**Publication: International Journal of Mathematical Combinatorics***Author: Al-Hawary, Talal*

*Date published: December 1, 2011*

(ProQuest: ... denotes formulae omitted.)

§1. Introduction

Graph theory has several interesting applications in system analysis, operations research and economics. Since most of the time the aspects of graph problems are uncertain, it is nice to deal with these aspects via the methods of fuzzy logic. The concept of fuzzy relation which has a widespread application in pattern recognition was introduced by Zadeh [8] in his landmark paper "Fuzzy sets" in 1965. Fuzzy graph and several fuzzy analogs of graph theoretic concepts were first introduced by Rosenfeld [6] in 1975. Sense then, fuzzy graph theory is finding an increasing number of applications in modelling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems.

Mordeson and Peng [2] defined the concept of complement of fuzzy graph and studied some operations on fuzzy graphs. In [7], the definition of complement of a fuzzy graph was modified so that the complement of the complement is the original fuzzy graph, which agrees with the crisp graph case. Moreover some properties of self-complementary fuzzy graphs ( fuzzy graphs that are isomorphic to their complements) and the complement of the operations of union, join and composition of fuzzy graphs that were introduced in [2] were studied. For more on the previous notions and the following ones, one can see [2]-[7].

A neutrosophic set based on neutrosophy, is defined for an element x(T, I, F) belongs to the set if it is t true in the set, i indeterminate in the set, and f false, where t,i and f are real numbers taken from the sets T, I and F with no restriction on T, I, F nor on their sum n = t + i + f. Particularly, if i" = 0, we get the fuzzy set. Formally, a fuzzy subset of a non-empty set V is a mapping s : V - > [0, 1] and a fuzzy relation µ on a fuzzy subset s, is a fuzzy subset of V x V. All throughout this paper, we assume that s is reflexive, µ is symmetric and V is finite.

Definition 1.1([6]) A fuzzy graph with V as the underlying set is a pair G : (s,µ) where s : V [arrow right] [0, 1] is a fuzzy subset and µ : V x V [arrow right] [0, 1] is a fuzzy relation on s such that µ(x, y) < s(x) A a(y) for all x, y G V, where A stands for minimum. The underlying crisp graph of G is denoted by G* : (s*,µ*) where s* = supp(a) = x G V : s(x) > 0} and µ* = supn(µ) = (x, y) ∈VxV: µ(x,y) > 0}.H = (σ' ,µ') is a fuzzy subgraph of G if there exists X⊆V such that, σ' : X [arrow right] [0, 1] is a fuzzy subset and µ' : X x X [arrow right] [0, 1] is a fuzzy relation on s' such that µ(x,y) ≤ σ(x) ∇ σ(y) for all x,y G X.

Definition 1.2([5]) A fuzzy graph G : (σ, µ) is complete if µ(x, y) = σ(x)Aσ(y) for all x,y ∈ V.

Next, we recall the following two results from [7].

Lemma 1.3 Let G : (σ, µ) be a self- complementary fuzzy graph. Then Σ^sub x,y∈v^ µ(x,y) = (1/2)Σ^sub x,y∈v^(σ(x)∀σ(y)).

Lemma 1.4 Let G : (σ,µ) be a fuzzy graph with µ(x,y) = (1/2)(σ(x) σ (y)) for all x, y ∈ V. Then G is self- complementary.

Definition 1.5([I]) Two fuzzy graphs G1 : (σ^sub 1^,µ^sub 1^) with crisp graph G*^sub 1^ : (V, E^sub 1^) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) with crisp graph G^sub 2^* : (V^sub 2^, E^sub 2^) are isomorphic if there exists a bisection h : V^sub 1^ [arrow right] V^sub 2^ such that σ^sub 1^(?) = σ^sub 2^(h(x)) and µ^sub 1^ (x, y) = µ^sub 2^(h(x) , h(y)) for all x, y ∈ V^sub 1^.

Lemma 1.6([3]) Any two isomorphic fuzzy graphs G^sub 1^ : (σ^sub 1^, µ^sup 1^) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) satisfy Σ^sub x∈v^sub 1^^σ^sub 1^(x) = Σ^sub x∈v^sub 2^^σ^sub 2^(x) and Σ^sub x∈v^sub 1^µ^sub 1^ te, y) = Σ^sub x∈v^sub 1^^σ^sub 2^

In this paper, we provide three new operations on fuzzy graphs, namely direct product, semi-strong product and strong product. We give sufficient conditions for each one of them to be complete and show that if any one of these product of two fuzzy graphs is complete, then at least one of the two fuzzy graphs must be complete. Moreover, we introduce and study the notion of balanced fuzzy graph and show that this notion is weaker than complete and we give necessary and sufficient conditions for the direct product, semi-strong product and strong product of two balanced fuzzy graphs to be balanced. Finally we prove that given a balanced fuzzy graph G, then any isomorphic fuzzy graph to G must be balanced.

§2. Complete Fuzzy Graphs

We begin this section by defining three new fuzzy graphs products.

Definition 2.1 The direct product of two fuzzy graphs G^sub 1^ : (σ^sub 1^, µ^sub 1^) with crisp graph G*^sub 1^ :

(V^sub 1^, E1) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) with crisp graph G^sub 2^* : (V^sub 2^,E2),where we assume that V^sub 1^DV^sub 2^ = 0, is defined to be the fuzzy graph ... with crisp graph G* : (V^sub 1^ ? V^sub 2^, E) where

...

Definition 2.2 The semi-strong product of two fuzzy graphs G^sub 1^ : (σ^sub 1^, µ^sub 1^) with crisp graph G*^sub 1^ : (V^sub 1^, E1) andG^sub 2^ : (σ^sub 2^,µ^sub 2^) with crisp graphG^ : (V^sub 2^,∩2),where we assume that V^sub 1^CW2 = Ø, is defined to be the fuzzy graph G^sub 1^-G^sub 2^ : (σ^sub 1^ * σ^sub 2^,µ^sub 1^.µ^sub 2^) with crisp graph G* : (V^sub 1^ x V^sub 2^, E) where

E = (u, V^sub 1^)(U, V^sub 2^) : u ∈ V^sub 1^, (V^sub 1^, V^sub 2^) ∈ E2] U [(U1, V^sub 1^)(U2, V^sub 2^) : (U1, U2) ∈ E1, (V^sub 1^, V^sub 2^) ∈ E2], (σ^sub 1^ * σ^sub 2^)(u,v) = G^sub 1^(U) Aa2(v), for all (u,v) ∈ V^sub 1^ ? V^sub 2^, (µ?µ^sub 2^)((% V^sub 1^)(U, V^sub 2^)) = σ^sub 1^ (x) ?µ^sub 2^(σ^sub 2^,σ^sub 2^) and (µ^sub 1^- )((u^sub 1^, V^sub 1^)(U2, V^sub 2^)) = µ^sub 1^ (σ^sub 1^, u^sub 2^) ∇ µ^,?^.

Definition 2.3 The strong product of two fuzzy graphs G^sub 1^ : (σ^sub 1^, µ^sub 1^) with crisp graph G*^sub 1^ : (V^sub 1^, E1) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) with crisp graph Gl1 : (V^sub 2^, E2), where we assume that V^sub 1^HV^sub 2^ = 0, is defined to be the fuzzy graph G^sub 1^^G^sub 2^ :(σ^sub 1^®σ^sub 2^,µµ^sub 2^) with crisp graph G* : (V^sub 1^ ? V^sub 2^, E) where

E = (u, V^sub 1^)(U, V^sub 2^) : u ∈ V^sub 1^, (V^sub 1^, V^sub 2^) ∈ E2] U (uí,w)(u2,w) :w ∈ V^sub 2^, (U1, U2) e E1] U [(U1, V^sub 1^)(U2, V^sub 2^) : (U1, U2) ∈ E1, (V^sub 1^, V^sub 2^) ∈ E2],

(σ^sub 1^ * σ^sub 2^)(u,v) = σ^sub 1^(quo;) ? σ^sub 2^(?), for all (u, ?) ∈ V^sub 1^ ? V^sub 2^, (µ^sub 1^ * µ^sub 2^)((u, V^sub 1^)(U, V^sub 2^)) =aí(u) ?µ^sub 2^(??,?2) , (µ^sub 1^ * µ^sub 2^)((v^sub 1^,)(σ^sub 2^,(w) = A(w) A (u^sub i^,u^sub 2^) and (µ^sub 1^ * µ^sub 2^)((V^sub 1^)(U2, V^sub 2^)) = µ^sub 1^(u^sub 1^,u^sub 2^) µ^sub 1^.

Next, we show that the direct product, the semi-strong product and the strong product of two complete fuzzy graphs are again fuzzy complete graphs.

Theorem 2.4 If G^sub 1^ : (s?, µ?) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) are complete fuzzy graphs, then G^sub 1^ ? G^sub 2^ is complete.

Proof If (U^sub 1^, V^sub 1^)(U^sub 2^, V^sub 2^) ∈ E ,then since Gi and G^sub 2^ are complete

(µ^sub 1^ * µ^sub 2^)((^, V^sub 1^)(U2, V^sub 2^)) = µ^sub 1^(u^sub 1^,u^sub 2^) A µ^sub 2^sub 2^(v^sub 1^,v^sub 2^) = σ^sub 1^(u^sub 1^) Aσ^sub 1^(U^sub 2^) Aσ^sub 2^(V^sub 1^) Aσ^sub 2^(V^sub 2^) = (σ^sub 1^ * σ^sub 2^)((u^sub 1^ v^sub 1^)) A (σ^sub 1^ * σ^sub 2^)((u^sub 2^, u^sub 2^)).

Hence, G^sub 1^ ∏ G^sub 2^ is complete.

Theorem 2.5 If G^sub 1^ : (σ^sub 1^, µ^sub 1^) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) are complete fuzzy graphs, then G^sub 1^ * G^sub 2^ is complete.

Proof If (u, v^sub 1^)(u, v^sub 2^) ∈ E, then

(µ^sub 1^ * µ^sub 2^)((u, v^sub 1^)(u, v^sub 2^)) = σ(u)⋁ µ^sub 2^ (v^sub 1^, v^sub 2^)

= σ^sub 1^ (u^sub 1^)⋁ sgr;^sub 2^(v^sub 1^) ⋁ σ^sub 2^(v^sub 2^) (since G^sub 2^ is complete)

= (σ^sub 1^ * σ^sub 2^)((u, v^sub 1^)) ⋁ (σ^sub 1^ * σ^sub 2^)((u, v^sub 2^)).

If (u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^) ∈ E, then since G^sub 1^ and G^sub 2^ are complete

(µ^sub 1^ * µ^sub 2^)((u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^)) = µ^sub 1^(u^sub 1^, u^sub 2^) ⋁ (v^sub 1^,v^sub 2^)

= σ^sub 1^(u^sub 1^) ⋁ σ^sub 1^(u^sub 2^) ⋁ σ^sub 2^(v^sub 1^) ⋁ σ^sub 2^(v^sub 2^)

= (σ * σ^sub 2^)((u^sub 1^, v^sub 1^)) ⋁ (σ * σ^sub 2^)((u^sub 2^,v^sub 2^)).

Hence, G^sub 1^ * G^sub 2^ is complete.

Theorem 2.6 If G^sub 1^ : (σ^sub 1^, µ^sub 1^) and G^sub 2^ : (σ^sub 2^, µ^sub 2^) are complete fuzzy graphs, then G^sub 1^ ⊗ G^sub 2^ is complete.

Proof If (u,v^sub 1^)(u,v^sub 2^) ∈ E, then

(µ^sub 1^ ⊗? µ^sub 2^)((u, v^sub 1^)(u, v^sub 2^)) = σ^sub 1^ (u) ⋁ µ^sub 2^ (v^sub 1^, v^sub 2^)

= σ^sub 1^(u^sub 1^) ⋁ σ^sub 2^(v^sub 1^) ⋁ σ^sub 2^(v^sub 2^) (since G^sub 2^ is complete)

= (σ^sub 1^ ⊗ σ^sub 2^)((u, v^sub 1^))⋁(σ^sub 1^ ⊗ σ^sub 2^)((u, v^sub 2^)).

If (u^sub 1^, w)(u^sub 2^, w) ∈ E, then

(µ^sub 1^ ⊗ µ^sub 2^)((u^sub 1^,w)(u^sub 2^,w)) = σ^sub 2^(w) ⋁ µ^sub 1^(u^sub 1^, u^sub 2^)

= σ^sub 2^(w) ⋁ σ^sub 1^(u^sub 1^) ⋁ σ^sub 1^ (u^sub 2^) (sinceG^sub 1^ is complete)

= (σ ⊗ σ^sub 2^)((u^sub 1^, w)) ⋁ (σ^sub 1^ ⊗ σ^sub 2^)((u^sub 2^, w)).

If (u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^) ∈ E, then since G^sub 1^ and G^sub 2^ are complete

(µ^sub 1^ ⊗ µ^sub 2^)(u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^)) = µ^sub 1^(u^sub 1^, u^sub 2^) ⋁ µ^sub 2^(v^sub 1^, v^sub 2^)

= σ^sub 1^(u^sub 1^) ⋁ σ^sub 1^(u^sub 2^) ⋁ σ^sub 2^(v^sub 1^) ⋁ σ^sub 2^(v^sub 2^)

= (σ^sub 1^ ⊗ σ^sub 2^)((u^sub 1^, v^sub 1^)) ⋁ (σ^sub 1^ ⊗ σ^sub 2^)((u^sub 2^, v^sub 2^)).

Hence, G^sub 1^ ⊗ G^sub 2^ is complete.

An interesting property of complement is given next.

Theorem 2.7 //Gi : (s?, µ?) and G2 : (s2,µ2) are complete fuzzy graphs, then ...

Proof Let G : ... . We only need to snow .... For any arc e joining nodes of V, we have the following cases:

Case 1 e = (u, v^sub 1^)(u, v^sub 2^) where (v^sub 1^, v^sub 2^) ∈ E^sub 2^. Then as G is complete by Theorem 2.6, .... On the other hand, (...)(e) = 0 since (u^sub 1^, v^sub 2^) ....

Case 2 e = (u, v^sub 1^)(u, v^sub 2^) where (v^sub 1^,v^sub 2^) ∈ E^sub 2^ and v^sub 1^ ≠ v^sub 2^. Since e ∈ E, µ(e) = 0 and ...(e) = σ(u, v^sub 1^)⋁σ(u, v^sub 2^) = σ^sub 1^ (u^sub 1^)⋁σ^sub 2^(v^sub 1^)⋁σ^sub 2^(v^sub 2^) and as (v^sub 1^, v^sub 2^) ... and as ... is complete, (...)(e) = σ(u^sub 1^) ⋁ σ^sub 2^(v^sub 1^) ⋁ σ^sub 2^(v^sub 2^).

Case 3 e = (u^sub 1^, w)(w^sub 2^, w) where (u^sub 1^, u^sub 2^) ∈ E^sub 1^. Since e ...

Case 4 e = (u^sub 1^, w)(w^sub 2^, w) where (u^sub 1^, u^sub 2^) ∈ E^sub 1^. Since e ...

Case 5 e = (u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^) where (u^sub 1^, u^sub 2^) ...

Case 6 e = (u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^) where (u^sub 1^, u^sub 2^) ∈ E^sub 1^ and v^sub 1^ ≠ v^sub 2^. Since ...

Case 7 e = (u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^) where (u^sub 1^, u^sub 2^) ∉ E^sub 1^ and (u^sub 1^, u^sub 2^) ∉ E^sub 2^. Since e ∉ E, µ(e) = 0 and ...

In all cases ...

By similar arguments to those in the preceding result and using Theorems 2.4 and 2.5, we can prove the following result.

Theorem 2.8 If G^sub 1^ : (σ^sub 1^, µ^sub 1^) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) are fuzzy complete graphs, then ...

Next, we show that if the direct product, the semi-strong product or the strong product of two fuzzy graphs is complete, then at least one of the two fuzzy graphs must be complete.

Theorem 2.9 If G^sub 1^ : (σ^sub 1^, µ^sub 1^) and G^sub 2^ : (σ^sub 2^, µ^sub 2^) are fuzzy graphs such that G^sub 1^[white square]G^sub 2^ is complete, then at least G^sub 1^ or G^sub 2^ must be complete.

Proof Suppose that G^sub 1^ and G^sub 2^ are not complete. Then there exists at least one (u^sub 1^ ,v^sub 1^) ∈ E^sub 1^ and (u^sub 2^, v^sub 2^) ∈ E^sub 2^ such that

µ^sub 1^ ((u^sub 1^, v^sub 1^)) < σ^sub 1^(u^sub 1^) ⋁ σ^sub 1^(v^sub 1^)) and

µ^sub 2^((u^sub 2^, u^sub 2^)) < σ^sub 2^(u^sub 2^) ⋁ σ^sub 2^(v^sub 2^))

Now

(µ^sub 1^∏ µ^sub 2^)((u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^)) = µ^sub 1^(u^sub 1^, u^sub 2^) ⋁ µ^sub 2^(v^sub 1^, v^sub 2^)

< σ^sub 1^(u^sub 1^) ⋁ σ^sub 1^(u^sub 2^) ⋁ σ^sub 2^(v^sub 1^) ⋁ σ^sub 2^(v2) (since G^sub 1^ and G^sub 2^ are complete).

But (σ^sub 1^ ∏ σ^sub 2^)((u^sub 1^, v^sub 1^)) = σ^sub 1^ (u^sub 1^) ⋁ σ^sub 2^(v^sub 1^) and (σ^sub 1^ ∏ σ^sub 2^)((u^sub 2^, u^sub 2^)) = σ^sub 1^(u^sub 2^) ⋁ σ^sub 2^(u^sub 2^) Thus

(σ ∏ σ^sub 2^)((u^sub 1^, v^sub 1^))⋁(σ ∏ σ^sub 2^)((u^sub 2^,u^sub 2^)) = σ^sub 1^(u^sub 1^) ⋁ σ^sub 1^(u^sub 2^) ⋁ σ^sub 2^(v^sub 1^) ⋁ σ^sub 2^(v^sub 2^)

> (µ^sub 1^ ∏ µ^sub 2^)((u^sub 1^, v^sub 1^)(u^sub 2^, v^sub 2^)).

Hence, G^sub 1^ ∏ G^sub 1^ is not complete, a contradiction.

The next result can be proved in a similar manor to the preceding one.

Theorem 2.10 If G^sub 1^ : (σ^sub 1^, µ^sub 1^) and G^sub 2^ : (σ^sub 2^,µ^sub 2^) are fuzzy graphs such that G^sub 1^ * G^sub 2^ or G^sub 1^ ⊗ G^sub 2^ is complete, then at least G^sub 1^ or G^sub 2^ must be complete.

§3. Balanced Fuzzy Graphs

We begin this section by defining the density of a fuzzy graph and balanced fuzzy graphs. We then show that any complete fuzzy graph is balanced, but the converse need not be true.

Definition 3.1 The density of a fuzzy graph G : (σ, µ) is

...

G is balanced if D(H) ≤ D(G) for all fuzzy non-empty subgraphs H of G.

Theorem 3.2 Any complete fuzzy graph is balanced.

Proof Let G be a complete fuzzy graph. Then

D(G) = ...

.If H is a non-empty fuzzy subgraph of G,then

D(H) = ...

Thus G is balanced.

The converse of the preceding result need not be true.

Example 3.3 The following fuzzy graph G : (σ, µ) is a balanced graph that is not complete.

We next provide two types of fuzzy graphs each with density equals 1.

Theorem 3.4 Every self- complementary fuzzy graph has density equals 1.

Proof Let G be a self-complementary fuzzy graph. Then by Lemma 1.3,

...

This completes the proof.

The converse of the preceding result need not be true.

Example 3.5 The following fuzzy graph G : (s,µ) has density equals 1, but it is not selfcomplementary.

Theorem 3.6 Let G : (σ, µ) be a fuzzy graph such that µ(u,v) = (1/2)((σ(u) ⋁ σ(v)), for all u, v ∈ V Then D(G) = I.

Proof Let G : (σ, µ) be a fuzzy graph such that µ(u, v) = (1/2)(σ(u)⋁σ(v)), for all u, v ∈ V. By Lemma 1.4, G is self-complementary and thus by the preceding Theorem D(G) = 1.

Next, we prove the following lemma that we use to give necessary and sufficient conditions for the direct product, semi-strong product and strong product of two fuzzy balanced graphs to be balanced.

Lemma 3.7 Let G^sub 1^ and G^sub 2^ be fuzzy graphs. Then D(G^sub i^) ≤ D(G^sub 1^ ∏ G^sub 2^) for i = 1,2 if and only if D(G^sub 1^) = D(G^sub 2^) = D(G^sub 1^ ∏ G^sub 2^).

Proof If D(G^sub i^) ≤ D(G^sub 1^ ∏ G^sub 2^) for i = 1, 2, then

...

Hence D(G^sub 1^) ≥ D(G^sub 1^ ∏ G^sub 2^) and thus D(G^sub 1^) = -D(G^sub 1^ ∏ G^sub 2^). Similarly, D(G^sub 2^) = D(G^sub 1^ ∏ G^sub 2^). Therefore, D(G^sub 1^) = D(G^sub 2^) = D(G^sub 1^ ∏ G^sub 2^).

Theorem 3.8 Let G1 and G2 be fuzzy balanced graphs. Then G^sub 1^ ∏ G^sub 2^ is balanced if and only if D(G^sub 1^) = D(G^sub 2^) = D(G^sub 1^ ∏ G^sub 2^).

Proof If G^sub 1^ ∏ G^sub 2^ is balanced, then D(G^sub i^) ≤ D(G^sub 1^ ∏ G^sub 2^) for i = 1,2 and by Lemma 3.7, D(G^sub 1^) = D(G^sub 2^) = D(G^sub 1^ ∏ G^sub 2^).

Conversely, if D(G^sub 1^) = D(G^sub 2^) = D(G^sub 1^ ∏ G^sub 2^) and H is a fuzzy subgraph of G^sub 1^ ∏ G^sub 2^, then there exist fuzzy subgraphs H^sub 1^ of G^sub 1^ and H^sub 2^ of G^sub 2^. As G^sub 1^ and G^sub 2^ are balanced and D(G^sub 1^) = D(G^sub 2^) = n^sub 1^/r^sub 1^, then D(H^sub 1^) = a^sub 1^/b^sub 1^ ≤ n^sub 1^/r^sub 1^ and D(H^sub 2^) = a^sub 2^/b^sub 2^ ≤ n^sub 1^/r^sub 1^. Thus a^sub 1^r^sub 1^ + a^sub 2^r^sub 1^ ≤ b^sub 1^n^sub 1^ + b^sub 2^n^sub 1^ and hence D(H) ≤ (a^sub 1^ + a^sub 2^)/(b^sub 1^ + b^sub 2^) ≤ n^sub 1^/r^sub 1^ = D(G^sub 1^ ∏ G^sub 2^). Therefore, G^sub 1^ ∏ G^sub 2^ is balanced.

By similar arguments to those in Lemma 3.7 and Theorem 3.8, we can prove the following result:

Theorem 3.9 Let G1 and G2 be fuzzy balanced graphs. Then

(1) G^sub 1^ * G^sub 2^ is balanced if and only If D(G^sub 1^) = D(G^sub 2^) = D(G^sub 1^ * G^sub 2^).

(2) G^sub 1^ ⊗ G^sub 2^ is balanced if and only IfD(G^sub 1^) = D(G^sub 2^) = D(G^sub 1^ ⊗ G^sub 2^).

We end this section by showing that isomorphism between fuzzy graphs preserve balanced.

Theorem 3.10 Let G^sub 1^ and G^sub 2^ be isomorphic fuzzy graphs. If G^sub 2^ is balanced, then G^sub 1^ is balanced.

Proof Let h : V^sub 1^ [arrow right] V^sub 2^ be a bijection such that σ^sub 1^(x) = σ^sub 2^(h(x)) and µ^sub 1^(x, y) - µ^sub 2^(h(x), h(y)) for all x,y ∈ V^sub 1^. By Lemma 1.6, ... is a fuzzy subgraph of Gi with underlying set W, then H^sub 2^ = (σ^sub 2^, µ^sub 2^) is a fuzzy subgraph of G^sub 2^ with underlying set h(W) where α'^sub 2^(h(x)) = σ'^sub 1^(x) and µ'^sub 2^(h(x) , h(y)) = µ'^sub 1^(x, y) for all x,y ∈ W. Since G^sub 2^ is balanced, D(H^sub 2^) ≤ D(G^sub 2^) and so

...

and so

...

Therefore, Gi is balanced.

Acknowledgement

The author would like to thank the referees for useful comments and suggestions.

1 Received september 2, 2011. Accepted November 8, 2011.

References

[1] K. R.Bhutani, On automorphism of fuzzy graphs, Pattern Recognition Letter, 9(1989), 159-162.

[2] J. N.Mordeson and C. S.Peng, Operations on fuzzy graphs, Information Sciences, 79(1994), 381-384.

[3] A.Nagoor Gani and J.Malarvizhi, Isomorphism on fuzzy graphs, Int. J. Comp, and Math. Sci., Vol.2, 4(2008), 190-196.

[4] A.Nagoor Gani and J.Malarvizhi, Isomorphism properties on strong fuzzy graphs, Int. J. Algorithms, Comp, and Math., Vol.2, 1(2009), 39-47.

[5] A.Nagoor Gani and K.Radha, On regular fuzzy graphs, J. Physical Sciences, 12(2008), 33-40.

[6] A. Rosenfeld, Fuzzy graphs, in L.A.Zadeh, K. S. Fu, K.Tanaka and M.Shirmura (Eds.), Fuzzy and Their Applications to Cognitive and Decision Processes, Academic Press, New York, 1975, 77-95.

[7] M.S.Sunitha and A. V. Kumar, Complements of fuzzy graphs, Indian J. Pure Appi. Math., Vol.33, 9(2002), 1451-1464.

[8] L.A.Zadeh, Fuzzy sets, Inform. Control, 8(1965), 338-353.

Author affiliation:

Talal AL-Hawary

(Department of Mathematics, Yarmouk University, Irbid-Jordan)

E-mail: talalhawary@yahoo.com