N. Arul Pandiyan

Degree of Fuzzy Digraphs

N. Arul Pandiyan

and K R. Balasubramanian Research Scholar

, Assistant Professor

1,2

Department of Mathematics,

H. H. Rajah’s College, Pudukkottai-622 001, Tamil Nadu, INDIA.

email: [email protected]

[email protected]

.

(Received on: October 24, 2018) ABSTRACT

In this paper, we discussed Degree of fuzzy digraphs. A fuzzy digraph can be obtained from two given fuzzy digraphs using Cartesian product, composition, union and join. The concepts of the degree of a vertex in fuzzy digraphs formed by these operations in terms of the degree of vertices in the given fuzzy digraphs.

Keywords: Degree of a vertex, Regular fuzzy digraph, Cartesian product of two fuzzy digraphs, composition of two fuzzy digraph, union and join of two fuzzy digraph.

1. INTRODUCTION

In 1905 left. A. Zadeh (6) introduced a fuzzy set to explain the concept of uncertainty in real life. The Theory of fuzzy graph was developed by Azriel Rosenfeld in 1975-

has been growing applications in various fields. The operation of Cartesian product, composition, union and join of two fuzzy digraphs are defined by Mordeson J.N and peng C.S

. In this paper we discussed the degree of a vertex in fuzzy digraphs. Which are obtained from two given fuzzy digraphs using these operations.

John N.Medeson and premchand S.Nair

analyzed the concept of fuzzy graphs and fuzzy hypergraphs. A. Nagoor Gani and M.Basheer Ahamed

proposed the concept of order and size in fuzzy graph. If the fuzzy digraph G

is formed from two fuzzy digraphs G

and G

we find the degree of vertices in Cartesian product, composition union and Join of G

and G

in terms of the degree of vertices of G

and G

under some restriction. The basic definitions, which can be found in

.

2. PRELIMINARIES

Definitions 2.1

A fuzzy graph G= (𝑉, 𝜎, 𝜇) where v is the vertices, 𝜎 is the fuzzy subset of a non empty set 𝑉 and 𝜇 is a symmetric fuzzy relation on 𝜎 →: 𝜇(𝑢𝑣) ≤ 𝜎(𝑢)Λ 𝜎 (𝑣) ∀

. The underlying crisp graph G= (𝑉, 𝜎, 𝜇) is denoted by G

: (V,E) where 𝐸 ⊆ 𝑉𝑋𝑉.

Definition 2:2

A fuzzy digraph G

= (𝑉, 𝜎

, 𝜇

) is a pair of function 𝜎

: v→[0,1] and 𝜇

: 𝑉𝑋𝑉 → [0,1] where 𝜇

≤ 𝜎

(𝑢)⋀𝜎

(𝑣)∀ 𝑢, 𝑣 ∈ 𝑣 and 𝜇

is a set of fuzzy directed edges called fuzzy arcs.

Definition 2.3

The order of fuzzy digraph G

is defined by O(G

) =

𝜎

(𝑢).

Definition 2.4

Let G

: (𝜎

, 𝜇

) be a fuzzy digraph. The degree of a vertex u is defined by 𝑑

(𝑢)=

𝜇

(𝑢, 𝑣) =

𝜇

(𝑢, 𝑣).

Definition 2.5

The Cartesian product of two fuzzy digraph G

and G

is defined as a fuzzy digraph G

= G

x G

: (𝜎

𝑥𝜎

, 𝜇

𝑥𝜇

) on 𝐺

: (V,E) where V=V

xV

and E={(𝑢

, 𝑢

) (𝑣

, 𝑣

))/𝑢

= 𝑣

, 𝑢

𝑣

or 𝑢

= 𝑣

, 𝑢, 𝑣, ∈ 𝐸}with,

(𝜎

𝑥𝜎

) (𝑢

, 𝑢

) = 𝜎

(𝑢

) ⋀ 𝜎

(𝑢) ∀ (𝑢

, 𝑢

) ∈ 𝑣

, 𝑣

.

(𝜇

𝑥𝜇

) ((𝑢

, 𝑢

)( 𝑣

, 𝑣

)) = 𝜎

(𝑢

) ⋀ 𝜇

(𝑢

, 𝑣

), if 𝑢

= 𝑣

, 𝑢

𝑣

∈ 𝐸

= 𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

), if (𝑢

𝑣

), 𝑢, 𝑣

∈ 𝐸

Definition 2.6

The composition of two fuzzy digraphs G

and G

is defined as a fuzzy digraph

G

=G

[G

]: (𝜎

° 𝜎

, 𝜇

° 𝜇

) on G*: (V,E) where V=V

xV

and E={(𝑢

, 𝑢

) (𝑣

, 𝑣

)) / 𝑢

= 𝑣

, 𝑢

𝑣

∈ 𝐸

or 𝑢

= 𝑣

, 𝑢

𝑣

∈ 𝐸

, or 𝑢

≠ 𝑣

, 𝑢

𝑣

∈ 𝐸

} with,

(𝜎

∘ 𝜎

) (𝑢

𝑢

) = 𝜎

(𝑢

)⋀ 𝜎

(𝑢

), ∀ (𝑢

, 𝑢

) ∈ (𝑣

𝑥 𝑣

) and (𝜇

° 𝜇

)((𝑢

𝑢

)(v

v

)) = 𝜎

(𝑢

)⋀ 𝜇

𝑢

𝑣

), if 𝑢

= 𝑣

, (𝑢

, 𝑣

) ∈ 𝐸

= 𝜎

(𝑢

)⋀ 𝜇

( 𝑢

𝑣

), if 𝑢

= 𝑣

, (𝑢

, 𝑣

) ∈ 𝐸

= 𝜎

(𝑢

)⋀ 𝜎

(𝑣

)⋀ 𝜇

(𝑢

, 𝑣

), if 𝑢

≠ 𝑣

, (𝑢

, 𝑣

) ∈ 𝐸

Definition : 2.7

The union of two fuzzy digraphs 𝐺

and 𝐺

is defined as a fuzzy digraph 𝐺

= 𝐺

∪ 𝐺

∶ (𝜎

∪ 𝜎

, 𝜇

∪ 𝜇

) on,

𝐺

(𝑉, 𝐸) where 𝑉 = 𝑉

∪ 𝑉

and 𝐸 = 𝐸

∪ 𝐸

, with ( 𝜎

∪ 𝜎

) (𝑢) = 𝜎

(𝑢), if 𝑢 ∈ 𝑉

− 𝑉

= 𝜎

(𝑢), if 𝑢 ∈ 𝑉

− 𝑉

= 𝜎

(𝑢) ∨ 𝜎

(𝑢)if 𝑢 ∈ 𝑉

∩ 𝑉

= 𝜇

(𝑢𝑣) if 𝑢𝑣 ∈ 𝐸

− 𝐸

and

(𝜇

∪ 𝜇

) (𝑢𝑣), if 𝑢𝑣 ∈ 𝐸

∩ 𝐸

= 𝜇

(𝑢𝑣), if 𝑢𝑣 ∈ 𝐸

− 𝐸

= 𝜇

(𝑢𝑣) ∨ 𝜇

(𝑢𝑣), if 𝑢𝑣 ∈ 𝐸

∩ 𝐸

.

Definitions: 2.8

Assume that 𝑉

∩ 𝑉

= ∅. The join (sum) of 𝐺

and 𝐺

id defined as a fuzzy digraph 𝐺

=𝐺

+ 𝐺

:(𝜎

+ 𝜎

𝜇

+ 𝜇

)on𝐺

∶ (𝑉, 𝐸) where 𝑉 = 𝑉

∪ 𝑉

and E=𝐸

∪ 𝐸

∪ 𝐸

Where 𝐸

is the set of all edges joining vertices of 𝑉

with verties of 𝑉

with, (𝜎

+ 𝜎

)(𝑢) = (𝜎

∪ 𝜎

(𝑢) ∀ 𝑢 ∈ 𝑉

∪ 𝑉

.and

(𝜇

+ 𝜇

) (𝑢𝑣) = (𝜇

∪ 𝜇

)(𝑢𝑣) if 𝑢 ∈ 𝐸

∪ 𝐸

= 𝜎

(𝑢) ⋁𝜎

(𝑢) if 𝑢𝑉 ∈ 𝐸.

3. DEGREE OF A VERTEX IN CARTESIAN PRODUCT

Definition: 3.1

The Cartesian product of two fuzzy digraph 𝐺

∶ (𝜎

𝜇

) and 𝐺

∶ (𝜎

𝜇

) is defined as a fuzzy digraph 𝐺

= 𝐺

x 𝐺

: (𝜎

x 𝜎

), ( 𝜇

x 𝜇

) on 𝐺

∶ (𝑉, 𝐸) where 𝑉

x 𝑉

and 𝐸 = { (𝑢

𝑢

)(𝑣

, 𝑣

)/ 𝑢

= 𝑣

, 𝑢

𝑣

∈ 𝐸 or 𝑢

= 𝑣

, 𝑢

𝑣

∈ 𝐸 } with,

(𝜎

x 𝜎

) (𝑢

𝑢

) = 𝜎

(𝑢

)⋀ 𝜎

(𝑢) ∀ 𝑢

, 𝑢

) ∈ 𝑣

x𝑣

(𝜇

x 𝜇

) ((𝑢

𝑢

) (𝑣

𝑣

)) = 𝜎

(𝑢

)⋀ 𝜇

(𝑢

𝑢

), if 𝑢

= 𝑣

, 𝑢

𝑣

∈ 𝐸

= 𝜎

(𝑢

)⋀ 𝜇

(𝑢

𝑣

) if 𝑢

= 𝑣

, 𝑢

𝑣

∈ 𝐸

. Theorem 3.2

Let 𝐺

: (𝜎

, 𝜇

) and 𝐺

∶ (𝜎

, 𝜇

) be two fuzzy digraphs if 𝜎

≥ 𝜇

and 𝜎

≥ 𝜇

, then 𝑑

2𝐷

(𝑢

𝑢

) = 𝑑

(𝑢

) + 𝑑

(𝑢

).

Proof:

By definition

For any (𝑢

𝑢

) ∈ 𝑉

x 𝑉

, 𝑑

2𝐷

(𝑢

𝑢

) =

1,𝑢₂) (𝑣1,𝑣₂)∈𝐸

(𝜇

x 𝜇

) ((𝑢

, 𝑢

)(𝑣

, 𝑣

)) =

1=𝑣₁,𝑢₂𝑣₂∈𝐸₂

𝜎

(𝑣

) ⋀ 𝜇

(𝑢

, 𝑣

) +

2=𝑣₂,𝑢₁𝑣₁∈𝐸₁

𝜎

(𝑢) ∧ 𝜇

(𝑢

𝑣

)

=

2𝑣_2∈𝐸2

𝜇

(𝑢

, 𝑣

) +

1𝑣_1∈𝐸1

𝜇

(𝑢

, 𝑣

) ∵ [𝜎

≥ 𝜇

and 𝜎

≥ 𝜇

]

= 𝑑

(𝑢

) + 𝑑

(𝑢

) Example 3.3

Consider the fuzzy digraphs 𝐺

∶ (𝜎

, 𝜇

) and 𝐺

∶ (𝜎

, 𝜇

).

Here 𝜎

≥ 𝜇

and 𝜎

≥ 𝜇

𝑑

x 𝐺

(𝑢

𝑢

) = 𝑑

(𝑢

) + 𝑑

(𝑢

) = 0.2 + 0.1 = 0.3 Then 𝐺

x 𝐺

is called the degree of a vertex in Cartesian product.

Theorem 3.4

Let 𝐺

∶ (𝜎

, 𝜇

) and 𝐺

∶ (𝜎

, 𝜇

) be two fuzzy digraphs and let 𝜎

(𝑢) = 𝐶

∀ 𝑢 ∈ 𝑉

, where 𝐶

is a constant 𝑖 = 1,2

i) if 𝜎

≤ 𝜇

Then 𝑑

x 𝐺

(𝑢

𝑢

) = 𝑑

(𝑢

) + 𝐶

𝑑

(𝑢

).

ii) if 𝜎

≤ 𝜇

Then 𝑑

x 𝐺

(𝑢

𝑢

) = 𝑑

(𝑢

) + 𝐶

𝑑

(𝑢

).

Where 𝑑

(𝑢

) is the degree of 𝑢

in 𝐺

Proof:

Since 𝜎

and 𝜎

are constants, 𝜇

≤ 𝜎

and 𝜇

≤ 𝜎

1) we have 𝜎

≤ 𝜇

.

Hence 𝜇

≤ 𝜎

≤ 𝜇

≤ 𝜎

. so 𝜎

≤ 𝜇

now by definition for any (𝑢

𝑢

) ∈ 𝑉

x 𝑉

𝑑

2𝐷

(𝑢

𝑢

) =

1,𝑢₂)(𝑣1,𝑣₂)∈𝐸

(𝜇

x 𝜇

)(𝑢

𝑢

)(𝑣

𝑣

)

= ∑

𝑢

= 𝑣

𝑢

𝑣

∈ 𝐸

𝜎

(𝑢

)⋀𝜇

(𝑢

𝑣

)

+ ∑

𝑢

= 𝑣

𝑢

𝑣

∈ 𝐸

𝜎

(𝑢

)⋀𝜇

(𝑢

𝑣

)

= ∑

𝑢

𝑣

𝜎

(𝑢

) + ∑

𝑢

𝑣

(𝜇

(𝑢

, 𝑣

) [∵ 𝜎

≤ 𝜇

and 𝜎

≥ 𝜇

]

=

2𝑣_2∈𝐸2

𝐶

= 𝐶

𝑑

2𝐷

(𝑢

) + 𝑑

(𝑢

) iii) We have 𝜎

≤ 𝜇

Hence 𝜇

≤ 𝜎

≤ 𝜇

≤ 𝜎

. so 𝜎

≤ 𝜇

now by definition for any (𝑢

𝑢

) ∈ 𝑉

x 𝑉

𝑑

2𝐷

(𝑢

𝑢

) =

1,𝑢₂)(𝑣_1,𝑣₂)∈𝐸

(𝜇

x 𝜇

)((𝑢

𝑢

)(𝑣

𝑣

))

= ∑

𝑢

= 𝑣

𝑢

𝑣

∈ 𝐸

𝜎

(𝑢

)⋀𝜇

(𝑢

𝑣

)

+

1=𝑣_1,𝑢₂𝑣₂∈𝐸₂

𝜎

(𝑢

)⋀𝜇

(𝑢

𝑣

)

= ∑

𝑢

𝑣

𝜎

(𝑢

) + ∑

𝑢

𝑣

(𝜇

(𝑢

, 𝑣

) [∵ 𝜎

≤ 𝜇

and 𝜎

≥ 𝜇

]

=

1𝑣_1∈𝐸1

𝐶

𝑑

2𝐷

(𝑢

𝑢

) = 𝐶

𝑑

(𝑢

) + 𝑑

(𝑢

) Hence the proof.

Example: 3.5

Consider the fuzzy digraphs 𝐺

: (𝜎

𝜇

) and 𝐺

: (𝜎

𝜇

)

Hence 𝜎

and 𝐺

are constants with 𝜎

≤ 𝐺

. Then,

𝑑

2𝐷

(𝑢

𝑢

) = 𝐶

𝑑

(𝑢

) + 𝑑

(𝑢

) = 0.3 x 2 + 0.2 = 0.6 + 0.2 = 0.8.

This can be verified in example

In 𝐺

x 𝐺

, (𝜎

x 𝜎

) (𝑢, 𝑣) = 0.2 ∀ (𝑢, 𝑣) ∈ 𝑉

x 𝑉

. 4. DEGREE OF A VERTEX IN COMPOSITION Definition 4.1

The composition of two fuzzy digraphs G

and G

is defined as a fuzzy digraph

G

=G

[G

]: (𝜎

∘ 𝜎

, 𝜇

∘ 𝜇

) on G*: (V,E) Where V=V

x V

and

E= {((𝑢

, 𝑢

) (𝑣

, 𝑣

))/ 𝑢

=𝑣

, 𝑢

𝑣

∈ 𝐸

or 𝑢

𝑣

, 𝑢

𝑣

∈ 𝐸

or 𝑢

≠ 𝑣

𝑢

𝑣

∈ 𝐸

} with (𝜎

∘ 𝜎

) (𝑢

, 𝑢

) = 𝜎

(𝑢

) ⋀ 𝜎

(𝑢

), ∀ (𝑢

, 𝑢

) ∈ 𝑣

, 𝑣

and

(𝜇

∘ 𝜇

)((𝑢

, 𝑢

) (𝑣

, 𝑣

)) = 𝜎

(𝑢

) ⋀ 𝜇

)(𝑢

, 𝑣

), if 𝑢

= 𝑣

(𝑢

𝑣

) ∈ 𝐸

= 𝜎

(𝑢

) ⋀ 𝜎

𝜇

(𝑢

, 𝑣

), if (𝑢

= 𝑣

), (𝑢

, 𝑣

) ∈ 𝐸

= 𝜎

(𝑢

) ⋀ 𝜎

(𝑣

) ⋀ 𝜇

(𝑢

, 𝑣

) if (𝑢

≠ 𝑣

)(𝑢

, 𝑣

) ∈ 𝐸

. Theorem 4.2

Let G

: (𝜎

, 𝜇

) and G

: (𝜎

, 𝜇

) be two fuzzy digraphs. If (𝜎

≥ 𝜇

) and (𝜎

≥ 𝜇

. Then

𝑑

(𝑢

, 𝑢

)= 𝑃

𝑑

(𝑢

) + 𝑑

(𝑢

).

Proof:

By definition, for any (𝑢

, 𝑢

) ∈ 𝑣

x 𝑣

𝑑

(𝑢

, 𝑢

) =

1,𝑢₂) (𝑣1,𝑣₂)∈𝐸

(𝜇

∘ 𝜇

)((𝑢

, 𝑢

) (𝑣

, 𝑣

))

=

1=𝑣₁,𝑢₂𝑢₂∈𝐸₂

𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

) +

2=𝑣₂,𝑢₁𝑢₁∈𝐸₁

𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

)

+

2≠𝑣₂,𝑢₁𝑢₁∈𝐸₁

𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

)

=

2𝑣2,∈𝐸2

𝜇

(𝑢

, 𝑣

) +

𝑢1𝑣1,∈𝐸1,𝑢2=𝑣2

𝜇

(𝑢

, 𝑣

)

+

1𝑣1,∈𝐸1,𝑢1 ≠𝑢2

𝜇

(𝑢

, 𝑣

)

= 𝑑

(𝑢

)+ |𝑉

|

𝜇

(𝑢

𝑣

) [∵ 𝜎

≥ 𝜇

𝑎𝑛𝑑 𝜎

≥ 𝜇

].

𝑑

1𝐷 [𝐺2𝐷]

(𝑢

, 𝑢

) = 𝑑

(𝑢

)+ P

𝑑

(𝑢

).

Example 4.3

Consider the fuzzy digraphs G

and G

.

𝑑

(𝑢

, 𝑢

) = 𝑑

(𝑢

)+ P

𝑑

(𝑢

)

= 0.1 + 2 x 0.2

= 0.1 + 0.4= 0.5.

This can be verified G

[G

] is called the degree of a vertex in composition.

Theorem 4.4

Let G

:(𝜎

, 𝜇

) and G

:(𝜎

, 𝜇

) be two fuzzy digraphs and let 𝜎

(u) = 𝐶

∀ 𝑢 ∈ 𝑣

where 𝐶

is a constant, i=1,2.

i) If 𝜎

≤ 𝜇

Then 𝑑

(𝑢

, 𝑢

)= P

𝑑

(𝑢

)+ C

𝑑

(𝑢

).

ii) If 𝜎

≤ 𝜇

Then 𝑑

(𝑢

, 𝑢

)= 𝑑

(𝑢

)+ P

C

𝑑

(𝑢

).

Proof:

Since 𝜎

and 𝜎

are constants, 𝜇

≤ 𝜎

and 𝜇

≤ 𝜎

.

i) We have 𝜎

≤ 𝜇

Hence 𝜇

≤ 𝜎

≤ 𝜇

≤ 𝜎

. So 𝜎

≥ 𝜇

By definition, for any (𝑢

, 𝑢

) ∈ 𝑣

x 𝑣

.

𝑑

(𝑢

, 𝑢

) =

1,𝑢₂)(𝑣1,𝑣₂)∈𝐸

(𝜇

∘ 𝜇

)(𝑢

, 𝑢

)(𝑣

, 𝑣

)

=

1=𝑣₁,𝑢₂𝑣_2∈𝐸2

𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

)+

2=𝑣₂,𝑢₁𝑣_1∈𝐸1

(𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

) +

2≠𝑣₂,𝑢₁𝑣_1∈𝐸1

(𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

)

=

2𝑣_2∈𝐸2

𝜎

(𝑢

) +

1𝑣_1∈𝐸1,𝑢₂=𝑣₂

𝜇

(𝑢

𝑣

)+

1𝑣_1∈𝐸1𝑢₂≠𝑣₂

𝜇

(𝑢

𝑣

)

=

2𝑣_2∈𝐸2

𝐶

+ |𝑉

|

1𝑣_1∈𝐸1

𝜇

(𝑢

𝑣

) [∵ 𝜎

≤ 𝜇

and 𝜎

≤ 𝜇

].

= C

𝑑

𝐺′2𝐷

(𝑢

)+ P

𝑑

(𝑢

).

ii) We have 𝜎

≤ 𝜇

.

Hence 𝜇

≤ 𝜎

≤ 𝜇

≤ 𝜎

. So 𝜎

≥ 𝜇

. By definition for any (𝑢

, 𝑢

) ∈ 𝑉

x 𝑉

.

𝑑

1𝐷 [𝐺2𝐷]

(𝑢

, 𝑢

) =

1,𝑢₂)(𝑣1,𝑣₂)𝜖𝐸

(𝜇

°𝜇

)((𝑢

, 𝑢

)(𝑣

, 𝑣

))

=

1=𝑣1𝑢2𝑣2∈𝐸2

𝜎

+

2=𝑣2,(𝑢1𝑣1)∈𝐸1

𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

) +

2≠𝑣₂,𝑢₁𝑣₁∈𝐸₁

𝜎

(𝑢

) ⋀ 𝜇

(𝑢

𝑣

)

=

2𝑣₂∈𝐸₂

𝜇

(𝑢

𝑣

) +

2=𝑣₂,(𝑢₁𝑣₁)∈𝐸₁

𝜎

(𝑢

)

+

2≠𝑣₂,𝑢₁𝑣₁∈𝐸₁

𝜎

(𝑢

)

= 𝑑

(𝑢) + |𝑉

|

1𝑣₁∈𝐸₁

𝐶

= 𝑑

(𝑢

) + 𝑃

𝐶

𝑑

(𝑢

).

[∵ 𝜎

≥ 𝜇

and 𝜎

≤ 𝜇

]

Example 4.5

Consider the fuzzy digraphs G

and G

Here 𝜎

and 𝜎

are constant with 𝜎

≤ 𝜇

Then,

𝑑

(𝑢

, 𝑢

) = 𝐶

𝑑

(𝑢

) + P

𝑑

(𝑢

)

= 0.3 + 1x 2 x 0.1

= 0.3 + 0.2

=0.5.

Then G

[G

] is called the degree of a vertex in composition.

5. DEGREE OF A VERTEX IN UNION

Definition 5.1

The union of two fuzzy digraphs G

and G

is defined as a fuzzy digraph

G

= (G

⋃ G

,) (𝜎

⋃ 𝜎

𝜇

⋃ 𝜇

) on G

* (V,E) Where V= V

⋃V

and E=E

⋃ E

with.

(𝜎

⋃𝜎

) (𝑢) = 𝜎

(𝑢) if 𝑢 ∈ V

-V

= 𝜎

(𝑢) if 𝑢 ∈ V

-V

= 𝜎

(𝑢) V 𝜎

(𝑢) if 𝑢 ∈ V

⋂ V

and

(𝜇

⋃𝜇

) (𝑢𝑣) = 𝜇

(𝑢𝑣) if 𝑢𝑣 ∈ E

-E

= 𝜇

(𝑢𝑣) if 𝑢𝑣 ∈ E

-E

= 𝜇

(𝑢𝑣) V 𝜇

(𝑢𝑣) if 𝑢𝑣 ∈ E

⋂ E

Proposition 5.2

For any 𝑢 ∈ V

⋃ V

we have three cases to consider.

Case 1:

Either 𝑢 ∈ V

or 𝑢 ∈ V

but not both.

So (𝜇

⋃𝜇

) (𝑢𝑣) = 𝜇

(𝑢𝑣) if 𝑢 ∈ 𝑣

i.e) if 𝑢𝑣 ∈ 𝐸

.

= 𝜇

(𝑢𝑣) if 𝑢 ∈ 𝑣

i.e) if 𝑢𝑣 ∈ 𝐸

.

Hence if 𝑢 ∈ 𝑣

Then 𝑑

( 𝑢) = ∑ 𝜇

(𝑢𝑣)

𝑢𝑉 ∈ 𝐸

= 𝑑

(𝑢) if 𝑢 ∈ 𝑣

, Then 𝑑

(𝑢)= ∑ 𝜇

(𝑢𝑣)

𝑢𝑉 ∈ 𝐸

= 𝑑

(𝑢) Case 2:

𝑢 ∈ 𝑣

∩ 𝑣

but no edge incident at 𝑢 lies in 𝐸

∩ 𝐸

. Then any edge incident at u is either in 𝐸

or in 𝐸

but not both. Also all these edges will be included in 𝐺

∪ 𝐺

.

Hence 𝑑

(u) =

(𝜇

∪ 𝜇

(𝑢𝑣)

=

1

𝜇

(𝑢𝑣) + =

2

𝜇

(𝑢𝑣)

= 𝑑

(𝑢) +𝑑

(𝑢).

Case 3:

𝑢 ∈ 𝑣

∩ 𝑣

and some edges incident at 𝑢 and are in 𝐸

∩ 𝐸

. Any edges 𝑢𝑣 which is in 𝐸

∩ 𝐸

appear only once.

In 𝐺

∪ 𝐺

and for this 𝑢𝑣 𝜇

(𝑢𝑣) = 𝜇

(𝑢𝑣)𝑣 𝜇

(𝑢𝑣) By definition 𝑑

(u) =

(𝜇

∪ 𝜇

(𝑢𝑣)

=

1−𝐸₂

𝜇

(𝑢𝑣) +

2−𝐸1

𝜇

(𝑢𝑣) +

1∩𝐸₂

𝜇

(𝑢𝑣) ∧ 𝜇

(𝑢𝑣)

=

1−𝐸2

𝜇

(𝑢𝑣) +

2−𝐸1

𝜇

(𝑢𝑣) +

𝜇

(𝑢𝑣)⋁ 𝜇

(𝑢𝑣) +

1∩𝐸₂

𝜇

(𝑢𝑣) ∧ 𝜇

(𝑢𝑣) −

1∩𝐸₂

𝜇

(𝑢𝑣) ∧ 𝜇

(𝑢𝑣)

=

1

𝜇

(𝑢𝑣) +

2

𝜇

(𝑢𝑣) −

1∩𝐸₂

𝜇

(𝑢𝑣) ∧ 𝜇

(𝑢𝑣)

= 𝑑

(𝑢) + 𝑑

(𝑢) −

1∩𝐸₂

𝜇

(𝑢𝑣) ∧ 𝜇

(𝑢𝑣).

Example 5.3

Consider the fuzzy digraphs 𝐺

: (𝜎

, 𝜇

) and 𝐺

: (𝜎

, 𝜇

).

Consider x:

Here 𝑥 ∈ 𝑣

. So by case 1, 𝑑

(x) = 𝑑

(𝑥)

=0.5+0.6= 1.

Consider V:

We have 𝑣 ∈ 𝑣

∩ 𝑣

but no edge incident at v lies in 𝐸

∩ 𝐸

. So by case 2, 𝑑

(𝑣) = 𝑑

(𝑣) + 𝑑

(𝑣) = 1 + 0.3 = 1.3.

Consider u :

We have 𝑣 ∈ 𝑣

∩ 𝑣

and 𝑢𝑤 ∈ 𝐸

∩ 𝐸

. So by case 3,

𝑑

(𝑢) = 𝑑

(𝑢) + 𝑑

(𝑢) − 𝜇

(𝑢𝑤) ∧ 𝜇

(𝑢𝑤) = 0.7+0.4-0.3

= 0.8.

Then 𝐺

∪ 𝐺

is called the Degree of a vertex in union.

6. DEGREE OF A VERTEX IN JOIN

Definition 6.1

Assume that 𝑣

∩ 𝑣

= ∅. The join (sum) of 𝐺

and 𝐺

is defined as a fuzzy digraph 𝐺

= 𝐺

+ 𝐺

: (𝜎

+ 𝜎

, 𝜇

+ 𝜇

) on G

:(𝑣

𝐸) where 𝑣 = 𝑣

∪ 𝑣

and E=𝐸

∪ 𝐸

∪ 𝐸

. Where 𝐸

is the set of all edges joining vertices of 𝑣

with vertices of𝑣

, with

(𝜎

+ 𝜎

) (𝑢) = (𝜎

∪ 𝜎

) (𝑢)∀ 𝑢 ∈ 𝑣

∪ 𝑣

and (𝜇

+ 𝜇

)(𝑢𝑣) = (𝜇

∪ 𝜇

)(𝑢𝑣) if 𝑢𝑣 ∈ 𝐸

∪ 𝐸

= 𝜎

(𝑢) ∨ 𝜎

(𝑢) if 𝑢𝑣 ∈ 𝐸

Proposition 6.2

Here 𝑣

∩ 𝑣

= ∅. So 𝐸

∩ 𝐸

= ∅.

So, (𝜇

∪ 𝜇

) (uv) =𝜇

(𝑢𝑣)if 𝑢𝑣 ∈ 𝐸

=𝜇

(𝑢𝑣)if 𝑢𝑣 ∈ 𝐸

.