Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—9

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Pacific Science Review A: Natural Science and Engineering

journal homepage: www.journals.elsevier.com/pacific-science-review-a-natural-science-and-engineering/

Some properties of m-polar fuzzy graphs

Ganesh Ghorai*, Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721 102, India

ARTICLE INFO

ABSTRACT

Article history: Received 6 June 2016 Accepted 27 June 2016 Available online xxx

Keywords: m-polar fuzzy sets Generalized m-polar fuzzy graphs Isomorphisms

Strong and self-complementary m-polar fuzzy graphs

In many real world problems, data sometimes comes from n agents (n > 2), i.e., "multipolar information" exists. This information cannot be well-represented by means of fuzzy graphs or bipolar fuzzy graphs. Therefore, m-polar fuzzy set theory is applied to graphs to describe the relationships among several individuals. In this paper, some operations are defined to formulate these graphs. Some properties of strong m-polar fuzzy graphs, self-complementary m-polar fuzzy graphs and self-complementary strong m-polar fuzzy graphs are discussed.

Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The origin of graph theory started with the Konigsberg bridge problem in 1735. This problem led to the concept of the Eulerian graph. Euler studied the Konigsberg 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 graph and a bipartite graph and Kuratowski proved that they are planar by means of recreational problems. Currently, concepts of graph theory are highly utilized by computer science applications, especially in areas of computer science research, including data mining, image segmentation, clustering, and networking. The introduction of fuzzy sets by Zadeh [21] in 1965 greatly changed the face of science and technology. Fuzzy sets paved the way for a new method of philosophical thinking, "Fuzzy Logic" which is now an essential concept in artificial intelligence. The most important feature of a fuzzy set is that it consists of a class of objects that satisfy a certain property or several properties. In 1994, Zhang [24,25] initiated the concept of bipolar fuzzy sets. Juanjuan Chen et al. [1] introduced the notion of the m-polar fuzzy set as a

* Corresponding author. E-mail addresses: ghoraiganesh@gmail.com (G. Ghorai), mmpalvu@gmail.com (M. Pal).

Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University.

generalization of bipolar fuzzy sets. The first definition of fuzzy graphs was proposed by Kafmann [7] from Zadeh's fuzzy relations [21—23]. However, Rosenfeld [11] introduced another group of elaborated definitions, including the fuzzy vertex, fuzzy edges, and several fuzzy analogues of theoretical graph concepts, such as paths, cycles, connectedness, and so on. Mordeson and Nair [10] defined the complement of a fuzzy graph. McAllister [9] characterized fuzzy intersection graphs. Samanta and Pal studied fuzzy tolerance graphs [14], fuzzy threshold graphs [15], bipolar fuzzy hypergraphs [16], irregular bipolar fuzzy graphs [17], fuzzy k-competition graphs, m step fuzzy competition graphs [18,19], and fuzzy planar graphs [20]. Later, Rashmanlou et al. [12,13] studied bipolar fuzzy graphs with categorical properties and Ghorai and Pal introduced product bipolar fuzzy line graphs [3]. In 2014, Juanjuan Chen et al. [1] defined m-polar fuzzy graphs. Ghorai and Pal introduced some operations and the density of m-polar fuzzy graphs [2], studied m-polar fuzzy planar graphs [4] and defined faces and the dual nature ofm-polar fuzzy planar graphs [5]. In this paper the Cartesian product, composition, union and join of two m-polar fuzzy graphs are defined. Some important properties of isomorphisms, strong m-polar fuzzy graphs, self-complementary m-polar fuzzy graphs and self-complementary strong m-polar fuzzy graphs are discussed.

http://dx.doi.org/10.1016/j.psra.2016.06.004

2405-8823/Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

2. Preliminaries

In this section, we briefly recall some definitions of undirected graphs, the notions of fuzzy sets, bipolar fuzzy sets and m-polar fuzzy sets. For further reference, see Refs. [6,8,10].

Definition 2.1. [6] A graph is an ordered pair G* = (V, E), where V is the set of vertices of G* and E is the set of all edges of G*. Two vertices x and y in an undirected graph G* are said to be adjacent in G* if xy is an edge of G*. A simple graph is an undirected graph that has no loops and no more than one edge between any two different vertices. A subgraph of a graph G* = (V, E) is a graph H = (W, F), where W4 V and F4E.

Definition 2.2. [6] Let Gl = (V1, E1) and G'2 = (V2, E2) be two simple graphs.

The Cartesian product G* = Gl x G2 = (V, E) of graphs Gl and G2. Then, V = Vi x V2 andE = {(x, x2)(x, y2) : xeVl, x2 y2 eE2}u{(xl, z) (yl,z): zeV2,xiyi eEl}. * *

Then, the composition of the graph Gl with G2 is denoted by GlG] = (Vi x V2,E0), where * E0 = Eu{(xi,x2*)(yl,y2)*: xi*Vi eE1, x2 sy2} and E is defined in Gl x G2. Note that Gl [G2] sG^Gl].

The union of two simple graphs Gl = (Vl, El) and G2 = (V2, E2) is the simple graph with the vertex set Vl uV2 and edge set El uE2 . The union ofGl and G2 is denoted by G* = Gl uG2 = (VluV2, EluE2).

The join of two simple graphs Gl = (Vl, El) and G2 = (V2, E2) is the simple graph with the vertex set Vl uV2 and edge set El uE2 uE', where E is the set of all edges joining the nodes ofVl and V2 and assume that Vl nV2 = 0. The join of Gl and G2 is denoted by G* = Gl + G2 = (ViuV2, EluE2uE').

Definition 2.3. [l] Throughout the paper, [0,l]m (the m-power of [0,l]) is considered to be a poset with point-wise order <, where m is an natural number. < is defined by x <yo for each i = l,2,...,m; pi(x) < pi(y), where x, ye[0,l]m and p,:[0,l]m/[0,l] is the ith projection mapping.

An m-polar fuzzy set (or a [0,l]m-set) on X is a mapping A:X/ [0,l]m. The set of all m-polar fuzzy sets on X is denoted by m(X).

Definition 2.4. Let A and B be two m-polar fuzzy sets in X. Then, AuB and AnB are also m-polar fuzzy sets in X defined by: for i = l, 2, ...,m andxeX

pi + (AuB)(x) = max{pi+A(x),pi+B(x)} and pi + (AnB)(x) = min{pi+A(x),pi + B(x)}.A4B if and only if pi +A(x) < pi+B(x) and A = B if and only if pi+A(x)= pi+B(x).

Definition 2.5. Let A be an m-polar fuzzy set on a set X. An m-polar fuzzy relation on A is an m-polar fuzzy set B of X x X such that B(x,y) < min{A(x), A(y)} for all x, yeX, i.e., for each i = l, 2,...,m, for all x, yeX, pi+B(x,y) < min{pi+A(x),pi+A(y)}. An m-polar fuzzy relation B on X is called symmetric if B(x, y) = B(y, x) for all x, yeX.

We assume the following: For a given set V, define an equivalence relation ~ onV x V - {(x, x):xeV} as follows: (xl, yl)~(x2, y2)o either (xl, yl) = (x2, y2) orxl = y2 and yl = x2. The quotient set obtained in

this way is denoted by V2, and the equivalence class that contains the element (x, y) is denoted as xy oryx.

Throughout this paper, G* = (V, E) represents a crisp graph and G is an m-polar fuzzy graph of G*.

3. Generalized m-polar fuzzy graphs

Juanjuan Chen et al. [l] defined the m-polar fuzzy graph in the following way: An m-polar fuzzy graph with an underlying pair (V, E) (where E4V x V is symmetric) is defined to be a pairG = (A, B), where A:V/[0,l]m andB:E/[0,l]m, satisfying B(xy) < min{A(x), A(y)} for all xyeE.

According to the above definition, B is actually an m-polar fuzzy set in E4V x V. However, when the definition is used, B is actually an m-polar fuzzy set defined in V2, satisfying B(xy) = 0 = (0,0,...,0) for all xy e (V2 - E). The above definition will cause some problems by calculating the complement of an m-polar fuzzy graphs. Therefore, a generalized m-polar fuzzy graph is defined below.

Definition 3.1. A generalized m-polar fuzzy graph of a graph G* = (V, E) is a pair G = (V, A, B), where A: V/ [0,l]m is an m-polar fuzzy set in V and B : V2 / [0, l]m is anm-polar fuzzy set in V2 such that B(xy) < min {A(x), A(y)} for all xyeV2 and B(xy)=0 for all xye(V2 - E) (0 = (0, 0,... ,0) is the smallest element in [0,l]m ). A is called the m-polar fuzzy vertex set of G, and B is called the m-polar fuzzy edge set of G.

Example 3.2. Let X = {Fl, F2, F3, F4} and M = {Ml, M2, M3} be the set of four friends and three movies, respectively. Suppose they planned to watch a movie. This situation can be represented as a 4-polar fuzzy graph G by considering the vertex set as M and the edge set as M x M. Let A be a 4-polar fuzzy set of M. The membership value of Mi represents the preference degrees of the movie Mi corresponding to the friends. Suppose A(Ml) = < 0.9, 0.4, 0.6, 0.l>, A(M2) = < 0.5, 0.3, 0.8, 0.l>, A(M3) = < 0.8, 0.9, 0.8, 0.2>. This means that the preference degrees of Ml corresponding to Fl, F2, F3 and F4 are 0.9, 0.4, 0.6 and 0.l, respectively, and is similar for the others. An edge between any two nodes represents the degrees of common features (i.e., love story, comedy, fighting, and horror) of the nodes. Let B(MlM2) = < 0.4, 0.2, 0.2, 0.l>, B(M2M3) = < 0.4, 0.2, 0.2, 0.2>, B(M3Ml) = <0.4, 0.2, 0.3, 0.l>. This means that the degrees of common features (i.e., love story, comedy, fighting, and horror) of the movies Ml and M2 are 0.4, 0.2, 0.2 and 0.l. In other words, both movies Ml and M2 have 40% love story, 20% comedy, 20% fighting and l0% horror. Similar to the others. It is easy to verify that G of. It Fig. l is a 4-polar fuzzy graph.

Hereafter, we assume an m-polar fuzzy graph to be a generalized m-polar fuzzy graph.

4. Cartesian product, composition, union and join on m-polar fuzzy graphs

In this section, four types of operations, such as the Cartesian product, composition, union and join have been defined on m-polar fuzzy graphs to construct new types of m-polar fuzzy graphs.

Definition 4.1. The Cartesian product G1 x G2 of two m-polar fuzzy graphs G1 = (V1, A1, B1) and G2 = (V2, A2, B2) of the graphs Gi = (V1, E1) and G2 = (V2, E2), respectively, is defined as a pair (V x V2,Ai xA2,Bi xB2), such thatfori = 1, 2,...,m

(i) pi + (Ai x A2)(xi;X2)=min{p,oAi(xi),pi +A2(x2)} for all (xi, X2)eVi x V2.

Fig. 1. Example of 4-polar fuzzy graph G.

(ii) pi°(B1 x B2)((x,x2)(x,y2)) = min{pi«A1(x),pi°B2(x2y2)} for all x 2 V1, for all x^y2 2E2.

(iii) pi + (B1 x B2)((x1,z)(y1,z)) = min{pi+B1(x1 y1),pi+A2(z)} for allz2V2, for all x1y12E1.

(iv) pi + (B1 x B2)((x1,y1 )(x2;y2))= 0 for all (x1,y^fey2)eV1 x V22 - E.

Example 4.2. Let G1 = (V1, E1) and G2 = (V2, E2) be the graphs such thatV1 = {a, b}, V2 = {c, d}, E1={ab} and E2={cd}. Consider the 3-polar fuzzy graphs G1 = (V1, A1, B1) and G2 = (V2, A2, B2) of the graphs G1 = (V1, E1) and G'2 = (V2, E2), respectively where

a I < 0.3,0.4,0.6 > < 0.3,0.5,0.7 >

A1 - s-a-,-b-

_ I < 0.1,0.2,0.5 >

| < 0.1,0.4,0.5 > < 0.2,0.6,0.6 > j B2 = I < 0.1,0.3,0.4 > j Then it is easy to verify the following:

(B1 X B2)((a, c)(a, d))= < 0.1,0.3,0.4 >, (Bt x B2)((a, c)(b, c)) = < 0.1, 0.2, 0.5 > ,

B X B2)((b, c)(b, d))= < 0.1,0.3, 0.4 >, B X B2)((a, d)(b, d)) = < 0.1, 0.2, 0.5 >,

B x B2)((a, c)(b, d))= < 0,0,0 >, (B1 x B2)((b, c)(a, d)) = < 0, 0, 0 >

Hence, G1 x G2 is a 3-polar fuzzy graph of G*1 x G*2 (see Fig. 2).

Proposition 4.3. The Cartesian product G1 x G2 = (V1 x V2,A1 x A2,B1 x B2) of two m-polar fuzzy graphs of the graphs G1 and G*2 is an m-polar fuzzy graph of G*1 x G*2.

Proof. LetxeV1, x^y22E2. Then, fori = 1, 2,...,m pi + (B\ x B2)((x,x2)(x,y2)) = min{pi+A1(x), pi + B2(x2y2)}

< minfpi oAj(x), minfpi +A2(x2),pi + A2(y2)}}

= min{min{pi + A1 (x), pi + A2 (x2)}, min{pi +A1 (x), pi+A2(y2)}} = min{pi + (A1 x A2)(x,x2),pi + (Aj x A2)(x,y2)}.

LetzeV2, x1y12E1. Then, fori = 1, 2,...,m pi + B x B2)((xt,z)(y\,z)) = min{pi + Bi(xiyi), pi + A2 (z)}

< min{min{pi+A1(xi),pi+Ai(y1)},pi+A2(z)}}

= min{min{pi +A1 (xt),pi +A2(z)}, minfa +A1 (yt), pi +A2(z)}} = min{pi + (A1 x A2)(xt,z),pi + (A1 x A2)(y 1,z)}. 2

Let (x1, y1)(x2, y2)eV1 x V2 - E. Then, fori = 1,2, ...,m

pi+(B1 x B2)((x1,y1)(x2,y2)) -0 < minfpi + (A1 x A2)(x1,y1>,

pi + {A1 x A2)(x2,y2)g.

Definition 4.4. The composition G1[G2] = (V1 x V2, A1+A2, B1+B2) of two m-polar fuzzy graphs G1 = (V1, A1, B1) and G2 = (V2, A2, B2) of the graphs G1 = (V1, E1) and G2 = (V2, E2) respectively is defined as follows: fori = 1, 2,...,m

(i) pi + (A1 +A2)(x1,x2) = min{pi +A1(x1),pi+A2(x2)} for all (x1, x2)eVi x V2.

(ii) pi + (B1+B2)((x,x2)(x,y2))= min{pi+A1 (x),pi+B2(x2y2)} for all x 2 V1, for all x^y2 2E2.

(iii) pi + (B1+B2)((x1,z)(y1,z))= min{pi+B1(x1 y1),pi +A2(z)} for all z2 V2, for all x1y12E1.

(iv) pi + (B1+B2)((x1,x2)(y 1,y2))= min{pi+A2(x2),pi +A2(y2), pi +B1(x1y1)} for all (x1, x2)(y1,y2) 2E0 - E.

(v) pi + (B1+B2)((x1,y 1 )(x2,y2))= 0 for all (x1,y 1 )(x2,y2)

2 V1 x V2 - E0.

Example 4.5. Let G1 and G'2 be the same as in Example 4.2. Let G1 = (V1, A1, B1) and G2 = (V2, A2, B2) be two 3-polar fuzzy graphs of the graphs G*1 and G*2, respectively, where

n ) < 0.2,0.4,0.5 > < 0.3,0.5,0.4 > 1 □

A1 - i-V-,-~r-h B1 -

A2 j < 0.1,0^4,0.5 > , < 0.2,0 7,0.6 > 1 ^ -

<0.2,0.3,0.4>

<0.1,0.2,0.3> ——

Then, we have,

(B1+B2)((a, c)(a, d))- < 0.1, 0.2, 0.3 >, (B^Xp, c)(b, d))

- < 0.1, 0.2, 0.3 >,

(B1+B2)((a, c)(b, c))- < 0.1,0.3,0.4 >, (B1+B2)((a, d)(b, d))

- < 0.2, 0.3, 0.4 >

(B1+B2)((a, c)(b, d))- < 0.1,0.3,0.4 >, (B^Xp, c)(a, d))

- < 0.1, 0.3, 0.4 > .

It can be easily determined that G1[G2] is a 3-polar fuzzy graph of G1 [G2] (see Fig. 3).

Proposition 4.6. The composition G1[G2] of two m-polar fuzzy graphs G1 and G2 is an m-polar fuzzy graph.

Fig. 2. Product of two 3-polar fuzzy graphs G1 and G2.

Proof. LetxeV^ x2y2eE2. Then, fori = l, 2,...,m,

pi + (Bl x B2)((x,x2)(x,y2)) = min{pi +Al (x) , pi » B2 (x2y2)} < min{pi +Al (x) , min{pi »A2 (x2) , pi » A2 (y2)}} = min{min{p; +Al (x) , pi » A2 (x2)} , min{pi » Ai (x) , pi » A2 (y2)}} = min{pi » (Al x A2)(x,x2),pi+(Al x A2)(x, y2)}.

Let zeV2, xlyl eEl. The proof is similar to the above. Let (xi,x2)(yl,y2)eE0 - E. Therefore, xlyl eEl and x2*y2. Then, we have for eachi = l, 2, ...,m,

p;°(Bl°B2)((xl,x2)(yl,y2)) = min{pi +A2 (x2) , p; » A2 (y2) , p; + Bi (xi yl)} < min{pi » A2 (x2) , p; » A2 (y2) , min{pi+Al (xi ) , p; » Al (yi )}} = min{min{{p;+Al(xl), p^fe)}, min{p;+Al(yl) , p;+A2 (y2)}} = min{p; » (Al x A2)(xl,x2),p; + (Al x A2)(yl,y2)}.

Hence, Gl[G2] is an m-polar fuzzy graph.

Definition 4.7. The union GluG2 = (VluV2,AluA2,BluB2) of two m-polar fuzzy graphs Gl = (Vl, Al, Bl) and G2 = (V2, A2, B2) of the graphs Gl = (Vl, El) and G2 = (V2 , E2), respectively is defined as follows: fori = l, 2,...,m

{ p; »Al (x) if xeVl - V2

1. p; + (Al uA2 )(x)= p;»A2 (x) if x e V2 - Vl

: max{p;»Al(x) , p, »A2(x)} if xeVlnV2. (p;°Bl (xy) if xyeEl - E2

2. p;»(Bl uB2 )(xy) = < p;»B2 (xy) if xy e E2 - El

:max{p;»Bl(xy),p,»B2(xy)} if xyeElnE2.

3. p; » (Bl uB2 )(xy) = 0 if xyeVl'^V22 - EluE2.

Example 4.8. Let Gl and G*2 be graphs such that Vl = {a, b, c, d},El = {ab, bc, ad, bd}, V2 = {a, b, c, f} and {ab, bc, bf, cf}. Consider the two 3-polar fuzzy graphs Gl = (Vl, Al, Bl) and G2 = (V2, A2, B2),

where Al = I <0.2, 0.4 , 0.3> <0.4 , 0.5 , 0.6> <0.3 ,0.6,0.2> <0.3 , 0.7 , 0.8> j b1 =

[ <0.l,0.3 , 0.2> <0.2 , 0.5 , 0.l> <0.2 ,0.3 ,0.2> <0.3,0.4,0.5> <0,0,0> <0 , 0 , 0> \ 1 ab , bc , ad , bd , cd , ac n

A^ = | <0.2 , 0.4 , 0.7> <0.2 , 0.5 , 0.6> <0.3 ,0.6,0.7> <0.4, 0.5 , 0.3>j and b2 =

<0.2 ,0.3 ,0.5> <0.2 ,0.5,0.4> <0.2 ,0.5 ,0.3> <0.1,0.4,0.3> <0,0 0> <0,0,0> at , He , "cf , Hf , af , ac

Clearly, G1uG2 is a 3-polar fuzzy graph (See Fig. 4).

Proposition 4.9. The union G1uG2 = (V1uV2, A1uA2, B1uB2) of two m-polar fuzzy graphs of the graphs Gl = (V1, E1) and G2 = (V2, E2 ) respectively is an m-polar fuzzy graph.

Proof. Let xy eElnE2. Then, for i = l, 2, ...,m

p;»(Bl uB2)(xy) = max{p;»Bl (xy) , p;»B2 (xy)} < max{min{p;»Al (x) , p;»Al (y)} , min{p; »A2 (x) , p;»A2 (y)}} = min{p;»(AluA2)(x),p;» (AluA2)(y)}.

Similarly, ifxyeEl -E2, then pi»(BluB2)(xy) < min{p;»(Al uA2 )(x), p;+ (AluA2)(y)} and if xyeE2 - El, then p;»(BluB2)(xy) < min{p;+ (AluA2)(x),p;»(AluA2)(y)}. This completes the proof.

Definition 4.10. The join Gl + G2 = (Al + A2, Bl + B2) of two m-po-lar fuzzy graphs Gl = (Vl, Al, Bl) and G2 = (V2, A2, B2) of the graphs Gl = (Vl, El) and G'2 = (V2, E2), respectively, is defined as follows:

(i) p;»(Al + A2)(x)=p;o(AluA2)(x) ifxeVluV2

(ii) pi0(Bl + B2)(xy)=p;»(BluB2)(xy) ifxyeEluE2

(iii) p;o(Bl + B2)(xy) =min{p; +Al(x),p,»A2(y)}ifxyeE,whereE is the set of all of the edges joining the nodes of Vl and V2 and assuming that Vl nV2 = 0. 2

(iv) p; o (Bl + B2 )(xy) =0 ifxy e Vl x V2 - El uE2 uE'.

Proposition 4.11. The join Gl + G2 = (Al + A2, Bl + B2) of two m-polar fuzzy graphs of the graphs Gl =( Vl, El) and G2 = (V2, E2) is an m-polar fuzzy graph of G\ + G2. Proof. Follows from the definition.

Proposition 4.12. Let Gl = (Vl, El) and G2 = (V2, E2) be crisp graphs and let Vl nV2 = _0. Let Al, A2, B and B2 be m-polar fuzzy

subsets of V1 , V2, V2

respectively. Then,

G, uG2 = ( V, uV2 , A, uA2 , B, uB2 ) is an m-polar fuzzy graph ofG, uG2 if and only if G, = (V,, A,, B,) and G2 = (V2, A2, B2) are m-polar fuzzy graphs of G, and G2, respectively.

Proof. Suppose G,uG2 is an m-polar fuzzy graph of G, uG2. LetxyeEThen, xy;E2 and x,ye V, - V2, and for i = ! 2,...,m pi +B, (xy) = pi » (B,uB2)(xy) < minfpi » (A,UA2) _

(x) ,pi»(A^Xy)} < min{PioAl(x) ,p^A^y)}. Let xyeVf -E,. Then, fori = ^2,... ,m, pi»B1 (xy) = p,- » B, uB2 (xy) = 0. This shows that G, = (V,, A,, B,) is an m-polar fuzzy graph of G,. Similarly, we can show that G2 = (V2, A2, B2) is an m-polar fuzzy graph of G*2. The converse follows from proposition 4.9.

Proposition 4.13. Let G, = (V,, E, ) and G2 = (V2, E2) be crisp graphs and let V1ny2 = 0j.et A,, A2, B, and B2 be m-polar fuzzy subsets ofV,, V2 , V2 and V|, respectively. Then, G, + G2 = (A, + A2, B, + B2) is an m-polar fuzzy graph ofG\ + G2 ifand only if G, = (V,, A,,

Fig. 3. Composition of two 3-polar fuzzy graphs G, and G2.

Fig. 4. Union of two 3-polar fuzzy graphs Gi and G2.

B1) and G2 - (V2, A2, B2) are m-polar fuzzy graphs of G1 and G2, respectively.

Proof. Follows from propositions 4.11 and 4.12.

5. Isomorphisms of m-polar fuzzy graphs

In this section, different types of isomorphisms of m-polar fuzzy graphs are defined.

Definition 5.1. Let G1 = (V1, A1, B1) and G2 = (V2, A2, B2) be two m-polar fuzzy graphs of the graphs G1 = (V1,E1) and G2 = (V2,E2), respectively. A homomorphism between G1 and G2 is a mapping f:V1/V2 such that for i = 1, 2, ...,m

(i) pi+A1(x1) <pi+A2(f(x1)) for all x12 V1, _

(ii) pi+B1(x1y1) <pi+B2(f(x1)f(y1)) forallx1y1 2f2.

Definition 5.2. Let G1 = (V1, A1, B1) and G2 = (V2, A2, B2) be two m-polar fuzzy graphs of the graphs G1 = (V1,E1) and G2 = (V2,E2), respectively. An isomorphism between G1 and G2 is a bijective mapping f:V1/V2 such that for i = 1, 2, ...,m

(i) pi+A1(x1)=pi+A2(f(x1)) for all x12V1, _

(ii) pi +B1 (x1y1) = pi +B2(f(x1 )f(y1)) for all x1y1 2 Vf2. In this case, we write G1 y G2.

Definition 5.3. Let G1 = (V1, A1, B1) and G2 = (V2, A2, B2) be two m-polar fuzzy graphs of the graphs G1 = (V1,E1) and G2 = (V2,E2), respectively. A weak isomorphism between G1 and G2 is a bijective mapping f :V1 / V2, which satisfies the following conditions:

p1+A1 (a) = 0.2=p1+A2(f(a)=d), p2+A1(a)=0.4=p2+A2(f(a)=d), p3 +A1 (a) =0.5 = p3 +A2(f(a) =d). p10A1 (b) =0.3 = p1 +A2(f(b) =c), p2 0A1 (b) = 0.5 = p2 0A2(f(b) = c), p3 0A1 (b)=0.7 = p3 0A2(f(b)=c). p10B1 (ab) =0.1 < 0.2 = p1 °B2(f(a)f(b)=dc), p2 0 B1 (ab)= 0.4=p20B2(f(a)f(b)=dc),

p3 0B1 (ab) =0.3<0.4 = p3 0B2(f(a)f(b)=dc).Hence B1(ab) s B2(f(a) f (b)). This shows that the map f is a weak isomorphism but not an isomorphism.

Definition 5.5. Let G1 = (V1, A1, B1) and G2 = (V2, A2, B2) be two m-polar fuzzy graphs of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively. A co-weak isomorphism between G1 and G2 is a bijective mapping f :V1 / V2 which satisfies the following:

(i) f is a homomorphism,

(ii) for each i = 1, 2,--,m, pi0B1(x1y1)=p,-0B2(f(x1y1)) for all x1y1 2 V-j2. In other words, a co-weak isomorphism preserves the weight of the arcs but not necessarily the weights of the nodes.

Example 5.6. Let G1 = (V1, E1) and G2 = (V2,E2) be as in Example 5.4. Consider the 3-polar fuzzy graphs G1 = (V1, A1, B1) and G2 = (V2, A2, B2) ofG1 and G2 (see Fig. 6). Consider the map f:V1/V2 defined by f(a) = d, f(b)=c. Then, we have the following:

p10A1(a) = 0.2<0.3 = p10A2(f(a) = d), p20A1(a) = 0.4<0.6 = p2 0A2(f(a)=d), p3 0A1 (a) = 0.5 = 0.5 = p3 0A2W a) = d). Therefore, A1(a) s A2(f(a) = d). Similarly, A1(b) s A2(f(b) = c). However, p10B1(ab)= 0.1 = p10B2(f(a)f(b)=dc), p2 0B1 (ab)= 0.4 = p2«B2 (f(a)f(b) = dc), and p3 0B1 (ab)= 0.2 = p30B2 (f(a)f(b)=dc). Therefore, B1(ab) = B2(f(a)f(b) = dc). Hence, the map f is a co-weak isomorphism, but not an isomorphism.

(i) f is a homomorphism, and

(ii) for each i = 1, 2,...,m, pi0A1 (x1) = pi0A2(f(x1)) for all x12 V1. In other words, a weak isomorphism preserves the weights of the nodes, but not necessarily the weights of the arcs.

Example 5.4. Consider the two 3-polar fuzzy graphs G1 and G2 (see Fig. 5) of the graphs G1 = (V1,E1) and G2 = (V2, E2), respectively, where V1 = {a, b}, V2 = {c, d}, E1 = {ab} and E2 = {cd}. Let us define a map f :V1 /V2 to be defined by f(a) = d, f(b) = c. Then, we have

6. Some properties of m-polar fuzzy graphs

The strong m-polar fuzzy graph is defined below.

Definition 6.1. An m-polar fuzzy graph G = (V, A, B) of the graph G* = (V, E) is called strongifpi0B(xy) = min{pi0A(x), pi0A (y)} for all xy2E, i = 1,2,...,m.

Example 6.2. Consider a graph G* = (V, E) such that V = {x, y, z},E = {xy, yz, zx}. Let G = (V, A, B) be the 3-polar fuzzy graph of G* where

Fig. 5. Weak isomorphism of G, and G2.

Fig. 6. Co-weak isomorphism of G, and G2.

_ J < 0.2 , 0.4 , 0.5 > < 0.3 , 0.5 ,0.6 > < 0.4 , 0.3 , Q.j > x , y , z

B = j < 02 ,0.4, 05 > , < 03 , 0y3 , ^ > , ^OMM^j. Hence, G is a strong 3-polar fuzzy graph (see Fig. 7).

Proposition 6.3. If G, and G2 are the strong m-polar fuzzy graphs of the graphs G, = (V,,E1) G2 = (V2,E2), respectively, then G, x G2, G1[G2] and G, + G2 are strong m-polar fuzzy graphs of the graphs G, x G2, G, [G2] and G, + G2.

Proof. Follows from the Proposition 4.3, 4.6 and 4.H.

Remark 6.4. The union of two strong m-polar fuzzy graphs is not necessarily a strong m-polar fuzzy graph. For example, let us consider the 3-polar fuzzy graphs G, and G2, as shown in Fig. 8.

Proposition 6.5. If G, x G2 is strong m-polar fuzzy graph, then at least G, or G2 must be strong.

Proof. Suppose that both G, and G2 are not strong m-polar fuzzy graphs. Then, there exists at least one x-iy-i eE, and at least one x2y2eE2 such that

(i) B^yO < min{Al(Xl), A^yO), and B2(x2y2) < minute),

A2(y2)}.

Without loss of generality, we assume that

(ii) B2(x2y2) < Bl(XlУl) < min{Al(Xl), Al(yl)} < Al(Xl).

Let E = {(x,x2)(x,y2) : xeV,,X2y2eE2}u{ (x,,z)(y,,z) : zeV2, x,y, eE,}. Consider (x, x2)(x, y2)eE. Then, by definition of G, x G2 and inequality (i) we have,

(Bl x B2)((x,x2)(x,y2)) =min{Al(x),B2(x2y2)}

< min^X) , A2(X2) , A2 (y2 )}

and (A, x A2)(x, X2) = min{Al(x), A2(X2)},(A, x A2)(x, y2) = min{Al(x), A2(y2)}. Thus, min^A, x A2)(x, X2),(A, x A2)(x, y2)} = min{Al(x), A2(x2), A2(y2)}.

Hence, (B, x B2)((x,X2)(x,y2)) = mi^A^x),B2^y2)} <min {(A, x A2)(x ,x2) , (A, x A2)(x, y2)}, i.e., G, x G2 is not a strong m-polar fuzzy graph, which is a contradiction. Hence, if G, x G2 is a strong m-polar fuzzy graph, then at least G, or G2 must be a strong m-polar fuzzy graph.

Proposition 6.6. If G,[G2] is a strong m-polar fuzzy graph, then at least G, or G2 must be strong.

Proof. Follows from previous propositions.

Proposition 6.7. Let G = (V, A, B) be a strong m-polar fuzzy graph of a graph G* = (V, E). If G = (V ,B) satisfies A = A and Bis defined by, for all xyeV2, i = X 2, ...,m

pi»B(xy)

0 if 0<pi°B(xy) <1

min{pi»A(x),pi»A(y)} if pi»B(xy) = 0.

Fig. 7. Strong 3-polar fuzzy graph G.

Then, G is a strong m-polar fuzzy graph of G* = (V , V2 - E). Proof. Obviously, the m-polar fuzzy sets A and B satisfy p; +B(xy) < min{p;oA(x),p;oA(y)} forallxyeV2, i = l, 2,...,m.

Now, let xyeV2 - (V2 - E)= E. As G is a strong m-polar fuzzy graph, therefore we have for i = l, 2,...,m, p;»B(xy) = min{p;»A(x) , p; +A(y)}.

If B(xy) = 0, then for each i = l, 2,...,m, p;»B(xy) = 0. Therefore, p;»B(xy) = min{p;»A(x), p;»A(y)} = p;»B(xy) = 0, i = l,2,...,m. Hence, B(xy) = 0.

If for i = l, 2,...,m, 0<p;»B(xy) <l then p;»B(xy)=0, i.e., B(xy) = 0. Hence, for all xyeV2 - (V2 - E) = E, B(xy) = 0. Therefore, G = (V, A ,B) is an m-polar fuzzy graph of G* = (V, V2 - E).

On the other hand, for all xyeV2 - E, we have by Definition 3.l, B(xy) = 0, i.e., for each i = l, 2, ...,m, p;»B(xy) = 0. Then, we have for each i = l, 2,...,m, p;»B(xy) = min{p;oA(x),p;»A(y)}. Therefore, G is a strong m-polar fuzzy graph of G* = (V , V2 - E).

Definition 6.8. The strong m-polar fuzzy graph G = (V,A ,B) defined above, is called the complement of the strong m-polar fuzzy graph G = (V, A, B).

Definition 6.9. A strong m-polar fuzzy graph G is called self-complementary if GyG.

Gi is strong

Fig. 8. The union of the two strong 3-polar graphs G1 and G2 is not strong.

Example 6.10. Let G* = (V, E) be a graph where V = {a, b, c, d}, E = {ab, ac, cd} and G = (V, A, B)(see Fig. 9) be a strong 3-polar fuzzy graph ofG* where A = <j < 0.1, 0.2 ,0.3 > < 0.1, 0.2 , 0.3 > < < 0.1,0.2 ,0.3 > < 0.1, 0.2, 0.3 >

D I < 0.1, 0.2 , 0.3 > < 0.1,0.2 ,0.3 > < 0.1,0.2 ,0.3 > < 0 ,0, 0 > < 0 , 0 ,0 > B = 1 ab , ac , cd , bd , ad ,

<0 bc° > }. Then, G is self complementary. Let G = (V,A , B) be the complement of G, where A = A, B ^ < 0 ab0 > , < 0 'a°c0 > ,

<°°o>, , < 0.1 , a-2 , 0.3 > , < 01, 0bc2 , 03 >j. Let us now define a

mapping f:V/ Vby f (a) = b, f(b) = c, f(c) = d, f (d) = a. Then, clearly, f is a bijective mapping and A(a) = A(f(a) = b), A(b) = A(f(b) = c), A(c) = A(f(c) = d), A(d)=A(f(d) = a). Additionally,

B(ab)= < 0.1,0.2 ,0.3 > = B(f(a)f(b) = bc), B(ac) = < 0.1,0.2 ,0.3 > = B(f(a)f(c) = bd),

B(cd)= < 0.1,0.2 ,0.3 > = B(f(c)f(d) = ad), B(bc)= < 0 ,0 ,0 > = B(f(b)f(c) = cd) ,

B(bd) = < 0 , 0 , 0 > = B(f(b)f(d) = ac), B(ad) = < 0 , 0 , 0 > = B(f(a)f(d) = ab).

Hence f is an isomorphism from G onto G, i.e., GyG, which means that G is self complementary.

Proposition 6.11. Let G = (V, A, B) be a strong m-polar fuzzy graph of the graph G* = (V, E) and G = (V, A , B) be the complement ofG. Then, pi0B(xy) = min{pi0A(x),p,0A(y)}-pi0B(xy) for all xy2V2, i = 1, 2,...,m.

Proof. Let xyeV2. If 0<pi+B(xy) < 1 for each i- 1, 2,...,m; then, xyeE by Definition 3.1. As G is strong, for i-1, 2,...,m, minfpi +A(x), pi °A(y)} - pi+B(xy) - 0 - pi +B(xy).

If for i -1, 2, ...,m, pi+B(xy)-0, then minfpi+A(x) , pi°A(y)} -pi+B(xy) - min{pi°A(x),pi+A(y)} - pi+B(xy). Hence the result.

Proposition 6.12. Let G be a self-complementary strong m-polar fuzzy graph. Then, for all xyeV2, i - 1, 2,...,m

]Tpi°B(xy)-l]T min{pi+A(x),pi +A(y)}.

Proof. Let G = (V, A, B) be a self-complementary strong m-polar fuzzy graph. Then, for all xy2E, i = 1, 2,...,m, pi0B(xy) = min{pi0A(x),p,0A(y)} and there exists an isomorphism f : G/G such that pi0A(x)= pi0A(x) for all x2V and pi0B(xy) = pj<>B(f(x)f(y)) for all xy2V2.

Letxy2V2. Then, by Proposition 6.11, for i = 1, 2,...,m,

pi'B(f(x)f(y)) = min{pioA(f(x)) , pioA(f(y))}~ pi'B(f(x)f(y))

i.e. , pt +B(xy) - min{pi+A(f(x)), prA($(y))} - pi + B(f(x)f(y)).

Therefore,

Epi°B(xy) + pi °B(f(x)f(y)

xsy xsy

-Y, min{pi°A(f(x)),pi°A(f(y))}

-J2 min{pi°A(x),pi°A(y)}

Fig. 9. Self-complementary 3-polar fuzzy graphs.

2J2Pi»B(xy) = J2 min{pi»A(x),p,»A(y)}

xsy xsy

Xpi°B(xy) = 2 X min{pi»A(x),p,»A(y)}

xsy 2 xsy

So, p;oB2(j(x)j(y)) = 0 = p;oBl(xy),_2 = l, 2,...,m. If for i = l, 2,...,m, 0 < p;oBl(xy)<l, then p, oB2(j(x)j(y)) = p; oBl(j(x)j

(y)) = 0.

Thus we have,

Pi_oB2(j(x)j(y)) = min{p;oA2(j(x)),pioA2(j(y))}_- 0 = min{p;o A2(j(x)) , p;oA2(j(y))}= min{p;oAl(j(x)) , p;oAl(j(y))} = p; oBl (xy). Hence Gl yG2.

Proposition 6.13. Let G = (V, A, B) be a strong m-polar fuzzy graph of G* = (V, E). If p,»B(xy)=^min{p,»A(x),p,»A(y)} for all xyeV2, i = ^ 2, ...,m, then G is self complementary.

Proof. If G = (V, A, B) is a strong m-polar fuzzy graph satisfying

pi»B(xy) =2 min{pi»A(x), pi»A(y)} for all xyeV2, i = ^ 2, ...,m, then the identity mapping I:V/V is an isomorphism from G to G. Clearly, I satisfies the first condition for isomorphism, i.e., A(x) = A(I(x)) for all xeV, and by Proposition 6.H, we have for all xyeV2, i = ^ 2, ...,m,

Pi»B(I(x)(y)) = p,»B(xy) = min{prA(x),pi»A(y)} -pi»B(xy) = min{pi»A(x),pi»A(y)} - 2min{pi»A(x),pi»A(y)}

= 2 min{pi»A(x) , pi»A(y)} = pi»B(xy).

i.e., pi » B(xy) = p, » B(xy) for all xyeV2, i = ^2, .. .,m i.e., I also satisfies the second condition for isomorphism. Therefore, GyG, i.e., G is self complementary.

From Proposition 6^2 and 6.0, we have the following result.

Corollary 1 Let G = (V, A, B) be a strong m-polar fuzzy graph of G* = (V, E). Then, G is self complementary if and only if pi»B(xy) =5 min{pi»A(x), pi»A(y)} for all xyeV2, i = ^ 2, ...,m.

Proposition 6.14. Let G, and G2 be two strong m-polar fuzzy graphs. Then, G, yG2 if and only if G, yG2.

Proof. Assume that G, y G2. Then, there exists a bijective mapping f:^/^ satisfying A,(x) = A2(f(x)) for all xeV, and

PioBl (xy) = pi»B2(f(x)f(y)) for all xy e Vf, i = ^ 2,.. ,,m.

Let xyeV^. If for i = ^ 2, ...,m, pioB1(xy) = 0, then pioB1(xy) = min{pioAl(x) , p, »A, (y)} = min{pi»A2 (f(x)) , p,»A2(f(y))} = p, »B2 (f(x)f(y)).

If for, 0 < pioB1(xy) < ^ then 0 < pi»B2(f(x)f(y)) < L Therefore,

PioBl(xy) =0 = pi»B2(fJx)f (y)). So, G, y G2.

Conversely, let G,yG2. Then, there exists a bijective mapping j:V1/V2 satisfying A,(x)= A2( j(x)) for all xeV, and Pi »Bi(xy) = p, »B2(j(x)j(y)) for all xyeV2;.

Let xyeV2. If for i = ^ 2, ...,m, pioB1(xy) = 0, then

7. Applications

Fuzzy graphs of the l-polar type are nothing more than the most familiar fuzzy graphs and have many applications for cluster analysis and solving fuzzy intersection equations, database theory, problems concerning group structure, and so on. The further possible applications of m-polar fuzzy graphs in real-world problems can be viewed in the case of bipolar fuzzy graphs, i.e., 2-polar fuzzy graphs. Bipolar fuzzy graphs have many applications in social networks, engineering, computer science, database theory, expert systems, neural networks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, medical diagnosis and so on. Additionally, m-polar fuzzy graphs (m > 2) are very useful in many decision-making situations. This occurs when a group of friends decides which movie to watch, when a company decides which product design to manufacture, and when a democratic country elects its leader. For instance, consider the case of a company. In a company, a group of people decides which product design to manufacture. In such a case, different product designs can be taken as nodes. An edge is drawn between two nodes if there is some m-polar fuzzy relationship between them. We assume that the membership value of each node represents the degrees of preference of the product design corresponding to the group of people of the company. The degrees of preference (within [0,l]) represent the individual preferences of the people. Thus, a node has multi-preference degrees corresponding to a product design. Similarly, the degree of relationship between the nodes measures the edge relationship value. Between two product designs, one design may have a better appearance, may be in very high demand, may be cheap, and so on. Therefore, there is multipolar information between two product designs. This type of network is an ideal example of m-polar fuzzy graphs. It is very important for a company to decide which product design to manufacture so that they can make as great a profit as possible. A very good product design is readily accepted by customers if it is also inexpensive. The determination of which product design to manufacture is called the decision-making problem. By taking the very good decision (very good product design), one company can spread their product all over the world, keeping in mind that the product design is very good, in demand, cheap, easily accessible, and so on. Moreover, the results of m-polar fuzzy graphs can be applicable in various areas of engineering, com-

p,»B2(j(x)j(y)) = PioBl (xy) = min{PioAl (x) ,p^A^y)} = min^p,»A^x),p^A^y)} = min\p{ »A2(j(x)) , p,»A2(j(y))} = min{p, »A2(j(x)) , p^jy))}

Again, p,»B2(j(x)j(y)) = min{pi»A2(j(x)),pi»A2(j(y))} -p,»B2 puter science, artificial intelligence, neural networks, social (j(x)j(y)) networks, and so on.

G. Ghorai, M. Pal / Pacific Science Review A:

8. Conclusions

Graph theory is an extremely useful tool for solving combinatorial problems in different areas, including algebra, number theory, geometry, topology, operations research, optimisation and computer science. Because research on or modelling of real world problems often involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information, uncertainty, and/or process limits, m-polar fuzzy graphs are very useful. The m-polar fuzzy models give increasing precision, flexibility, and comparability to the system compared to the classical, fuzzy and bipolar fuzzy models. Therefore, we have studied several important results of m-polar fuzzy graphs. Our next plan is to extend our research work to m-polar fuzzy intersection graphs, isomorphisms on m-polar fuzzy graphs, m-polar fuzzy interval graphs, m-polar fuzzy hyper-graphs, and so on.

Acknowledgements

Financial support for the first author offered under the Innovative Research Scheme, UGC, New Delhi, India (Ref. No.VU/Inno-vative/Sc/15/2015) is gratefully acknowledged.

We are highly thankful to the Editor and the anonymous referees for their insightful comments and valuable suggestions.

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