On Matrices associated with L-Fuzzy Graphs · -fuzzy graphs and lattice matrices that will be...

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1799-1810

© Research India Publications

http://www.ripublication.com

On Matrices associated with L-Fuzzy Graphs

Pramada Ramachandran

P. G. Department of Mathematics, St. Paul’s College, Kalamassery Kochi-683 503, Kerala, India.

K. V. Thomas

P. G. Department of Mathematics & Research Centre Bharata Mata College, Thrikkakara

Kochi-682 021, Kerala, India.

Abstract

In this paper, we define different lattice matrices associated with an L -fuzzy

graph (LFG), namely the L -fuzzy degree matrix, L -fuzzy incidence matrix

and L -fuzzy adjacency matrix.We show that the join and meet of L -fuzzy

adjacency matrices of two LFGs also yield L -fuzzy adjacency matrices. We

prove that the diagonal entries of the composition of the L -fuzzy adjacency

matrix with itself are exactly the membership degrees of the corresponding

vertices. Finally, we establish the relation between the L -fuzzy incidence

matrix and L -fuzzy adjacency matrix of an LFG and show that the L -fuzzy

adjacency matrices of isomorphic LFGs are related.

Keywords: L -fuzzy graphs, lattice matrix, isomorphic LFGs, L -fuzzy

adjacency matrix, L -fuzzy incidence matrix

Mathematics Subject Classification. 05C50, 05C72

1. INTRODUCTION

Graph Theory was first introduced to the world by Leonhard Euler in 1736. It was

quickly accepted as one of the most convenient ways to model relationships between

objects and to study them. The field has seen tremendous growth.

1800 Pramada Ramachandran and K. V. Thomas

The concept fuzzy graphs was introduced by Azriel Rosenfeld in 1975 [11].Today,

fuzzy graphs are the basic mathematical structure in such areas of research that include

clustering analysis, group structure, database theory [8] , control systems [1] and even

decision theory [3].The work of Mordeson and Malik [6] prompted the authors to

introduce the concept of L -fuzzy graphs (LFGs) in [9] . Different types of

isomorphisms on LFGs, their properties and the complement of an LFG were studied.In

[10], the strong product of LFGs was introduced and some of its properties were

studied. The concept of lattice matrices appeared first in 1964 [5] and they have been

studied using the graph theoretic approach [2].In [12],work has been done on the

characteristic roots of lattice matrices.

This paper aims to define and study certain lattice matrices associated with an LFG.The

book [8] is a primary reference in this paper. Throughout this work,we choose ’L’ to be

a finite complete lattice with least element ’0’and greatest element ’1’, (L, , , 0, 1),

with partial order , unless otherwise stated.For all fundamental results in graph

theory, we refer to [3] and for all results regarding lattices, we refer [6].In section 2 of

this paper, we list basic definitions from the theory of fuzzy graphs, L -fuzzy graphs

and lattice matrices.In section 3, we define the L -fuzzy degree matrix, L -fuzzy

adjacency matrix and L -fuzzy incidence matrix of an LFG and arrive at the main

results.We conclude in section 4 with a summary of the work done and possible future

work.

2. PRELIMINARIES

In this section, we review some basic definitions and results of fuzzy graph theory, L-fuzzy graphs and lattice matrices that will be needed in the sequel.

Definition 2.1. [9] An L -fuzzy graph (LFG) ),,(= VGL with the underlying set

V is a nonempty set V together with a pair of functions LV : and

LVV : such that

( , ) ( ) ( ), u,v in V.u v u v

* and * , respectively denote the supports of and . The underlying graph of

the LFG ),,(= VGL is the crisp graph ),(= ** G .

Throughout this paper, we choose L -fuzzy graphs whose underlying graphs are finite,

simple graphs [3], ie; without loops or parallel edges.

Definition 2.2. [9] The LFG ),,(= VGL is said to be strong if it satisfies the

condition

*( , ) = ( ) ( ), u,v in .u v u v

On Matrices associated with L-Fuzzy Graphs 1801

Definition 2.3. [9] The LFG ),,(= VGL is said to be complete if it satisfies the

condition

( , ) = ( ) ( ), u,v in V.u v u v

Definition 2.4. [9] By an isomorphism of LFGs ),,(= 1111

1 VGL and

),,(= 2222

2 VGL, we mean a bijective mapping 21: VVh together with a bijective

mapping 21: LLl such that

)]([=)]([ 21 uhul

and

1 2 1[ ( , )] = [ ( ), ( )], u,v in .l u v h u h v V

Symbolically, we write 22

11

LL GG . When 21 = LL , l becomes the identity map.

Definition 2.5. [9] An LFG ),,(= VH L is said to be a partial fuzzy subgraph of the

LFG ),,(= VGL if ( ) ( ) and ( , ) ( , ), , inu v u v u v u v V

.

Definition 2.6. [9] Consider the LFG ),,(= VGL . We define the order ‘ p ’ and

size ‘ q ’ of LG as ),(=and)(=,

vuqupVvuVu

.

Definition 2.7 .[9] Consider the LFG ),,(= VGL . Then we define the degree ‘

)(ud ’ of a vertex ‘u ’ in LG as

),(=)(,

vuudVvuv

.

Definition 2.8. [5] A lattice matrix is a matrix whose entries are from a lattice L .We

denote the set of all such nm matrices by )(LM nm and by )(LM n if only square

matrices of order n are considered.An element of this set is denoted by nmijaA ][= .

Definition 2.9 .[5] Let )(LM n be the set of all nn matrices over L (Lattice

Matrices). We shall denote by ija the element of L which stands in the th),( ji entry

of )(LMA n .For )(][=,][=,][= LMcCbBaA nnnijnnijnnij , we define


a) ),1,2,=,( njibaBA ijij

b) The join ),1,2,=,(== njibacCBA ijijij

c) The meet ),1,2,=,(== njibacCBA ijijij

d) The composition ),1,2,=,(==1=

njibacCBA kjiknkij

e) The transpose ),1,2,=,(== njiacCA jiijT

f) The complement ),1,2,=,(,== njiacCA Cjiij

C , provided L is

complemented

g) The identity

ji

jiiiI ijnnijn

if0,

=if1,=,][=

3. THE MATRICES ASSOCIATED WITH LFGS

In this section,a vertex of the underlying graph is denoted by iu and the edge between

iu and ju by .juiue We begin by introducing the (lattice) matrices associated with

LFGs and see some of their properties.

Consider the LFG ),,(= VGL with underlying graph ),( ** such that || * =n

and || * =m. We enlist the elements of nuuu ,.....,,as 21

* and those of

meee ,.....,,as 21

* .

Definition 3.1. The L -fuzzy degree matrix of ),,(= VGL is the lattice matrix with

rows and columns corresponding to nuuu ,.....,, 21 .It is denoted by nnijdD ][=' where

= ( ),if i=jij id u

= 0 otherwise

Definition 3.2. The L -fuzzy incidence matrix of ),,(= VGL is the lattice matrix

with rows corresponding to nuuu ,.....,, 21 and columns corresponding to meee ,.....,, 21

.It is denoted by mnijcC ][=' where

),(= jij ec if the jth edge has one end ui

= 0 otherwise


Definition 3.3. The L -fuzzy adjacency matrix of ),,(= VGL is the lattice matrix

with rows and columns corresponding to nuuu ,.....,, 21 .It is denoted by nnijaA ][='

where

0,=ija if i = j

),(=juiue if i ≠ j

We note that these matrices can change on a change in the labeling of the vertices or

edges.

Illustrated below in figures 1 and 2 are a lattice and an associated LFG respectively.The

labels of the elements in the lattice are indicated within them. In the LFG, the

membership degrees of the vertices are indicated within them and those of the edges by

their sides:

Figure 1 Figure 2

Let the edges 52

,5143133221

,,,, uuuuuuuuuuuu eeeeee be respectively labeled

654321 ,,,,, eeeeee .


Then the associated L - fuzzy matrices are:

D =

10000

0000

00100

0000

0000

a

ab

, C =

cabbcb

cbcacc

0000

00000

000

000

000

and A =

000

0000

00

00

00

cab

bbccbcacc

As a direct consequence of these definitions, we have the following observations:

Observation 3.4. Let ACD and, respectively be the L -fuzzy degree matrix, the L-fuzzy incidence matrix and the L -fuzzy adjacency matrix of the LFG ),,(= VGL

. Then

1. D and A are symmetric matrices.

2. Each column of C can have exactly two equal non zero entries- the membership

degree

of the edge in the rows corresponding to its end vertices.

3. A column in D or A with all entries 0 indicates an isolated vertex.

4. The join of all the entries in a column ( or row ) of A gives the degree of the

corresponding vertex.

5. The join of all elements in A gives the size of the LFG.

6. The join of the diagonal entries of D gives the order of the LFG.

7. Given any square symmetric lattice matrix of order 'n' with diagonal entries 0, we

can construct an LFG whose L- fuzzy adjacency matrix is the same.

Next, we define the’ intersection’ and ’union’ of two LFGs .We recall the definition of

union and intersection of two fuzzy sets from [8].

Definition 3.5. Let ),,(= 1111 VGL and ),,(= 2222 VGL be any two LFGs . We

define the ( L -fuzzy) ’intersection’ and the ( L -fuzzy) ’union’ of these LFGs as

follows:

The ’intersection’ ),,(=)( 21212121 VVGG L

The ’union’ ),,(=)( 21212121 VVGG L


The following theorem shows that the union and intersection are also LFGs:

Theorem 3.6. The ’intersection’ and ’union’ of two LFGs as defined above are also

LFGs.

Proof. We shall show that the conditions for the L - fuzzy sets to be an LFG are

satisfied for each kind of edge:

Consider the underlying graph ),( 2121 EEVV of LGG )( 21 .

Then for each ,21 EEejuiu we have

)()(=)( 2121 juiujuiujuiu eee

)]()([)]()([ 2211 jiji uuuu by the definition of LFGs

)]()([)]()([= 2121 jjii uuuu by the commutativity and associativity of

lattice elements

)]([])([= 2121 ji uu

Hence, 21212121 )]([)]([)( EEeuuejuiujijuiu

Thus, the intersection of two LFGs is also an LFG.

Next,consider the underlying graph ),( 2121 EEVV of LGG )( 21 .

Then for each ,21 EEejuiu we have

)()(and)()( 2211 ijuiuijuiu ueue , by the property of LFGs

Hence

)(=)()()(()((=)(( 21211121 iiijuiujuiujuiu uuueee

Thus, )()(( 2121 ijuiu ue Similarly, )()(( 2121 jjuiu ue

Together, by the property of lattices,

)()()(( 212121 jijuiu uue

Thus, the union of two LFGs is an LFG.


This leads us to:

Proposition 3.7. Let ),,(= 111 VGL and ),,(= 222 VGL be two LFGs with the

same underlying graph with adjacency matrices. Let their L - fuzzy adjacency matrices

be 1A and 2A respectively. Then 21 AA and 21 AA are both the adjacency

matrices of LFGs.

Proof. Let ][=and][= 2

2

1

1 ijij aAaA . Then by definition,

].[=and][= 21

21

21

21 ijijijij aaAAaaAA

Consider the LFGs LL GG 21 and LL GG 21 . By theorem 3.2, these LFGs have the

adjacency matrices 21 AA and 21 AA respectively .

Remark 3.8. 21 GG has the same underlying graph as ),,(= 111 VGL and

),,(= 222 VGL . If L is a regular lattice, ie; 000,,, babaLba , then

21 GG also has the same underlying graph.

Hence,unlike in the ordinary fuzzy case, where the sum or difference of adjacency

matrices do not yield adjacency matrices, the join and meet of L -fuzzy adjacency

matrices yield L -fuzzy adjacency matrices.

Next, we consider BA , where A and B are L -fuzzy adjacency matrices. Obviously,

this composition need not even be symmetric, so it does not usually represent the L -

fuzzy adjacency matrix of an LFG.However, we do have:

Theorem 3.9. Let ),,(= 111 VGL and ),,(= 222 VGL be any two LFGs with the

same underlying graph ),(= EVG . Let their L -fuzzy adjacency matrices be 1A and

2A respectively. Then if constantanotherkconstantak ,= and ,= 2211 , then C

nIAA )( 21 is the L - fuzzy adjacency matrix of .21

LL GG

Proof. We have 1A =

0

0

0

111

111

111

kkk

kkkkkk

and 2A =

0

0

0

222

222

222

kkk

kkkkkk

.


Then

21 AA =

212121

212121

212121

kkkkkk

kkkkkkkkkkkk

and

CnIAA )( 21 =

0

0

0

212121

212121

212121

kkkkkk

kkkkkkkkkkkk

Now consider the LFG ),,(= 212121 VGG LL .

Then, Eejuiu , we have

)()(=)( 2121 juiujuiujuiu eee

21= kk

Obviously, this LFG has the L -fuzzy adjacency matrix CnIAA )( 21 .

Theorem 3.10. Let ),,(= VGL have the L -fuzzy adjacency matrix A . Then, the

diagonal entries of AA give the degrees of the corresponding vertices.

Proof. Let A =

0

0

0

321

22321

11312

nnn

n

n

aaa

aaaaaa

. Then AA = nnijb ][ , where

)(=1= kjik

nkij aab

)(=Now,1=

kiik

n

kii aab

kiik1=

a=a)(= ikik

n

kaa by symmetry

)(=1=

ik

n

ka


0=a)(= ii1=,

ik

n

kika

by definition

)(=1=,

kuiu

n

kike

by definition

)(= iud by definition

Finally, we have the following theorem:

Theorem 3.11. Let ),,(= 1111 VGL and ),,(= 2222 VGL be any two LFGs with

isomorphic underlying graphs. Let their L -fuzzy adjacency matrices be 1A and 2A

respectively. Then if the LFGs are isomorphic, one lattice matrix can be obtained from

the other by permutations of rows and columns.

Proof. Suppose ),,(= 1111 VGL and ),,(= 2222 VGL are isomorphic. Then by

definition, there exists a bijection h' between 1V and 2V that preserves the

membership degrees of the vertices and edges.Hence, 21 and AA are of the same order.

Let the vertices of LG1 be labeled nuuu ,....,, 21 . Then the row(/column) corresponding

to iu in 1A is identical to the row(/column) corresponding to )( iuh in 2A . Further,

the bijection ensures that this correspondence is exhaustive and unique. Thus 1A can

be obtained from 2A by permutations of rows and columns.

Remark 3.12. The converse of the above theorem need not be true.For, even if 1A can

be obtained from 2A by permutations of rows and columns, the LFGs can differ in the

membership degrees of the vertices, though not inthe edges.

This is illustrated in the following example:

Let the lattice L' be as in figure 1. Then the LFGs in figure 4 and figure 5 have the

same L - fuzzy adjacency matrix A =

00

00

00

00

babc

cbab

. However,it is not possible to

construct an isomorphism between the LFGs.


Figure 4 Figure 5

4 CONCLUSION

We have defined in this paper the L -fuzzy degree matrix, L -fuzzy incidence matrix

and L -fuzzy adjacency matrix of an LFG. Certain properties of these matrices have

been proved.Using these as basic building blocks,we intend to pursue this line of work

to make studies on the energy of an LFG and its spectrum.

Acknowledgement. The first author is thankful to the UGC for the award of Teacher

Fellowship under the XII Plan.

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