An Introduction to Fuzzy strong graphs, Fuzzy soft graphs ... · computer science applications,...

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2235-2254

© Research India Publications

http://www.ripublication.com

An Introduction to Fuzzy strong graphs, Fuzzy soft

graphs, complement of fuzzy strong and soft graphs

Dr. K. Kalaiarasi1 and L.Mahalakshmi2

1PG and research Department of Mathematics, Cauvery College for Women,

Trichy-18, Tamil Nadu, India.

2PG and research Department of Mathematics, Cauvery College for Women,

Trichy-18, Tamil Nadu, India.

Abstract

In this paper, we introduce some new concepts of fuzzy soft graphs with the

notations of complement and µ-complement fuzzy soft graphs introduced.

Investigating some of their applications, we show that the complement of

strong fuzzy soft graph is strong fuzzy soft graph as well as the complement of

a complete fuzzy soft graph is complete fuzzy soft one .Finally, we state and

prove some results related to these concepts.

Keywords: Graph, Fuzzy graph, Fuzzy soft graph, complement of fuzzy soft

graph, complete fuzzy soft graph, strong fuzzy soft graph, Isolated fuzzy soft

graph, Pseudo fuzzy soft graph, simple fuzzy soft graph.

INTRODUCTION:

The origin of the graph theory started with the Konignberg bridge problem in

1735.This problem led to the concept of the Eulerian graph. Euler studied the

Konignberg bridge problem and constructed a structure that solves the problem that is

referred to as an Eulerian graph. In 1840 Mobius proposed the idea of a complete

2236 Dr. K. Kalaiarasi and L.Mahalakshmi

graph and a bipartite graph and kuratowski proved that they are planar by means of

recreational problems. currently, concept of graph theory are highly utilize by

computer science applications, especially in area of research, including data mining

image segmentation, clustering and net working.The Introduction of fuzzy set by

Zadeh[20] in 1965 greatly changed the face of science and Technology. Fuzzy sets

paved the way for a new method of philosophy thinking, “Fuzzy logic” Which is now

an essential concept in artificial intelligence. The most important features of a fuzzy

set is that it consists of a class of objects that satisfy a certain property or several

properties .In 1999 Molodtsov[15] introduced the concept of soft set theory to solve

imprecise problem in the field of engineering, social science, economics, medical

science and environment. In recent times, a number of research studies contributed

into fuzzification of soft set theory. As a result many research were more active doing

research of soft set.

In 1999 ,Molodtsov [15]introduced the concept of soft set theory to solve imprecise

problems in the field of engineering, social science, economics, medical science and

environment. Molodtsov[15]applied this theory to several directions such as

smoothness of function, game theory, operation research, probability and

measurement theory. In resent times, a number of research studies contributed into

fuzzification of soft set theory. As a result ,many researchers were more active doing

research on soft set. In 2001 Maji et al [12,13] initiated the concept of fuzzy soft sets

which is a combination of fuzzy set and soft set .In 1975 Rosenfeld[17] introduced the

concept of fuzzy graph theory. In 1994,Moderson introduced the concept of

complement of fuzzy graphs, and in 2002,sunitha and vijayakumar [19]gave a

modified definition of complement of fuzzy graph.

In 2015,Mohinta and Samantv [14] introduced the fuzzy soft graph, union,

intersection of two fuzzy soft graph with a few properties related to finite union and

intersection of fuzzy soft graphs. Akram and Nawaz[1,2,4] introduced the notations of

fuzzy soft graph, strong fuzzy soft graph, complete fuzzy soft graph, regular fuzzy

soft graph and investigated some of their properties. Akram and Nawaz[1] developed

the concepts of soft graphs, vertex-induced soft graphs, edge-induced soft graphs and

describe some operations on soft graphs.

In 2016,Akram and Nawaz[2] presented concept of fuzzy soft graphs, certain types of

irregular fuzzy soft graphs and described application of fuzzy soft graphs in social

network and road network. Akram and Zafar[3] introduced notations of fuzzy soft

cycles.

In this paper, we introduce some new concepts of fuzzy soft graphs, complement

fuzzy soft graphs, and some applications of µ-complement fuzzy soft graphs.

An Introduction to Fuzzy strong graphs, Fuzzy soft graphs, complement… 2237

2. PRELIMINARIES:-

Definition:2.1

Let U be an initial universe set and E be the set of parameters. Let )(UP denotes

the power set of U . A pair ),( EF is called a soft set over U where F is a mapping

given by )(: UPEF .

Definition:2.2

Let V { nxxx ,......, 21 } non empty set. E (parameters set) and EA also let

(i) )(: VFA (collection of all fuzzy subsets in V )

eee )( (say)

and e )(:],1,0[: ie XXV

),( A : Fuzzy soft vertex.

(ii) )(: VVFA (collection of all fuzzy subsets in )( VV

eee )( (say)

and ]1,0[: vve

),(),( jieji xxxx

),( A : Fuzzy soft edge.

Then )),(),,(( AA is called a fuzzy soft graph if and only if

)()(),( jeiejie xxxx for all Ae and for all nji ,....,2,1, and this fuzzy

soft graph is denoted by VAG , .

Definition:2.3

A fuzzy soft graph VAG , = )),(),,(( AA is called a strong fuzzy soft graph if

)()(),( jeiejie xxxx for all Aexx ji ,),( and is complete fuzzy soft

graph if )()(),( jeiejie xxxx for all Aexx ji ,),( .


Definition:2.4

Let )),(),,((, AAG VA be a fuzzy soft graph. Then the order of VAG , is defined

as:

)( ,VAGO = ))(( i

Ae Vx

e xi

and the size of VAG , is defined as:

)( ,VAGS = )),(( j

Ae Vxx

ie xxji

Definition:2.5

Let )),(),,((, AAG VA be a fuzzy soft graph. The degree of a vertex u is defined

as:

)(,

udVAG = )),((

,

vuAe vuVv

e

Definition:2.6

A Fuzzy soft edge joining a fuzzy soft vertex to itself is called a fuzzy soft loop.

Definition:2.7

Let )),(),,((, AAG VA be a fuzzy soft graph. If for all Ae there is more than

one fuzzy soft edge joining two fuzzy soft vertices,then the fuzzy soft graph is called

a fuzzy soft pseudo graph and these edges are called fuzzy soft multiple edges.

Definition:2.8:

Let )),(),,((, AAG VA is called a fuzzy soft simple graph if it has neither fuzzy

soft loops nor fuzzy soft multiple edges for all .Ae

Definition:2.9:

Let )),(),,((, AAG VA be a fuzzy soft graph. Then VAG , is called isolated fuzzy

soft graph if 0),( jie xx for all .,, Aevvxx ji


Definition:2.10

An edge in a fuzzy soft graph )),(),,((, AAG VA is said to be an effective fuzzy

soft edge if )()(),( jeiejie xxxx for all .,, Aevvxx ji

Definition:2.11

If two fuzzy soft vertices have a fuzzy soft edge joining them , then they are called

fuzzy soft adjacent vertices.And if two fuzzy soft edges are incident with a common

fuzzy soft vertex, then they are called fuzzy soft adjacent edges.

Definition:2.12

Let )),(),,((, AAG VA be a fuzzy soft graph. The complement of VAG , is

defined as �̅�A,V where ),( jie xx .,,),,()()( Aevxxxxxx jijiejeie .The

order of �̅�A,V is equal to the order of VAG , .

Definition:2.13

Let )),(),,((, AAG VA be a fuzzy soft graph. The - complement of VAG , is

defined as )),(),,((, AAG VA , where ),()()(),( jiejeiejie xxxxxx

when ),( jie xx 0 and ),( jie xx 0 when ),( jie xx 0 , ., ji xx

Proposition 3.1

S (�̅�A,V)+ )( ,VAGS 2 )()( je

Ae xx

ie xxji

Proof:

(i) From the definition of �̅�A,V, (O �̅�A,V )=

)( ,VAGO

(ii) ),( jie xx )( ie x )( je x AeVxx ji ,,

),( jie xx )( ie x )( je x - ),( jie xx AeVxx ji ,,

Also,

),( jie xx )( ie x )( je x AeVxx ji ,,


From above two equations we have,

),( jie xx + ),( jie xx 2( )( ie x )( je x ) AeVxx ji ,,

),(),(( jiji

Ae xx

e xxxxji

2 )()( je

Ae xx

ie xxji

),(( ji

Ae xx

e xxji

Ae xx

jie

ji

xx ),( 2 )()( je

Ae xx

ie xxji

Hence, S (�̅�A,V)+ )( ,VAGS 2 )()( je

Ae xx

ie xxji

3. EXAMPLES:

Example 3.1:

Consider a fuzzy soft graph VAG , , where },......,{V 21 nxxx and

},......,,{ 21 neeeE . Here VAG , described by Table3.1 and ),( jie xx 0

)},(),,(),,(),,/{(),( 41313221 xxxxxxxxvvxx ji for all Ee .

Table 3.1:

It represent of a fuzzy soft graph.

1x 2x 3x

4x

1e 0.1 0.5 0.6 0.7

2e 0.3 0.8 0.2 0.4

3e 0.5 0.1 0.6 0.3

4e 0.2 0.5 0.7 0.1

µ ),( 21 xx ),( 32 xx ),( 43 xx ),( 14 xx

1e 0.1 0.1 0.2 0.3

2e 0.2 0.5 0.4 0.1

3e 0.4 0.7 0.3 0.2

4e 0.8 0.6 0.2 0.4


Table-3.1: Tabular Representation of a fuzzy soft graph

The order of fuzzy soft graph GA,V is

)( ,VAGO = ))(( i

Ae Vx

e xi

= )()()()( 4321 xxxx eee

Ae

e

= (0.1+0.5+0.6+0.7)+(0.3+0.8+0.2+0.4)+(0.5+0.1+0.6+0.3)+(0.2+0.5+0.7+0.1)

= 1.9+1.7+1.5+1.5

)( ,VAGO = 6.6

The size of fuzzy soft graph GA,V is


)( ,VAGS = )),(( j

Ae Vxx

ie xxji

= ),(),(),(),( 32413121 xxxxxxxx eee

Ae

e

= (0.1+0.1+0.2+0.3)+(0.2+0.5+0.4+0.1)+(0.4+0.7+0.3+0.2)+(0.8+0.6+0.2+0.4)

= 0.7+1.2+1.6+2.0

)( ,VAGS = 5.5

The degree of a vertex x1 is

)(,

udVAG = )),((

,

vuAe vuVv

e

)( 1,xd

VAG = ),(),(),({ 413121 xxxxxx ee

Ae

e

= (0.1+0.1+0.2)+(0.2+0.5+0.4)+(0.4+0.7+0.3)+(0.8+0.6+0.2)

= 0.4+1.1+1.4+1.6

)( 1,xd

VAG = 4.5

Example 3.2:

Consider a fuzzy soft graph VAG , , where },......,{V 21 nxxx and

},......,,{ 21 neeeE . Here VAG , described by Table3.1 and ),( jie xx 0

)},(),,(),,(),,(),,(),,/{(),( 114131323121 xxxxxxxxxxxxvvxx ji for all Ee .

1x 2x 3x 4x

1e 0.1 0.5 0.6 0.7

2e 0.2 0.8 0.1 0.2

3e 0.3 0.5 0.6 0.8

4e 0.4 0.1 0.1 0.1


µ ),( 21 xx ),( 31 xx ),( 32 xx ),( 31 xx ),( 41 xx ),( 11 xx

1e 0.1 0.1 0.2 0.3 0.4 0.4

2e 0.2 0.5 0.4 0.1 0.2 0.3

3e 0.4 0.7 0.3 0.2 0.2 0.1

4e 0.5 0.6 0.8 0.7 0.6 0.8

Table 3.2:Tabular Representation of a fuzzy soft graph

Similarly we draw e2,e3,e4. The above figure contains a fuzzy soft loops

corresponding to parameters. It also contains fuzzy soft multiple edges corresponding

to the parameter.

Therefore VAG , is a pseudo fuzzy soft graph but it is not simple fuzzy soft graph.

Example 3.3:

Consider the fuzzy graph )),(),,((, AAG VA in example 3.1.Thus the complement

of GA,V.(i.e) �̅�A,V = )),(),,(( AA is shown in the below figure.


The order of fuzzy complement soft graph.(i.e) �̅�A,V is

(O �̅�A,V )= ))(( i

Ae Vx

e xi

= )()()()( 4321 xxxx eee

Ae

e

= (0.1+0.7+0.5+0.6)+(0.3+0.8+0.2+0.4)+(0.5+0.3+0.6+0.1)+(0.2+0.5+0.7+0.1)

= 1.9+1.7+1.5+1.5

(O �̅�A,V ) = 6.6

The size of fuzzy complement soft graph.(i.e) �̅�A,V is

S (�̅�A,V) = )),(( j

Ae Vxx

ie xxji

= ),(),(),(),( 32413121 xxxxxxxx eee

Ae

e

= (0.1+0.1+0.2)+(0.4+0.3+0.2)+(0.6+0.2)+0

S (�̅�A,V) = 2.1

The degree of a vertex 1x is


)( 1,

xdVA

G = )),((

,

vuAe vuVv

e

= ),(),(),({ 413121 xxxxxx ee

Ae

e

= (0.1+0.1+0.2)+0+(0.7+0.3)+(0.7+0.3)

= 0.4+1.0+1.0

)( 1,

xdVA

G = 2.4

Example 3.4:

Consider the fuzzy graph )),(),,((, AAG VA in example 3.1.And its complement

fuzzy soft graph �̅�A,V in example 3.3.

Then (O �̅�A,V )=

)( ,VAGO =6.7

Also S (�̅�A,V)=2.1 ,

)( ,VAGS =5.5

So S (�̅�A,V)+ )( ,VAGS =2.1+5.5=7.6

Now,

))}()(())()(())()(())()({([2)()( 32413121 xxxxxxxxxjxiAe

e

Ae xx

e

ji

=2{(0.1+0.1+0.1+0.5)+(0.3+0.2+0.3+0.2)+(0.1+0.5+0.3+0.1)+(0.2+0.5+0.1+0.2)}

=2{0.8+1.0+1.0+1.0}

=7.6

Hence,

S (�̅�A,V)+ )( ,VAGS 2 )()( xjxi e

Ae xx

e

ji


Example:3.5

Consider a fuzzy soft graph VAG , , where },......,{V 21 nxxx and },......,,{ 21 neeeE .

Here VAG , described by Table3.3 and ),( jie xx 0

)},(),,(),,(),,(),,(),,(),,/{(),( 14544362323121 xxxxxxxxxxxxxxvvxx ji for all

Ee .

1x 2x 3x 4x 5x 6x

1e 1 0.8 0.6 0.4 0.2 0.1

2e 0.3 0.4 0.5 0.6 0.7 0.8

3e 0.1 0.5 0.9 1 0.3 0.2

4e 0.5 0.2 0.6 0.8 0.7 0.4

µ ( ), 21 xx ( ), 32 xx ( ), 62 xx ( ), 43 xx ( ), 54 xx ( ), 14 xx ( ), 31 xx

1e 0.3 0.1 0.3 0.4 0.8 0.7 0.9

2e 0.4 0.5 0.2 0.2 0.6 0.5 0.8

3e 0.8 0.7 0.8 0.1 0.4 0.3 0.5

4e 0.6 0.9 0.1 0.3 0.2 0.1 0.6


Table 3.3 Representation of a fuzzy soft graph


The -complement

VAG , of a fuzzy soft graph VAG , shown as in below:

4. MAIN RESULTS:

Theorem:4.1

The complement of a strong fuzzy soft graph is also strong fuzzy soft graph

Proof:

Let GA,V be a strong fuzzy soft graph. By definition of the complement:



= )( ie x )( je x - )( ie x )( je x , ),( jie xx >0

)( ie x )( je x , ),( jie xx =0

= 0 , ),( jie xx >0

= )( ie x )( je x , ),( jie xx =0

= 0, ),( jie xx 0

),( jie xx 0

),( jie xx )(xie ^ )(xje , ),( jie xx 0 AeVxx ji ,,

Where ji xx , denoted the edge for all ji xx , .

Theorem:4.2

The complement of a complete fuzzy soft graph is is also a complete fuzzy soft

graph.

Proof:

Let VAG , be a strong fuzzy soft graph.By definition of the complement:


= )( ie x )( je x - )( ie x )( je x , ),( jie xx >0

)( ie x )( je x , ),( jie xx =0

= 0 , ),( jie xx >0

= )( ie x )( je x , ),( jie xx =0

= 0, ),( jie xx 0

),( jie xx 0

),( jie xx )(xie ^ )(xje , ),( jie xx 0 AeVxx ji ,,

Where ji xx , denoted the edge for all ji xx , .

Thus the complement of a complete fuzzy soft graph is also a complete fuzzy soft

one.


Theorem:4.3

Let GVA ,

={( )},(),, to be a fuzzy soft graph. Then GVA ,

is an isolated fuzzy

soft graph if and only if �̅�A,V is a complete fuzzy soft graph.

Proof:

Given GVA ,

={( )},(),, to be a fuzzy soft graph.Let �̅�A,V be its complement.

Given GVA ,

to be isolated fuzzy soft graph.Then,

),( jie xx =0 AeVVxx ji ,,

Since

),( jie xx )( ie x )( je x - ),( jie xx AeVVxx ji ,,

),( jie xx )( ie x )( je x , AeVVxx ji ,,

Hence �̅�A,V to be a complete fuzzy soft graph.

Conversely,

Given �̅�A,V to be a complete fuzzy soft graph.

),( jie xx )( ie x )( je x , AeVVxx ji ,,

Since



= ),( jie xx - ),( jie xx AeVVxx ji ,,

=0 AeVVxx ji ,,


Hence, GVA ,

is an isolated fuzzy soft graph.

Theorem:4.4

Let GVA ,

={( )},(),, to be a fuzzy soft graph. Then GVA ,

is an isolated fuzzy

soft graph if and only if �̅�A,V is a strong fuzzy soft graph.


Proof:

Given GVA ,

={( )},(),, to be a fuzzy soft graph.Let �̅�A,V be its complement.

Given GVA ,

to be isolated fuzzy soft graph.Then,


Since


),( jie xx )( ie x )( je x AeVVxx ji ,,

Hence �̅�A,V to be a Strong fuzzy soft graph.

Conversely,

Given �̅�A,V to be a strong fuzzy soft graph.

),( jie xx )( ie x )( je x AeVVxx ji ,,

Since


),( jie xx = )( ie x )( je x - ),( jie xx AeVVxx ji ,,

= ),( jie xx - ),( jie xx AeVVxx ji ,,

=0 AeVVxx ji ,,


Hence, GVA ,

is an isolated fuzzy soft graph.

Theorem:4.5

Let GVA ,

={( )},(),, to be a fuzzy soft graph.Then G

VA, has isolated nodes if

and only if GVA ,

is a strong fuzzy soft graph.

Proof:

Let GVA ,

be a strong fuzzy soft graph.

(i.e) ),( jie xx = )( ie x )( je x Aexx ji ,,


Otherwise

),( jie xx =0

By definition of ),( jie xx and by above equations


Hence, G

VA, is an isolated fuzzy soft graph.

Conversely,

Assume G

VA, be an isolated fuzzy soft graph.


),( jie xx =0 Aexxandxx jiji ,),(,

By definition if ji xx , , ),( jie xx = )(xie ^ )(xje - ),( jie xx

Since ),( jie xx =0

0= )(xie ^ )(xje - ),( jie xx ji xx , , Ae

),( jie xx = )(xie ^ )(xje ji xx , , Ae

Hence, GVA ,

={( )},(),, is a strong fuzzy soft graph.

Theorem:4.6

Let GVA ,

={( )},(),, to be a fuzzy soft graph.Then G

VA, has isolated nodes if

and only if GVA ,

is a complete fuzzy soft graph.

Proof:

Let GVA ,

be a complete fuzzy soft graph.

(i.e) ),( jie xx = )(xie ^ )(xje Aexx ji ,,

Otherwise

),( jie xx =0


By definition of ),( jie xx and by above equations


Hence, G

VA, is an isolated fuzzy soft graph.

Conversely,

Assume G

VA, be an isolated fuzzy soft graph.


),( jie xx =0 Aexxandxx jiji ,),(,

By definition if ji xx , , ),( jie xx = )(xie ^ )(xje - ),( jie xx

Since ),( jie xx =0

0= )(xie ^ )(xje - ),( jie xx ji xx , , Ae

),( jie xx = )(xie ^ )(xje ji xx , , Ae

Hence, GVA ,

={( )},(),, is a complete fuzzy soft graph.

5. CONCLUSION:

In this paper, basic definitions of fuzzy soft graph, complement and complement

of fuzzy soft graphs are introduced with some of their properties.Also, some results

about strong fuzzy soft graphs,complete fuzzy soft graph and isolated fuzzy soft graph

with their complements constructed.We plan to extend our research of fuzzification to

weak complementary and complement weak fuzzy soft graphs and its

applications.

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