] introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. Since then, the theory of fuzzy sets has became a vigorous area of research in different disciplines including logic, topology, algebra, analysis, information theory, artificial intelligence, operations research, neural networks and planning etc [8], [12], [13], [14], [16]. The fuzzy graph theory as a generalization of Euler , s graph theory was first introduced by Rosenfeld [21] 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 [6] gave some remarks on fuzzy graphs, and some operation of fuzzy graphs were introduced by Mordeson and Peng [17]. The complement of a fuzzy graph was defined by Mordeson and Nair [16] and further studied by Sunitha and VijayaKumar [22]. The concept of

∗

Correspondence to: Ali Asghar Talebi, Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. Email: [email protected] † Received: 25 September 2012, revised: 9 April 2013, accepted: 20 August 2013. http://www.i-asr.com/Journals/jarpm/

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c

2014 Institute of Advanced Scientific Research

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Some operations on vague graphs

weak isomorphism, co-weak isomorphism and isomorphism between fuzzy graphs was introduced by Bhutani in [8]. Nagoorgani and Chandrasekaran [18], defined µ-complement of a fuzzy graph, which is slightly different from the definition of complement of a fuzzy graph discussed by Sunitha and Vijayakumar in [22]. The association of µ-complement with morphism properties is studied in [19] by Nagoorgani and Malarvizhi. In 2011 Akram and Dudek [3] defined interval-valued fuzzy graphs and give some operation on it. Akram and Davvaz discussed the properties of strong intuitionistic fuzzy graphs and also they introduced the concept of intuitionistic fuzzy line graphs in [1]. In [2], Akram introduced the concept of bipolar fuzzy graphs and studied some properties on it. Hedayati and Jafari, studied Connections between interval valued fuzzy graphs and fuzzy groups in [10]. Recently Akram, Feng, Sarwar and Jun studied certain types of vague graphs in [4].

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Preliminaries

A vague set A on a set X is a pair (tA , fA ) where tA and fA are real valued functions defined on X → [0, 1], such that tA (x) + fA (x) ≤ 1 for all x ∈ X. The interval [tA (x), 1 − fA (x)] is called the vague value of x in A and it is denoted by VA (x). Earlier Atanassov [5] studied Intuitionistic fuzzy sets which are mathematically equivalent to Vague sets. We prefer the terminology of vague sets in line with Gau and Buehrer [9]. Isomorphism on vague graphs studied in [23] by Talebi. A vague graph is a pair of functions G = (σ, µ) where σ is a vague set of a set of nodes N and µ is a vague relation on σ. In this paper, we introduce strong product, categorical product and µ-complement on vague graphs and discusses some properties of µcomplement. A fuzzy graph with S, a non empty finite set as the underlying set is a pair G = (σ, µ) where σ : S → [0, 1] is a fuzzy subset of S, µ : S × S → [0, 1] is a symmetric fuzzy relation on the fuzzy subset σ such that µ(x, y) ≤ \min(σ(x), σ(y)), ∀ x, y ∈ S. A fuzzy relation µ is symmetric if µ(x, y) = µ(y, x) for all x, y ∈ S. The underlying crisp graph of the fuzzy graph G = (σ, µ) is denoted as G∗ = (σ ∗ , µ∗ ) where σ ∗ = {x ∈ S | σ(x) > 0} and µ∗ = {(x, y) ∈ S × S | µ(x, y) > 0}. If µ(x, y) > 0, then x and y are called neighbors. For simplicity, an edge (x, y) will be denoted by xy. We give here a review of some definitions which are in [9], [20], [11], [7], [22], [19], [15] and [23]. Definition 2.1. Given a fuzzy graph GX = (σ, µ), with the underlying set S, the order of G is defined and denoted as O(G) = σ(x) and size of G is defined and denoted as x∈S

S(G) =

X

x,y∈S

µ(xy). The degree of a node u is defined as d(u) =

X

µ(xy).

y6=x∈S

Definition 2.2. A fuzzy graph G is said to be a complete fuzzy graph if µ(x, y) = σ(x) ∧ σ(y), ∀ x, y ∈ σ ∗ , it is denoted as Kσ = (σ, µ). Definition 2.3. A homomorphism of fuzzy graphs h : G1 → G2 is a map h : V1 → V2 which satisfies σ1 (x) ≤ σ2 (h(x)), ∀ x ∈ V1 and µ1 (xy) ≤ µ2 ((h(x)h(y)), ∀ x, y ∈

Ali Asghar Talebi, Narges Mehdipoor and Hossein Rashmanlou

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V1 . A weak isomorphism h : G1 → G2 is a map h : V1 → V2 which is a bijective homomorphism that satisfies σ1 (x) = σ2 (h(x)), ∀ x ∈ V1 . A co-weak isomorphism h : G1 → G2 is a map h : V1 → V2 which is a bijective homomorphism that satisfies µ1 (xy) = µ2 ((h(x)h(y))∀ x, y ∈ V1 . An isomorphism h : G1 → G2 is a map h : V1 → V2 which is bijective and satisfies σ1 (x) = σ2 (h(x))∀ x ∈ V1 and µ1 (xy) = µ2 ((h(x)h(y)), ∀ x, y ∈ V1 , and we denote G1 ∼ = G2 . Definition 2.4. A vague set A on a set X is a pair (tA , fA ) where tA : X → [0, 1], fA : X → [0, 1], with tA (x) + fA (x) ≤ 1 for all x ∈ X. The mapping tA : X → [0, 1], defines the degree of membership function and the mapping fA : X → [0, 1] defines the degree of non-membership function of the element x ∈ X, the functions tA and fA should satisfy the condition tA ≤ 1 − fA . The interval [tA (x), 1 − fA ] is called the vague value of x ∈ X, and it is denoted by VA (x). i.e., VA (x) = [tA (x), 1 − fA ]. A vague set A of set X with tA (x) = 0 and fA (x) = 1, ∀x ∈ X is called zero vague set of X. A vague set A of a set X with tA (x) = 1 and fA (x) = 0, ∀x ∈ X is called the unit vague set of X. A vague set A of a set X with tA (x) = α and fA (x) = 1 − α, ∀x ∈ X is called the α−vague set of X, where α ∈ [0, 1]. A vague set A is contained in a vague set B, A ⊆ B if and only if tA ≤ tB and fA ≥ fB . Let X and Y be any non-empty sets, a vague relation µ from X to Y is defined as a vague set of X × Y , it is denoted by µ = (tµ , fµ ) where tµ : X × Y → [0, 1], fµ : X × Y → [0, 1] which satisfies the condition tµ (x, y) + fµ (x, y) ≤ 1, ∀(x, y) ∈ X × Y . A vague relation µ on a set N is a vague relation from N to N . If σ is a vague set on a set N , then a vague relation µ on σ is a vague relation on N , which satisfies tµ (x, y) ≤ \min{tσ (x), tσ (y)}, fµ (x, y) ≥ \max{fσ (x), fσ (y)}, for all x, y ∈ N . Let N be a non-empty set, members of N are called nodes. A vague graph G = (σ, µ) with N as the set of nodes, is a pair functions (σ, µ), where σ is a vague set of N and µ is a vague relation on σ. Definition 2.5. A vague graph is complete if   Vµ (xy) = imin{Vσ (x), Vσ (y)} t (xy) = \min{tσ (x), tσ (y)}  µ fµ (xy) = \max{fσ (x), fσ (y)}.

∀x, y ∈ σ ∗

Definition 2.6. Let G1 = (σ1 , µ1 ) and G2 = (σ2 , µ2 ) be two vague graphs. A homomorphism f : G1 → G2 is a mapping f : N1 → N2 such that:

(a)Vσ1 (x1 ) ≤ Vσ2 (f (x1 )), i.e., tσ1 (x1 ) ≤ tσ2 (f (x1 )), fσ1 (x1 ) ≥ fσ2 (f (x1 )). (b)Vµ1 (x1 y1 ) ≤ Vµ2 (f (x1 )f (y1 )), i.e., tµ1 (x1 y1 ) ≤ tµ2 (f (x1 )f (y1 )), fµ1 (x1 y1 ) ≥ fµ2 (f (x1 )f (y1 )). A bijective homomorphism with the property (c)Vσ (x1 ) = Vσ2 (f (x1 )), i.e., tσ (x1 ) = tσ2 (f (x1 )), fσ1 (x1 ) = fσ2 (f (x1 ))

∀x1 , y1 ∈ N1

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is called a weak isomorphism. 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)Vµ1 (x1 y1 ) = Vµ2 (f (x1 )f (y1 )), i.e., tµ1 (x1 y1 ) = tµ2 (f (x1 )f (y1 )), fµ1 (x1 y1 ) = fµ2 (f (x1 )f (y1 )).

∀x1 , y1 ∈ N1

is called a weak co-isomorphism. A bijective mapping f : G1 → G2 satisfying (c) and (d) is called an isomorphism. Definition 2.7. A vague graph G is said to be self weak complementary vague graph if G is weak isomorphic with it’s complement G, where Vσ (x) = Vσ (f (x)), for all x ∈ N , and Vµ (xy) ≤ Vµ (f (x)f (y)), ∀x, y ∈ N , i.e. tσ (x) = tσ (f (x)) fσ (x) = fσ (f (x)) and   tµ (xy) ≤ tµ (f (x)f (y) 

∀x, y ∈ N

fµ (xy) ≥ fµ (f (x)f (y)).

Definition 2.8. Let G = (σ, µ) be a fuzzy graph. The µ-complement of G is defined as Gµ = (σ, µµ ), where µ µ = σ(x) ∧ σ(y) − µ(x, y) if µ(x, y) > 0 0 if µ(x, y) = 0. Definition 2.9. Let G1 = (V1 , σ1 , µ1 ), G2 = (V2 , σ2 , µ2 ) be two fuzzy graphs on V1 , V2 , respectively. Then the categorical product of G1 and G2 denoted by G1 × G2 is the fuzzy graph on V1 × V2 defined as follows: G1 × G2 = (σ1 × σ2 , µ1 × µ2 ) where σ1 × σ2 (u1 , u2 ) = σ1 (u1 ) ∧ σ2 (u2 ) and µ1 × µ2 ((u1 , u2 )(v1 , v2 )) =

µ1 (u1 v1 ) ∧ µ2 (u2 v2 ) 0

if u1 6= v1 , u2 6= v2 otherwise.

Definition 2.10. Let G1 = (V1 , σ1 , µ1 ), G2 = (V2 , σ2 , µ2 ) be two fuzzy graphs on V1 , V2 respectively. Then the strong product of G1 and G2 , denoted by (G1 timesG2 ), is the fuzzy graph on V1 × V2 defined as follows:


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G1 timesG2 = (σ1 timesσ2 , µ1 timesµ2 ) where σ1 timesσ2 (u1 , u2 ) = σ1 (u1 ) ∧ σ2 (u2 ) and

  σ1 (u1 ) ∧ µ2 (u2 v2 ) σ2 (u2 ) ∧ µ1 (u1 v1 ) µ1 timesµ2 ((u1 , u2 )(v1 , v2 )) =  µ2 (u2 v2 ) ∧ µ1 (u1 v1 )

3

if u1 = v1 , u2 6= v2 if u2 = v2 , u1 = 6 v1 otherwise.

Vague graphs and operations on vague graphs

In this section, we define strong product and categorical product on vague graphs, and investigate some of their properties. N. Ramakrishna introduced the following definition [20]. Definition 3.1. Let N be a non-empty set, members of N are called nodes. A vague graph G = (σ, µ) with N as the set of nodes, is a pair functions (σ, µ), where σ is a vague set of N and µ is a vague relation on σ. Definition 3.2. A vague graph G = (σ, µ) is a strong vague graph if Vµ (uv) = imin{Vσ (u), Vσ (v)}, ∀uv ∈ µ∗ , i.e., tµ (uv) = \min{tσ (u), tσ (v)}, fµ (uv) = \max{fσ (u), fσ (v)},

∀uv ∈ µ∗ .

Remark 3.3. Let G = (σ, µ) be a vague graph. We denote the underlying (crisp) graph of G = (σ, µ) by G∗ = (σ ∗ , µ∗ ) where σ ∗ is referred to as the non-empty set N of nodes and µ∗ = E ⊆ N × N . i.e., σ ∗ = {u ∈ N | Vσ (u) > 0}, i.e., u ∈ σ ∗ ⇔ Vσ (u) > 0 µ∗ = {uv ∈ N × N | Vµ (uv) > 0}, i.e., uv ∈ µ∗ ⇔ Vµ (uv) > 0. Definition 3.4. Let G1 = (N1 , σ1 , µ1 ), G2 = (N2 , σ2 , µ2 ) be two vague graphs with G∗1 = (σ1∗ , µ∗1 ) and G∗2 = (σ2∗ , µ∗2 ). Let G∗ = G∗1 × G∗2 = (N, E) be the categorical product of two graphs where N = N1 × N2 and E = {(u1 , u2 )(v1 , v2 ) | u1 6= v1 , u2 6= v2 )}. The categorical product of two vague graphs G = G1 × G2 = (σ1 × σ2 , µ1 × µ2 ) is a vague graph defined by

Vσ1 ×σ2 (u1 , u2 ) = imin{Vσ1 (u1 ), Vσ2 (u2 )}. tσ1 ×σ2 (u1 , u2 ) = \min{tσ1 (u1 ), tσ2 (u2 )}. fσ1 ×σ2 (u1 , u2 ) = \max{fσ1 (u1 ), fσ2 (u2 )}. For all u1 v1 ∈ E1 and u2 v2 ∈ E2 ,

∀(u1 , u2 ) ∈ N ∀(u1 , u2 ) ∈ N ∀(u1 , u2 ) ∈ N

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Vµ1 ×µ2 ((u1 , u2 )(v1 , v2 )) =

imin{Vµ1 (u1 v1 ), Vµ2 (u2 v2 )} 0

tµ1 ×µ2 ((u1 , u2 )(v1 , v2 )) =

\min{tµ1 (u1 v1 ), tµ2 (u2 v2 )} 0

fµ1 ×µ2 ((u1 , u2 )(v1 , v2 )) =

\max{fµ1 (u1 v1 ), fµ2 (u2 v2 )} 0

u1 6= v1 , u2 6= v2 otherwise. u1 6= v1 , u2 6= v2 otherwise. u1 6= v1 , u2 6= v2 otherwise.

Example 3.1. Suppose that N1 = {v1 , v2 , v3 } be a set of nodes, then G1 = (σ1 , µ1 ) is a vague graph by σ1 = {(v1 , 0.1, 0.2), (v2 , 0.4, 0.5), (v3 , 0.25, 0.5)}. and µ1 = {(v1 v2 , 0.1, 0.5), (v2 v3 , 0.2, 0.6)}. Let N2 = {u1 , u2 , u3 } be a set of nodes, then G1 = (σ1 , µ1 ) is a vague graph by σ2 = {(u1 , 0.3, 0.5), (u2 , 0.1, 0.7), (u3 , 0.25, 0.65)}, and vague relation on N2 is given by µ2 = {(u1 u2 , 0.1, 0.8), (u2 u3 , 0.1, 0.7)}. The categorical product of them is G = G1 × G2 , that σ = σ1 × σ2 = {((u1 , v1 ), 0.1, 0.5), ((u1 , v2 ), 0.3, 0.6), ((u1 , v3 ), 0.25, 0.65), ((u2 , v1 ), 0.1, 0.8), ((u2 , v2 ), 0.1, 0.7), ((u2 , v3 ), 0.1, 0.7), ((u3 , v1 ), 0.1, 0.65), ((u3 , v2 ), 0.2, 0.7), ((u3 , v3 ), 0.25, 0.65)}. µ = {(u1 , v1 )(u2 , v2 ), 0.1, 0.8), (u3 , v1 )(u2 , v2 ), 0.1, 0.8), (u2 , v3 )(u3 , v2 ), 0.1, 0.7), (u2 , v2 )(u3 , v3 ), 0.1, 0.75), (u1 , v2 )(u2 , v1 ), 0.2, 0.65), (u2 , v1 )(u3 , v2 ), 0.1, 0.85), (u1 , v2 )(u2 , v3 ), 0.1, 0.8), (u1 , v3 )(u2 , v2 ), 0.05, 0.8)}.


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Figure 1. Categorical product G = G1 × G2 Example 3.2. Suppose that N1 = {x1 , x2 } is a set of nodes, and G1 = (σ1 , µ1 ) is a vague graph with σ1 = {(x1 , 0.3, 0.5), (x2 , 0.1, 0.55)}. Vague relation on N1 is given by µ1 = {(x1 x2 , 0.1, 0.65)}. Let N2 = {y1 , y2 } is a set of nodes, and G2 = (σ2 , µ2 ) is a vague graph with σ2 = {(y1 , 0.2, 0.65), (y2 , 0.15, 0.27)}, and vague relation on N2 is given by µ2 = {(y1 y2 , 0.5, 0.8)}. The categorical product of them is G = G1 × G2 , that σ = σ1 × σ2 = {((x1 , y1 ), 0.1, 0.65), ((x1 , y2 ), 0.15, 0.6), ((x2 , y1 ), 0.1, 0.7), ((x2 , y2 ), 0.1, 0.6)}. µ = {(x1 , y1 )(x2 , y2 ), 0.5, 0.8), (x1 , y2 )(x2 , y1 ), 0.2, 0.85)}.

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Figure 2. Categorical product G = G1 × G2 Definition 3.5. Let G1 = (N1 , σ1 , µ1 ), G2 = (N2 , σ2 , µ2 ) be two vague graphs with G∗1 = (σ1∗ , µ∗1 ) and G∗2 = (σ2∗ , µ∗2 ). Let G∗ = G∗1 × G∗2 = (N, E) be the strong product of two graphs where N = N1 × N2 and S E S = {(u, u2 )(u, v2 ) | u ∈ N1 , u2 v2 ∈ E2 )} {(u1 , w)(v1 , w) | w ∈ N2 , u1 v1 ∈ E1 )} {(u1 , u2 )(v1 , v2 ) | u1 6= v1 , u2 6= v2 )}. The strong product of two vague graphs G = G1 timesG2 = (σ1 timesσ2 , µ1 timesµ2 ) is a vague graph defined by Vσ1 timesσ2 (u1 , u2 ) = imin{Vσ1 (u1 ), Vσ2 (u2 )}, ∀(u1 , u2 ) ∈ N , i.e., tσ1 timesσ2 (u1 , u2 ) = \min{tσ1 (u1 ), tσ2 (u2 )}, ∀(u1 , u2 ) ∈ N . fσ1 timesσ2 (u1 , u2 ) = \max{fσ1 (u1 ), fσ2 (u2 )}, ∀(u1 , u2 ) ∈ N . Vµ1 timesµ2 ((u, u2 )(u, v2 )) = imin{Vσ1 (u), Vµ2 (u2 v2 )}, ∀u ∈ N1 , u2 v2 ∈ E2 , u2 6= v2 . Vµ1 timesµ2 ((u1 , w)(v1 , w)) = imin{Vµ1 (u1 v1 ), Vσ (w)}, ∀u1 v1 ∈ E1 , w ∈ N2 , u1 6= v1 . Vµ1 timesµ2 ((u1 , u2 )(v1 , v2 )) = imin{Vµ1 (u1 v1 ), Vµ2 (u2 v2 )}, ∀u1 v1 ∈ E1 , u2 v2 ∈ E2 , u1 6= v1 , u2 6= v2 , i.e., tµ1 timesµ2 ((u, u2 )(u, v2 )) = \min{tσ1 (u), tµ2 (u2 v2 )}, ∀u ∈ N1 , u2 v2 ∈ E2 , u2 6= v2 . tµ1 timesµ2 ((u1 , w)(v1 , w)) = \min{tµ1 (u1 v1 ), tσ (w)}, ∀u1 v1 ∈ E1 , w ∈ N2 , u1 6= v1 . tµ1 timesµ2 ((u1 u2 )(v1 v2 )) = \min{tµ1 (u1 v1 ), tµ2 (u2 v2 )}, ∀u1 v1 ∈ E1 , u2 v2 ∈ E2 , u1 6= v1 , u2 6= v2 . fµ1 timesµ2 ((u, u2 )(u, v2 )) = \max{fσ1 (u), fµ2 (u2 v2 )}, ∀u ∈ N1 , u2 v2 ∈ E2 , u2 6= v2 . fµ1 timesµ2 ((u1 , w)(v1 , w)) = \max{fµ1 (u1 v1 ), fσ (w)}, ∀u1 v1 ∈ E1 , w ∈ N2 , u1 6= v1 .


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fµ1 timesµ2 ((u1 , u2 )(v1 , v2 )) = \max{fµ1 (u1 v1 ), fµ2 (u2 v2 )}, ∀u1 v1 ∈ E1 , u2 v2 ∈ E2 , u1 6= v1 , u2 6= v2 . Example 3.3. Suppose that N1 = {u1 , u2 } be a set of nodes, G1 = (σ1 , µ1 ) is a vague graph by σ1 = {(u1 , 0.2, 0.55), (u2 , 0.15, 0.75)}, and vague relation on N1 is given by µ1 = {(u1 u2 , 0.1, 0.8)}. Let N2 = {v1 , v2 } be a set of nodes G2 = (σ2 , µ2 ) is a vague graph by σ2 = {(v1 , 0.25, 0.65), (v2 , 0.3, 0.6)}. Vague relation on N2 is given by µ2 = {(v1 v2 , 0.2, 0.7)}. The strong product of them is denoted by G = G1 timesG2 . σ = σ1 × σ2 = {(u1 , v1 , 0.2, 0.65), (u1 , v2 , 0.1, 0.6), (u2 , v1 , 0.15, 0.75), (u2 , v2 , 0.1, 0.8)}. µ = {(u1 , v1 )(u2 , v2 ), 0.1, 0.8), (u1 , v2 )(u2 , v1 ), 0.2, 0.85), (u1 , v1 )(u1 , v2 ), 0.2, 0.7), (u1 , v1 )(u2 , v1 ), 0.1, 0.8), (u1 , v2 )(u2 , v2 ), 0.1, 0.8), (u2 , v1 )(u2 , v2 ), 0.15, 0.75)}.

Figure 3. Strong product G1 timesG2 Example 3.4. Suppose that N1 = {x1 , x2 } be a set of nodes G1 = (σ1 , µ1 ) is a vague graph by σ1 = {(x1 , 0.25, 0.4), (x2 , 0.1, 0.65)}. Vague relation on N1 is given by µ1 = {(x1 x2 , 0.1, 0.7)}. Let N2 = {y1 , y2 , y3 } be a set of nodes G1 = (σ1 , µ1 ) is a vague graph by σ2 = {(y1 , 0.1, 0.3), (y2 , 0.15, 0.55), (y3 , 0.2, 0.6)}.

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Vague relation on N2 is given by µ2 = {(y1 y2 , 0.15, 0.6), (y1 y3 , 0.05, 0.65)}. The strong product of them is denoted by G = G1 timesG2 . σ = σ1 × σ2 = {(x1 y1 , 0.1, 0.4), (x1 y2 , 0.15, 0.6), (x2 y1 , 0.1, 0.7), (x2 y2 , 0.1, 0.65)(x1 y3 , 0.2, 0.6), (x2 y3 , 0.1, 0.7)}. µ = {(x1 , y1 )(x1 , y2 ), 0.15, 0.6), (x1 , y3 )(x2 , y3 ), 0.1, 0.7), (x2 , y1 )(x2 , y2 ), 0.1, 0.65), (x1 , y1 )(x2 , y3 ), 0.1, 0.75), (x1 , y3 )(x1 , y1 ), 0.25, 0.65), (x1 , y1 )(x2 , y1 ), 0.1, 0.7), (x1 , y3 )(x2 , y1 ), 0.1, 0.7), (x2 , y1 )(x2 , y3 ), 0.1, 0.65), (x1 , y2 )(x2 , y2 ), 0.1, 0.7), (x1 , y1 )(x2 , y2 ), 0.05, 0.7), (x1 , y2 )(x2 , y1 ), 0.02, 0.85)}.

Figure 4. Strong product G1 timesG2 Theorem 3.6. If G1 = (σ1 , µ1 ), G2 = (σ2 , µ2 ) are strong vague graphs, then the categorical product G1 × G2 , strong product G = G1 timesG2 are also strong vague graphs. Proof. Suppose that G1 = (σ1 , µ1 ), G2 = (σ2 , µ2 ) be two strong vague graphs. First, we prove that G1 × G2 is strong vague graph. Vµ1 ×µ2 ((u1 , u2 )(v1 , v2 )) = imin{Vµ1 (u1 v1 ), Vµ2 (u2 v2 )}, ∀u1 v1 ∈ E1 , u2 v2 ∈ E2 , u1 6= v1 , u2 6= v2 = imin{imin{Vσ1 (u1 ), Vσ2 (u2 )}, imin{Vσ1 (v1 ), Vσ2 (v2 )}} = imin{Vσ1 ×σ2 (u1 , u2 ), Vσ1 ×σ2 (v1 , v2 )}. Hence, G1 × G2 is strong vague graph. Now, we prove that G1 timesG2 is strong vague graph. For all u ∈ N1 , u2 v2 ∈ E2 , u2 6= v2 , we have Vµ1 timesµ2 ((u, u2 )(u, v2 )) = imin{Vσ1 (u), Vµ2 (u2 v2 )} = imin{Vσ1 (u), imin{Vσ2 (u2 )Vσ2 (v2 )}} = imin{Vσ1 (u), Vσ2 (u2 ), Vσ2 (v2 )}} = imin{imin{Vσ1 (u), Vσ2 (u2 )}, imin{Vσ1 (u), Vσ2 (v2 )}} = imin{Vσ1 ×σ2 (u, u2 ), Vσ1 ×σ2 (u, v2 )}.


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For all w ∈ N2 , u1 v1 ∈ E1 , u1 6= v1 , we have Vµ1 timesµ2 ((u1 , w)(v1 , w)) = imin{Vµ1 (u1 v1 ), Vσ (w)} = imin{imin{Vσ1 (u1 )Vσ1 (v1 ), Vσ1 (w)}} = imin{Vσ1 (u), Vσ2 (u2 ), Vσ2 (v2 )}} = imin{imin{Vσ1 (u1 )Vσ2 (w)}, imin{Vσ1 (v1 )Vσ2 (w)}} = imin{Vσ1 ×σ2 (u1 , w), Vσ1 ×σ2 (v1 , w)}. Also if ∀u1 v1 ∈ E1 , u2 v2 ∈ E2 , u1 6= v1 , u2 6= v2 , then, by the same argument as above, we may prove that Vµ1 timesµ2 ((u1 , u2 )(v1 , v2 )) = imin{Vµ1 (u1 v1 ), Vµ2 (u2 v2 )} = imin{Vσ1 ×σ2 (u1 , u2 ), Vσ1 ×σ2 (v1 , v2 )}. Therefore G1 timesG2 is strong vague graph.

4

µ-complement, self µ-complement and self µ-weak complement on vague graphs

In this section, we define µ-complement, self µ-complement and self µ-weak complement on vague graphs and investigate some of their properties. Definition 4.1. Let G = (σ, µ) be a X vague graph. Then the degree X of a node x ∈ N is defined by d(x) = (dt (x), df (x)) = Vµ (xy), i.e., dt (x) = tµ (xy) and df (x) = y6=x

X

y6=x

fµ (xy).

y6=x

Definition 4.2. Let G = (σ, Xµ) be a vague graph. Then X the order of G is defined X to be O(G) = (Ot (G), Of (G)) = Vσ (x), i.e., Ot (G) = tσ (x) and Of (G) = fσ (x). x∈N

x∈N

x∈N

Definition 4.3. [23] The size to be S(G) = (St (G), Sf (G)) = X Xof G = (σ, µ) is definedX Vµ (xy), i.e., St (G) = tµ (xy) and Sf (G) = fµ (xy). x,y∈N

x,y∈N

Definition 4.4. Let G = (σ, µ) be a vague G is defined as Gµ = (σ, µµ ), where Vµµ = imin{Vσ (x), Vσ (y)} − Vµ (xy) 0 tµµ = tσ (x) ∧ tσ (y) − tµ (xy) 0 fµµ = fµ (xy) − fσ (x) ∨ fσ (y) 0

x,y∈N

graph. The µ-complement of vague graph if Vµ (xy) > 0 if Vµ (xy) = 0 if tµ (xy) > 0 if tµ (xy) = 0 if fµ (xy) > 0 if fµ (xy) = 0.

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Example 4.1. Suppose that N = {u1 , u2 , u3 , u4 , u5 } is a set of nodes, and G = (σ, µ) is a vague graph, where σ = (tσ , fσ ) is a vague set on N given by σ = {(u1 , 0.5, 0.2), (u2 , 0.6, 0.3), (u3 , 0.25, 0.4), (u4 , 0.2, 0.7), (u5 , 0.1, 0.6), and vague relation on N is given by µ = {(u1 u2 , 0.2, 0.5), (u2 u3 , 0.02, 0.7), (u1 u5 , 0.05, 0.7), (u4 u5 , 0.05, 0.75), (u1 u3 , 0.1, 0.6), (u2 u5 , 0.02, 0.7)}. Then, Gµ = (σ, µµ ) is µ-complementary vague graph of G with σ = {(u1 , 0.5, 0.2), (u2 , 0.6, 0.3), (u3 , 0.25, 0.4), (u4 , 0.2, 0.7), (u5 , 0.1, 0.6), and vague relation on N is given by µµ = {(u1 u2 , 0.3, 0.2), (u2 u3 , 0.23, 0.3), (u1 u5 , 0.05, 0.1), (u4 u5 , 0.05, 0.05), (u1 u3 , 0.15, 0.2), (u2 u5 , 0.08, 0.1)}.

Figure 5. Vague graph G


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Figure 6. Vague graph Gµ Note, if G = (σ, µ) is a vague graph and Gµ = (σ, µµ ) is the µ-complement of G, then we have: i) Node set of Gµ is same as that of G. ii) The number of elements in the edge set of Gµ is less than or equal to the number of elements in the edge set of G. iii) If G∗ is a complete vague graph, then Gµ is complete vague graph. iv) O(G) = O(Gµ ) v) S(Gµ ) =

X

Vµµ (xy) =

xy∈µ∗

S(Gµ ) +

X

imin{Vσ (x), Vσ (y)} − Vµ (xy)

x,y∈N

Vµ (xy) =

xy∈µ∗

⇒ S(Gµ ) + S(G) =

X

X

imin{Vσ (x), Vσ (y)},

xy∈µ∗

X

imin{Vσ (x), Vσ (y)}

xy∈µ∗

Theorem 4.5. Let G = (σ, µ) be a vague graph, then all nodes Gµ are isolated if and only if G = (σ, µ) is a strong vague graph. Proof. First, suppose that G is a strong vague graph. Then, Vµ (xy) = imin{Vσ (x), Vσ (y)}, ∀(xy) ∈ µ∗ , i.e., tµ (xy) = \min{tσ (x), tσ (y)}, fµ (xy) = \max{fσ (x), fσ (y)}, ∀xy ∈ µ∗ . By the definition of µµ (xy) and (1), Vµµ (xy) = tµµ (xy) = fµµ (xy) = 0 ∀(x, y) ∈ N × N . µ Therefore, G has only isolated nodes. Conversely, let Gµ has only isolated nodes. Hence Vµµ (xy) = tµµ (xy) = fµµ (xy) = 0 ∀(x, y) ∈ N × N . Then ∗ V µµ (xy) = tµµ (xy) = fµµ (xy) = 0, ∀xy ∈ µ , i.e.,  Vµ (xy) = imin{Vσ (x), Vσ (y)} t (xy) = \min{tσ (x), tσ (y)} ∀xy ∈ µ∗  µ fµ (xy) = \max{fσ (x), fσ (y)}.

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Therefore, G = (σ, µ) is a strong vague graph.

Theorem 4.6. If there is a isomorphism between two vague graphs G1 = (σ1 , µ1 ) and G2 = (σ2 , µ2 ), then their µ-complements, Gµ1 and Gµ2 are also isomorphic. Proof. Assume that G1 and G2 are isomorphic, there exists a bijective map h : N1 → N2 , satisfying  tσ (x) = tσ2 (h(x))    1 fσ1 (x) = fσ2 (h(x)) ∀x ∈ N1 and xy ∈ E1 t (xy) = t (h(x)h(y))  µ2   µ1 fµ1 (xy) = fµ2 (h(x)h(y)).

By the definition of µ-complement, we have  tµµ1 (xy) = \min{tσ1 (x), tσ1 (y)} − tµ1 (xy) = \min{tσ2 (h(x)), tσ2 (h(y))} − tµ2 (h(x)h(y))   1   = t µ2 (h(x)h(y)) µ2 µ1 (xy) = fµ (xy) − \max{fσ (x), fσ (y)} = fµ (h(x)h(y)) − \max{fσ (h(x)), fσ (h(y))} f  1 1 1 2 2 2 µ    = 1f µ (h(x)h(y)). 2 µ 2

µ Hence, Gµ1 ∼ = G2 . The proof of the converse is as above.

Theorem 4.7. If there is a co-weak isomorphism h between vague graphs G1 , G2 where fσ1 (x) = fσ2 (h(x)),∀x ∈ N1 , then there can be homomorphism between Gµ1 and Gµ2 . Proof. By assumption , h : N1 → N2 is a bijective map that satisfies: tσ1 (x) ≤ tσ2 (h(x)) fσ1 (x) = fσ2 (h(x)) ∀x ∈ N1 and tµ1 (xy) = tµ2 (h(x)h(y)) ∀x, y ∈ N1 fµ1 (xy) = fµ2 (h(x)h(y)).

(1) (2)

Using (1) and (2), for all x, y ∈ N1 , we have tµµ1 (xy) = tσ1 (x) ∧ tσ1 (y) − tµ1 (xy) 1 ≤ tσ2 (h(x)) ∧ tσ2 (h(y)) − tµ2 (h(x)h(y)) = tµµ2 (h(x)h(y)). 2

Hence, tµµ1 (xy) ≤ tµµ2 (h(x)h(y)). 1

2

Also, fµµ1 (xy) = fµ1 (xy) − fσ1 (x) ∨ fσ1 (y) 1

(3)


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= fµ2 (h(x)h(y)) − fσ2 (h(x)) ∨ fσ2 (h(y)) = fµµ2 (h(x)h(y)). 2

Hence, fµµ1 (xy) = fµµ2 (h(x)h(y)). 1

2

(4)

By (3) and (4), h is a bijective homomorphism between Gµ1 and Gµ2 . Definition 4.8. A vague graph G is said to be self µ-complementary if G ∼ = Gµ . Theorem 4.9. Let G = (σ, µ) be a self µ-complementary vague graph. Then X 1 X Vµ (uv) = imin{Vσ (u), Vσ (v)}. S(G) = 2 ∗ uv∈µ u,v∈N

Proof. Let G = (σ, µ) be a self µ-complementary vague graph, then there exists an isomorphism h : N → N such that : Vσ (u) = Vσ (h(u)). ∀u ∈ N and Vµ (uv) = Vµµ (h(u)h(v)). ∀u, v ∈ N, uv ∈ µ∗ Now by definition of Gµ , we have, for all uv ∈ µ∗ , we have Vµµ (h(u)h(v)) = imin{Vσ (h(u)), Vσ (h(v))} − Vµ (h(u)h(v)). i.e. Vµ (uv) = imin{Vσ (u), Vσ (v)} − Vµ (h(u), h(v)). Hence, X X X Vµ (uv) + Vµ (h(u)h(v)) = imin{Vσ (u), Vσ (v)}. u6=v

⇒ 2

u6=v

X

u6=v

Vµ (uv) =

u6=v

X

imin{Vσ (u), Vσ (v)}.

u6=v

Therefore, X 1X Vµ (uv) = imin{Vσ (u), Vσ (v)}. 2 u6=v

u6=v

Definition 4.10. A vague graph G is said to be self weak µ-complementary vague graph if G is weak isomorphic with Gµ . Theorem 4.11. Let G = (σ, µ) be a self weak µ-complementary vague graph, then X 1X S(G) = Vµ (xy) ≤ imin{(Vσ (x), Vσ (y))}. 2 ∗ xy∈µ x6=y

Proof. Let G be a self weak µ-complementary vague graph, then G is weak isomorphic with Gµ . So, there exists an isomorphism h : N → N , a bijective mapping satisfying : Vσ (x) = Vσ (h(x)) Vµ (xy) ≤ Vµµ (h(x)h(y)). Using the definition of µ-complement, for all (xy) ∈ µ∗ ,

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⇒ Vµµ (h(x)h(y)) = imin{Vσ (h(x)), Vσ (h(y))} − Vµ (h(x)h(y)) Vµ (xy) ≤ Vµµ (h(x)h(y)) = imin{Vσ (h(x)), Vσ (h(y))} − Vµ (h(x)h(y)) = imin{Vσ (x), Vσ (y)} − Vµ (h(x), h(y)). ⇒ VX ≤ imin{VσX (x), Vσ (y)}. µ (xy) + Vµ (h(x)h(y)) X ⇒ Vµ (xy) + Vµ (h(x)h(y)) ≤ imin{(Vσ (x), Vσ (y))}. x6=y

⇒ 2

X

x6=y

Vµ (xy) ≤

x6=y

⇒ S(G) =

X

x6=y

imin{(Vσ (x), Vσ (y))}.

x6=y

X

Vµ (xy) ≤

x6=y

1X imin{(Vσ (x), Vσ (y))}. 2 x6=y

Theorem 4.12. Let G = (σ, µ) be a vague graph. If  1 t (xy) ≤  µ 2 (tσ (x) ∧ tσ (y)) 

then G is a self weak µ-complementary vague graph. tσ (x) = tσ (h(x)) Proof. Consider the identity map h : N → N , fσ (x) = fσ (h(x)).

By definition of tµµ (x) and fµµ (x), we have: tµµ (xy) = tσ (x) ∧ tσ (y) − tµ (xy) fµµ (xy) = fµ (xy) − fσ (x) ∨ fσ (y), Hence, tµµ (xy) ≥ tσ (x) ∧ tσ (y) − 12 (tσ (x) ∧ tσ (y)) = 12 (tσ (x) ∧ tσ (y)) ≥ tµ (xy) fµµ (xy) ≤ 32 (fσ (x) ∨ fσ (y)) − fσ (x) ∨ fσ (y) = 12 (fσ (x) ∨ fσ (y)) ≤ fµ (xy), tµ (xy) ≤ tµµ (h(x)h(y)) i.e. fµ (xy) ≥ fµµ (h(x)h(y)).

5

∀x, y ∈ N

fµ (xy) ≥ 32 (fσ (x) ∨ fσ (y)),

∀x ∈ N

∀xy ∈ µ∗ ∀xy ∈ µ∗

Conclusions

Graph theory is an extremely useful tool in solving the combinatorial problems in different areas including geometry, algebra, number theory, topology, operations research, optimization, and computer science. The concept of vague sets is due to Gau and Buehrer who studied the concept with the aim of interpreting the real life problems in better way than the existing mechanisms such as Fuzzy sets. In this paper we defined strong product, categorical product and µ-complement on vague graphs and investigated some properties of µ-complement. We also studied some definition of self µ-complementary and self µ-weak complementary on vague graphs.


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