Regular and Isometric Fuzzy Graphs

Vol. 14, No. 2, 2017, 379-384

ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 11 December 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/ijfma.v14n2a19

379

International Journal of

Regular and Isometric Fuzzy Graphs

K.Radha1 and P.Indumathi2

PG &Research Department of Mathematics Periyar E.V.R College, Trichy-23, Tamilnadhu, India.

1

Email:[email protected]; 2Email:[email protected]

Received 28 November 2017; accepted 9 December 2017

Abstract. In this paper, a condition for two regular fuzzy graphs to be isometric is obtained. Also a condition for two fuzzy graphs to be isometric is obtained using degree set.

Keywords: Regular fuzzy graph, isometric fuzzy graphs, degree set.

AMS Mathematics Subject Classification (2010): 03E72, 05C07

1. Introduction

Fuzzy graph theory was introduced by Rosenfeld in 1975. During the same time Yeh and bang have also introduced various connectedness concepts in fuzzy graphs. Though it is very young, it has been growing fast and has numerous application in various fields. Nagoorgani and Malarvizhi discussed the concept of isometry in fuzzy graphs and studied its properties. Radha and Kumaravel introduced the concept of edge regular fuzzy graphs. Radha and Rosemine introduced the concept of degree set in fuzzy graphs. Tom and Sumitha introduced sum distance in fuzzy graph. In this paper, we obtain conditions for regular fuzzy graphs to be isometric. Also we introduce isometric fuzzy subgraphs and distance preserving fuzzy graphs and prove that any fuzzy graph on a tree is distance preserving.

2. Preliminaries

In this section, we go through some basic definitions which can be found in [1-11].

Definition 2.1. Let V be a non-empty finite set and E ⊆V ×V. A fuzzy graph

) , ( : σ µ

G is a pair of functions σ :V → [0,1] and µ:E→[0,1] such that

) ( ) ( ) ,

(x y σ x σ y

µ ≤ Λ for all x,y∈V.

Definition 2.2. The order and size of a fuzzy graph G :(σ ,µ) are defined by

∑

∈

=

V x

x

G

O

(

)

σ

(

)

and

∑

∈

= E xy

xy G

S( )

µ

( ).

Definition 2.3 A fuzzy graph G :(σ,µ) is strong, ifµ(xy)=σ(x)Λσ(y) for all

.

K.Radha and P.Indumathi

380

Definition 2.4. A fuzzy GraphG :(σ,µ) is complete, if µ(xy)=σ(x)Λσ(y) for all

. ,y V

x ∈

Definition 2.5. Let G :(σ,µ)be a fuzzy graph onG*:(V,E) . The degree of vertex

x

is

∑

≠

= y x G x xy

d ( )

µ

( ).If each vertex in G has same degree k _{, then} Gis said to be a

regular fuzzy graph or k – regular fuzzy graph.

Definition 2.6. Let G*:(V,E)be a graph and lete = uvbe an edge in G*. Then the degree of an edge e=uv∈Eis defined by dG*(uv)=dG*(u)+dG*(v)−2. If each edge in

*

G has same degree, then G*said to be edge regular. The degree of an edge xy∈E is )

( 2 ) ( ) ( )

(xy xz zy xy

d

y z z

x

G =

∑

µ +

∑

µ − µ

≠ ≠

. If each edge in Ghas same degree k , then Gis

said to be an edge regular fuzzy graph or k - edge regular fuzzy graph.

Definition 2.7. Ifµ(x,y)>0then

x

and

y

are called neighbours,

x

and

y

are said to lie on the edge

e

=

xy

. A path

ρ

in a fuzzy graphG :(σ,µ)is a sequence of distinct

nodes

v

₀

,

v

₁

,

v

₂

,...

v

_nsuch that

µ

(

v

_i

,

v

_i−1

)

>

0

,

1

≤

i

≤

n

.

Here‘n’ is called the length of

the path. The consecutive paris

(

v

i

,

v

i−1

)

are called arcs of the path.

Definition 2.8. If

u,

v

are nodes inG :(σ,µ)and if they are connected by means of a

path then the strength of that path is defined as ( 1, )

1

i i m

i

v v₋

=

Λ

µ i.e., it is the strength of the weakest arc.

If

u,

v

are connected by means of paths of length 'k'then

µ

k

( v

u

,

)

is defined as

}.

,

,

,

,

/

)

,

(

..

)

,

(

)

,

(

)

,

(

sup{

)

,

(

u

v

u

v

₁

v

₁

v

₂

v

₂

v

₃

v

_{k 1}

v

u

v

₁

v

₂

v

_3...,

v

_k1

v

V

k

=

Λ

Λ

Λ

Λ

∈

− −

µ

µ

µ

µ

µ

If u,v∈V the strength of connectedness between

u

and

v

is,

,...}

3

,

2

,

1

/

)

,

(

sup{

)

,

(

=

=

∞

k

v

u

v

u

µ

k

µ

.

Definition 2.9. A fuzzy graph G :(σ,µ)is connected if

µ

∞

(

u

,

v

)

>

0

for all u,v∈V .

An arc

uv

is said to be a strong arc if

µ

(

u

,

v

)

≥

µ

∞

(

u

,

v

)

. A node

u

is said to be an isolate node if µ(u,v)=0,∀u≠v.

Definition 2.10. The

µ

- distance δ(u,v)is the smallest

µ

- length of any

u

−

v

path,

where the

µ

- length of path

ρ

:

u

0

,

u

1

,

u

2

,...,

u

nis

∑

= −

=

n

i

u

i

u

i

l

1

(

1

,

)

1

)

(

µ

ρ

. The

381

v

is defined as e(v)=maxu(

δ

(u,v)). The diameterdiam(G)=∨{e(v)|v∈V} , radius

} | ) ( { )

(G e v v V

r =∧ ∈ . A node whose eccentricity is minimum in a connected fuzzy

graph is called a central node. A connected fuzzy graph is called self – centered if each node is a central node.

Definition 2.11. Let G_i:(σ_i,µ_i) be the fuzzy graphs with underlying crisp graphs

Gi*:(Vi, Ei) for i= 1,2. G2 is said to be isometric from G1 if for each v∈V1 there is a

bijection

φ

v

:

V

1

→

V

2 such that δ1(u,v)=δ2(φv(u),φv(v)), for every u∈V1. If they are

isometric from each other they are said to be isometric. This relation is termed as an isometry relation.

Definition 2.12. Let G:(σ, µ) be a connected fuzzy graph. For any path P, the length of the path L(P) is defined as the sum of the weights of the arcs in P. The sum distance between two vertices u and v is ds(u, v) = min{L(P) / P is a u-v path}.

3. Main results

Theorem 3.1. Let G₁:(σ₁,µ₁) and G2:(σ2,µ2) be two fuzzy graphs such that G1* and

*

2

G are k-regular graphs,

µ

1 and

µ

2 are constant functions of same constant value c

and V1 = V2 = n. If ,

2 1

−

≥ n

k then G1 and G2 are isometric fuzzy graphs.

Proof: Let us construct an isometric mapping from G1 to G2 as follows

Choose u∈V1 and v∈V2 arbitrarily.

Define : φu:V1→V2 by φu(u)=v

Since G₁* and G₂* are k-regular, k- vertices say, u1,u2,...,uk are at distance 1 from u in G1* and K vertices say, v1,v2,...,vkare at distance 1 from v inG2*

Define : φu(ui)=vi, for i=1,2,...,k

Since G₁* and G₂* are k – regular with , 2

1

−

≥n

k d(G_i*)≤2. Therefore, the

remaining n−1−k, vertices say, u_k+1,u_k+2,...,u_n−1 in G₁* are at distance 2 from

u

and the remaining n−1−k vertices , say v_k+1,v_k+2,...,v_n−1 are at distance 2fromv in

G2*.

Define

φ

u

(

u

i

)

=

v

i

,

i

=

k

+

1

,...

n

−

1

Since µ₁ and µ₂ are constant functions of same constant value c, using sum distance,

u u u

dG1( , i)=

µ

1(

u

i

)

=

c

,

i

=

1

,

2

,...

...,

k

v

dG₂( ,vi)=µ1(vvi)=c, i=1,2,...,k

/

d

_G

(

u

,

u

_i

)

d

_G

(

v

,

v

_i

)

,

i

1

,

2

,...

...,

k

2

1

=

∀

=

K.Radha and P.Indumathi 382

1

1

,

2

)

,

(

1

u

u

C

i

k

,...,n

-d

_{G i}

=

=

+

Similarly dG₂(v,vi) = 2C, i = k +1,...,n -1

/

d

G₁

(

u

,

u

i

)

=

d

G₂

(

v

,

v

i

),

i

=

k

+

1

,...,n

-

1

Therefore

(

,

)

(

,

),

1

,

2

,

1

2

1

u

u

d

v

v

i

...,n-d

_{G i}

=

_{G i}

=

Since

u

and

v

are arbitrary, G₂ is isometric from G₁. Similarly, it can be proved that G1 is isometric from G2

Theorem 3.2. Let G1:(σ1,µ1) and G2 :(σ2,µ2) be two m-regular fuzzy graphs such

that µ₁and µ₂are constant functions of same constant value c and

V

₁

=

V

₂

=

n

. If

2 1

−

≥ n

K then G₁ and G₂are isometric fuzzy graphs.

Proof: Since G₁ is m−regular

,

)

(

1

u

m

d

_G

=

∀

u∈v₁

, ) ( 1 1 m uv E uv = ∑ ⇒

∈ µ ∀u∈v1

, 1 m c E uv = ∑ ⇒

∈ ∀u∈v1

,

)

(

1

*

u

m

cd

G

=

⇒

1 v u∈ ∀ , ) ( 1 * c m u d

G =

∀

u∈v1

Similarly ( ) ,

2 * c m u d

G = 2

V u ∈

∀ / G1* andG2* are

c

m _{- regular. Then proceeding as}

in theorem 3.1, G1 and G2 are isometaic fuzzy graphs.

Theorem 3.3. Let G₁:(σ₁,µ₁) and G₂:(σ₂,µ₂)be two fuzzy graphs such that

1

µ and

2

µ are constant functions of same constant value c. If G₁ and G₂ are isometric fuzzy graphs, then they have the same degree set and the same eccentricity set.

Proof: Since G₂ is isometric from G₁there exists a one-to-one map φ from V₁ to V₂ such that,

)),

(

),

(

(

)

,

(

2

2

u

v

d

u

v

d

_G

=

_G

φ

φ

∀u,v∈V_{1 (2.1)}

If

u

and

v

are adjacent in G₁, then using sum distance in fuzzy graphs,

C

uv

v

u

d

_G

(

,

)

=

(

)

=

1

µ

From (2.1)

(

(

),

(

))

(

(

),

(

))

2

u

v

C

u

v

d

_G

φ

φ

=

=

µ

φ

φ

) (u

φ and φ(v) are adjacent is G₂

))

(

(

)

(

* 2 * 1

u

d

u

d

G

383

) ( )

(

1 1

uv v

d

E uv

G = ∑_∈ µ

= C

E uv∈ ₁

∑

= *

(

)

1

u

cd

G

= *

(

(

))

2

u

cd

G

φ

=

(

(

))

2

u

d

G

φ

The degree of each vertex in

G

1 is the degree of some vertex in G 2

The degree set of

G

₁is contained in the degree set of

G

₂. (2.2) Also by the distance preserving property,

}

{

(

,

)

/

1

max

)

(

1

v

d

u

v

u

V

e

_G

=

∈

}

{

(

(

),

(

))

/

1

max

d

u

v

u

∈

V

=

ϕ

ϕ

))

(

(

2

v

e

_G

ϕ

=

(2.3) The eccentricity set of G₁is contained in the eccentricity set of G₂

Since G₁ is isometric from G_{2 proceeding as above.}

The degree set of G₂ is contained is the degree set of G₁ and the eccentricity set of G₂is contained in the eccentricity set of G₁.

From (2.2), (2.3)

1

G and G₂have the same degree set and the same eccentricity set.

Definition 3.4. Let G :(σ,µ)be a fuzzy graph. A fuzzy subgraph H of G is isometric if

)

,

(

)

,

(

u

v

d

u

v

d

_H

=

_G every

u

,

v

∈

H

Definition 3.5. A fuzzy graph G :(

σ

,

µ

)is distance preserving if it has an isometric

fuzzy subgraph with each possible number of vertices upto V .

Theorem 3.6.If G :(

σ

,

µ

)is a fuzzy graph on a tree, then G is a distance preserving fuzzy graph.

Proof: Let G :(

σ

,

µ

) be a fuzzy graph on a tree withn vertices. In a tree, there exists a unique path between any two vertices.

If

G

1 is a fuzzy subgraph obtained from G by removing a verter of degree 1 in

* G then

using sum distance

(

,

)

(

,

)

1

u

v

d

u

v

d

_G

=

_G ∀u,v∈v(G₁)

If

G

₂is a fuzzy subgraph obtained from G₁ by removing a vertex of degree 1 in

G

₁*

then

(

,

)

(

,

)

2

u

v

d

u

v

K.Radha and P.Indumathi

384

This procedure gives an isometric fuzzy subgraph

G

_i of a G with

V

−

i

,

,

1

,...,

2

,

1

−

−

V

i

number of vertices.

Hence Gis a distance preserving fuzzy graph. 4. Conclusion

The conditions obtained for isometric fuzzy graphs the concepts of isometric fuzzy subgraphs and distance preserving fuzzy graphs will be helpful in obtaining many more properties of isometric fuzzy graphs.

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