Pacific Science Review A: Natural Science and Engineering xxx (2015) 1—9

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Pacific Science Review A: Natural Science and Engineering

journal homepage: www.elsevier.com/locate/PSRA

On some operations and density of m-polar fuzzy graphs

Ganesh Ghorai*, Madhumangal Pal

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

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Keywords:

m-polar fuzzy graphs Direct product Semi-strong product Strong product

Balanced m-polar fuzzy graph Product m-polar fuzzy graph

The theoretical concepts of graphs are highly utilized by computer science applications, social sciences, and medical sciences, especially in computer science for applications such as data mining, image segmentation, clustering, image capturing, and networking. Fuzzy graphs, bipolar fuzzy graphs and the recently developed m-polar fuzzy graphs are growing research topics because they are generalizations of graphs (crisp). In this paper, three new operations, i.e., direct product, semi-strong product and strong product, are defined on m-polar fuzzy graphs. It is proved that any of the products of m-polar fuzzy graphs are again an m-polar fuzzy graph. Sufficient conditions are established for each to be strong, and it is proved that the strong product of two complete m-polar fuzzy graphs is complete. If any of the products of two m-polar fuzzy graphs G1 and G2 are strong, then at least G1 or G2 must be strong. Moreover, the density of an m-polar fuzzy graph is defined, the notion of balanced m-polar fuzzy graph is studied, and necessary and sufficient conditions for the preceding products of two m-polar fuzzy balanced graphs to be balanced are established. Finally, the concept of product m-polar fuzzy graph is introduced, and it is shown that every product m-polar fuzzy graph is an m-polar fuzzy graph. Some operations, like union, direct product, and ring sum are defined to construct new product m-polar fuzzy graphs.

Copyright © 2015, 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

Presently, science and technology is characterised by complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models is based on fuzzy sets. Graph theory has numerous applications to problems in computer science, electrical engineering, systems analysis, operations research, economics, networking routing, and transportation. Considering the fuzzy relations between fuzzy sets, Rosenfeld [18] introduced the concept of fuzzy graphs in 1975 and later developed the structure of fuzzy graphs. Mordeson and Nair [15] defined the complement of fuzzy graphs, which was further studied by Sunitha and Kumar [21]. The concepts of weak isomorphism, co-weak isomorphism and isomorphism between fuzzy graphs was

* 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.

introduced by Bhutani in Ref. [3]. Several researchers have worked on fuzzy graphs. Samanta and Pal introduced several types of fuzzy graphs, such as fuzzy planar graphs [29], fuzzy competition graphs [26,27], fuzzy tolerance graphs [22], and fuzzy threshold graphs [23]. Some more work on fuzzy graphs can be found on [4,12,16,17].

In 1994, Zhang [31—33] developed the concept of bipolar fuzzy sets as a generalization of fuzzy sets. The idea behind such description is connected with the existence of "bipolar information" (i.e., positive information and negative information) about the given set. Positive information represents what is granted to be possible, whereas negative information represents what is considered to be impossible. In 2011, using the concepts of bipolar fuzzy sets, Akram [1] introduced bipolar fuzzy graphs and defined different operations. Using this definition of bipolar fuzzy graphs, research is ongoing. Some work on bipolar fuzzy graphs may be found on [6,7,24,25,28,30]. Talebi and Rashmanlou [2] studied the complement and isomorphism of bipolar fuzzy graphs. Rashman-lou et al. [19,20] studied bipolar fuzzy graphs and bipolar fuzzy graphs with categorical properties.

In 2014, Juanjuan Chen et al. [5] introduced the notion of m-polar fuzzy sets as a generalization of bipolar fuzzy sets and showed that bipolar fuzzy sets and 2-polar fuzzy sets are cryptomorphic

http://dx.doi.org/10.1016/j.psra.2015.12.001

2405-8823/Copyright © 2015, 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/).

mathematical notions and that we can obtain one from the other. The idea behind this is that "multipolar information" (not just bipolar information, which corresponds to two-valued logic) exists because data of real world problems sometimes come from multiple agents. For example, the exact degree of telecommunication safety of mankind is a point in [0,l]n (nz7x109) because different persons have been monitored different times. There are many other examples, such as truth degrees of a logic formula that are based on n logic implication operators (n>2), similarity degrees of two logic formulas that are based on n logic implication operators (n>2), ordering results of a magazine, ordering results of a university, and inclusion degrees (accuracy measures, rough measures, approximation qualities, fuzziness measures, and decision preformation evaluations) of a rough set.

Ghorai and Pal [8] introduced the notion of generalized m-polar fuzzy graphs as a generalization of bipolar fuzzy graphs and defined different operations. In Ref. [9], they studied the complement and isomorphism of m-polar fuzzy graphs. Because of the importance of m-polar fuzzy graphs mentioned in Refs. [8,9], we investigated m-polar fuzzy graphs. In this paper, three new operations are defined on the m-polar fuzzy graph, including direct product, semi-strong product and strong product. Any of the products of m-polar fuzzy graphs are again an m-polar fuzzy graph. Sufficient conditions are established for each one to be strong, and it is proved that the strong product of two complete m-polar fuzzy graphs is complete. If any of the products of two m-polar fuzzy graphs G1 and G2 are strong, then at least Gi or G2 must be strong. Moreover, the density of an m-polar fuzzy graph is defined, the notion of balanced m-polar fuzzy graphs is studied, and the necessary and sufficient conditions for the preceding products of two m-polar fuzzy balanced graphs to be balanced are established. Finally, the concept of product m-polar fuzzy graphs is introduced, and it is proved that every product m-polar fuzzy graph is an m-polar fuzzy graph. Some operations, like union, direct product, and ring sum, are defined to construct new product m-polar fuzzy graphs.

2. Preliminaries

In this section, we recall some definitions of fuzzy graphs, m-polar fuzzy sets, and m-polar fuzzy relations, which are defined below. For further study, see Refs. [5,8,9,11,13—15].

Definition 2.1. [15] A fuzzy graph with V as the underlying set is a triplet G = (V,s,m), where s:V / [0,1] is a fuzzy subset of V and m: V x V / [0,1] is a fuzzy relation on s, such that m(x,y) < s(x) a s(y) for all x,y e V, where a stands for the minimum.

The underlying crisp graph of G is denoted by G* = (s*,m*), where s* = {x 2 V:s(x) > 0} and m* = {(x,y) e VxV:m(xy) > 0}.

Definition 2.2. [10] A fuzzy graph G = (V,s,m) is complete if m(x,y) = s(x) a s(y) for all x,y e V.

The main purpose of this paper is to study m-polar fuzzy graphs based on m-polar fuzzy sets, which is defined below.

Throughout the paper, [0,1]m (m-power of [0,1]) is considered to be a poset with point-wise order <, where m is a natural number. < is defined by x < yo for each i = 1,2,...,m; pi(x) < pi(y) where x,y e [0,1]m and p,:[0,1]m / [0,1] is the i-th projection mapping.

Definition 2.3. [5] An m-polar fuzzy set (or a [0,1]m-set) on X is a mappingA:X / [0,1]m. The set of all m-polar fuzzy sets on X is denoted by m(X).

Definition 2.4. [8] 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:

p, »(AuB((x) = {pioA(x)vpi»B(x)} and

pi°(AnB)(x) = { +A(x)Api+B(x)} fori = 1,2,...,m andx e X (v stands for maximum).

A4B if and only if for each i = 1,2,...,m and x e X, pi+A(x)<pi+B(x).

A = B if and only if for each i = 1,2,...,m and x e X,

pi+A(x)=pi+B(x).

Definition 2.5. [8] 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 ofX x X such that B(x,y) < min{A(x),A(y)} for all x, yeX i.e., for each i = 1,2, ...,m, for all x,y e X, 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,y e X.

Definition 2.6. [10] The semi-strong product of two fuzzy graphs G1 = (s1,m1) and G2 = (s2,m2) of the graphs G1* = (V1,E1) and G2* = (V2,E2), where it is assumed that V1 nV2 = 0, is defined to be the fuzzy graph G1»G2 = (s1»s2,m1»m2) of the graph G* = (V1 x V2,E) respectively, such that E = {(u, v1)(u, v2)|ueV1, v1v2 eE2}u{(u1, v1) (u2, V2)|u1 u2 eE1, V1V2 eE2},

(s1 • s2)(u, v) = s1(u)As2(v) for all (u, v)eV1 x V2, (m1 • m2)((u,v1)(u,V2)) = s1(u)Am2(v1V2) and O1 • m2)((u1> v1 )(u2; v2)) = m1 (u1u2)Am2(v1v2). Definition 2.7. [10] The strong product of two fuzzy graphs G1 = (s1,m1) and G2 = (s2,m2) of the graphs G1 =(V1, E1) and G2 = (V2, E2) respectively, where it is assumed that V1nV2 = 0, is defined to be the fuzzy graph G1 5 G2 = (s1 5 s2, m1 5 m2) of the graph G* = (V1 x V2,E), such that E = {(u,v1)(u,v2)|ueV1,v1v2eE2} u{(u1, w)(u2, w)|weV2, u1u2eE1}u{(u1, v1)(u2, v2 )|u1u2 eE1, v1 v2 e E2 },

(s1 5s2)(u, v) = ct1(u)act2(v) for all (u,v) e V1xV2, (m15m2)((u, v1)(u, v2)) = s1(u)Am2(v1v2), (m1 5m2)((u1, w)(u2, w)) = s2(w)Am1(u1u2) and (m15m2)((u1> v1 )(u2; v2)) = m1(u1u2)Am2(v1v2). Definition 2.8. [10] The direct product of two fuzzy graphs G1 = (s1,m1) and G2 = (s2,m2) of the graphs G1 =(V1, E1) and G2 = (V2, E2) respectively, such that V1nV2 = 0, is defined to be the fuzzy graph G1nG2 = (^1 ns?, m1nm2) of the graph G* = (V1 x V2,E), such thatE = {(u1, v1)(u2, v2)|u1u2eE1, v1v2 2E2},

(s1 ns2)(u, v) = s1(u)As2(v)for all (u,v) e V1 x V2, and

(m1 nm2)((u1> v1 )(u2; v2)) = m1(u1u2)Am2(v1v2). For a given set V, define an equivalence relation onV x V - {(x, x) : xe V} as follows:

(x1,y1) ~ (x2,y2) o either (x^yO = (x2,y2) orx1 = y2 andy1 = 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* represents a crisp graph, and G is an m-polar fuzzy graph of G*.

3. m-polar fuzzy graphs

In this section, we briefly recall some basic definitions related to m-polar fuzzy graphs.

Definition 3.1. [8] An m-polar fuzzy graph of a graph G* = (V,E) is a pair G = (A,B) where A:V / [0,1]m is an m-polar fuzzy set in V and

B : V2/[0,1]m is an m-polar fuzzy set in V2, such that for each

i = 1,2,...,m; pi +B(xy) < min{pi+A(x),pi+A(y)} for all xyeV2 and

B(xy) = 0 forallxyeV2 - E, (0 = (0,0,...,0) is the smallest element in [0,1]m).

A is called the m-polar fuzzy vertex set of G, and B is called the m-polar fuzzy edge set of G.

Fig. 1. Example of 3-polar fuzzy graph G

Example 3.2 Let us consider the graph G* = (V,E) where V = {u,v,w} and E = {uv,vw,uw}. A 3-polar fuzzy graph G of G* is shown in Fig. 1.

Definition 3.3. [8] The m-polar fuzzy graph H = (P,C,D) is called an m-polar fuzzy subgraph of G = (V,A,B) induced by P if P4V,

C(x) = A(x) for all x e P and D(xy) = B(xy ) for all xy e P2.

Example 3.4 H is a 3-polar fuzzy subgraph of G of Example 3.2 (see Fig. 2).

Definition 3.5. [8] An m-polar fuzzy graph G = (A,B) of the graph G* = (V,E) is called strong ifB(xy) = min{A(x),A(y)} for all xy e E, i.e., foreachi = 1,2,...,m;pi°B(xy) = min{pi°A(x),pi+A(y)}forallxy e E.

Definition 3.6. [9] An m-polar fuzzy graph G = (A,B) of the graph G* = (V,E) is said to be complete if B(xy) = min{A(x),A(y)} for all x,y e V, i.e., for each i = 1,2, ...,m, pi°B(xy) = min{pi°A(x),pi°A(y)} for allx,y e V.

Definition 3.7. [9] Let G = (A,B) be an m-polar fuzzy graph of a graph G* = (V,E). The complement of G is an m-polar fuzzy graph G =

(A, B) of G* = (V, V2) where A = A and B is defined by pi+B(xy) = min{pi+A(x), p,°A(y)} -pi+B(xy), for each i = 1,2,...,m

and for allxyeV2.

Definition 3.8. [8] A strong m-polar fuzzy graph G is called self-complementary if G y G.

G* = (V x V2,E) where, E = {(u1,vt)(u2,v2)|u1U2eE1, v^eE2} 4V1 x V2

and for each i = 1,2, ...,m

pi ° (A1 nA2)(u,v) =p,-oA1(u)Api»A2(v) forall (u,v) e V1XV2, p,-o(B1nB2)((U1, V1), (U2; V2)) = pi»B1(U1U2)Ap,-»B2 (v1 V2)for all u1u2 e E1and v1v2eE2,

pio(BmB2)((w,x); (y,z)) = 0 forall (w,x)(y,z)e(V1gTV2 - E). Below, the direct product of m-polar fuzzy graphs is explained with an example.

Example 4.2 Let G1 = (V1, E1) and G2 = (V2, E2) be two crisp graphs such that V1 = {u,v}, V2 = {w,x}, E1 = {uv} and E2 = {wx}. Consider two 3-polar fuzzy graphs G1 = (A1,B1) and G2 = (A2,B2) of the graphs G1 = (V1, E1) and G2 = (V2, E2). Using the definition of direct product, G1 nG2 is calculated (see Fig. 3).

It is easy to see that G1nG2 is a 3-polar fuzzy graph.

Theorem 4.3. The direct product G1 nG2 of two m-polar fuzzy graphs G1 and G2 is an m-polar fuzzy graph.

Proof: Let (u1,v1)(u2,v2) e E. Thenu1u2 e E1 andv1v2 e E2. Hence for each i = 1,2,,„,m; we have

pi + (B1 nB2)((u1, V1), (u2, V2))

= pi°Bi(uiu2)Api°B2(viV2)

< pi°A1 (u1)Api°A1(u2)Api°A2(v1)Api°A2(v2) (because G1 and G2 are m-polar fuzzy graphs)

= pi ° (A1 nA2)(ut, V1 )Api ° (A1 nA2)(u2, V2)

Also, forall (w,x)(y,z)e(V1 x V2 -E), i = 1,2,_,m;

pi ° (B1 nB2)((w,x), (y,z)) = 0 < pi ° (A1 nA2)(w,y)Api ° (A1 nA2)(x,z). This shows that G1 nG2 is an m-polar fuzzy graph.

Theorem 4.4. If G1 = (A1,B1) and G2 = (A2,B2) are two strong m-polar fuzzy graphs, then G1 nG2 is also strong.

Proof: Let (u1,v1)(u2,v2) e E. Because G1 and G2 are strong, we have for each i = 1,2,...,m

pi° (BtnB2)((u1, V1), (u2, V2))

4. Products on m-polar fuzzy graphs

Here, the direct product of two m-polar fuzzy graphs is defined.

Definition 4.1. Let G\ = (A1,B1) and G2 = (A2,B2) be two m-polar fuzzy graphs of the graphs G1J = (Vand G'2 = (V2,£2) respectively, such that Vi nV2 = 0. The direct product of Gi and G2 is defined to be the m-polar fuzzy graph G1nG2 = (A1nA2, B1nB2) of the graph

< 0.3,0.4,0.C >

< 0,3,0.3,0.2 >

< 0.4,0.0,0.3 >

Fig. 2. 3-polar fuzzy subgraph H of G

= PÍ + BÍ(UÍU2)API + B2(VÍV2) = Pi+A (Uj )APi +A1 (U2)APi +A2 (V! )APi +A2 (V2)

= Pi + (Al nA2)(U! , V! )APi + (Aj nA2)(U2, V2).

Hence, G1 nG2 is a strong m-Polar fUzzy graPh. Now, the semi-strong ProdUct is defined between two m-Polar fUzzy graPhs to constrUct a new m-Polar fUzzy graPh.

Definition 4.5. The semi-strong ProdUct of two m-Polar fUzzy graPhs Gi = (Ai,Bi) ofGj = (VJEj) and G2 = (A2B2) ofG'2 = (V2,E2), where it is assUmed that Vi nV2 = 0, is defined to be the m-Polar fUzzy graPh Gi»G2 = (Ai»A2,Bi»B2) of G* = (Vi x V2,E), where

E = {(U, vi)(U, V2)|UeVi, vi V2eE2>u{(Ui, vj)(U2, V2)|UiU2eEi, viV2eE2}4VigTv22

satisfying the following: for each i = i,2,...,m

Pi + (Ai • A2)(U, v) =Pi+Ai(U)APi+A2(v) forall (U,v) e VixV2, Pi + (Bi * B2)((U, vi)(U, V2)) = Pi+Ai(U)APi + B2(viV2) for all U e Vi and viv2 e E2,

Fig. 3. Direct product of Gj and G2

pi + (B1 • B2)((u1,v1)(u2,v2))= p, + B1(u1u2)ap, + B2(v1v2) for all u1u2 e E1 and v1v2 e E2, andpi + (B1 • B2)((w, x)(y, z)) = 0 for all

(w, x)(y, z)e(V1gTV22 - E).

We demonstrate this product to construct a new m-polar fuzzy graphs, which is explained in the following example.

Example 4.6 Consider the 3-polar fuzzy graphs G1 = (A1,B1) and G2 = (A2,B2) as in Example 4.2; then, G1»G2 is calculated using the above definition (see Fig. 4).

It is easy to see that G1$G2 is a 3-polar fuzzy graph.

Theorem 4.7. If G1 = (A1,B1) and G2 = (A2,B2) are m-polar fuzzy graphs, then G1$G2 is an m-polar fuzzy graph.

Proof: Let (u,v1)(u,v2) e E. Then, u e V1 and v1v2 e E2. Because G2 is an m-polar fuzzy graph, we have for each i = 1,2, ...,m

p, + (B1 • B2)((u, v1 )(u, v2)) = pi+A1(u)Api + B2(v1v2)

< P, + A1 (U)AP, +A2 (v1 )ap, + A2 (v2)

= P, + (A1 • A2)(u, v1 )AP, +(A1 • A2)(u, v2).

Let (u1,v1)(u2,v2) e E. Then u1u2 e E1 and v1v2 e E2. G1 and G2 being m-polar fuzzy graphs, we have for each i = 1,2,... ,m

pi + (B1 • B2)((u1, v1)(u2,v2)) = P, + B1 (u^Ap^^)

< P, +A (u )AP, + A1 (U2)AP, + A2 (v1 )AP, + A2 (v2)

= pi + (A1 • A2)(u1, v1 )AP, » (A1 • A2)(u2, v2).

Finally, for all (w, x)(y, z)e(V1 x V| - E), i = 1,2,^,m;

Pi + (Bj • B2)((W,X)(y,Z)) = 0 <Pi + (Aj .A2)(w,y)APi + (Aj .A2)(x,z).

Theorem 4.8. If G1 = (A1,B1) and G2 = (A2,B2) are strong m-polar fuzzy graphs, then G1»G2 is a strong m-polar fuzzy graph.

Proof: Let (u,v1)(u,v2) e E. Then u e V1 and v1v2 e E2»G2 being strong, we have for each i = 1,2,...,m

Pi + (Bj • B2)((u, vj)(u, V2)) = Pi+Aj(u)APi + B2(vj V2)

= P, 0 A1 (U)aP, + A2 (v1) AP, + A2 (v2)

= p, + (A1 • A2)(u, v1 )Ap,0 (A1 • A2)(u, v2).

If (u1,v1)(u2,v2) e E. Then u1u2 e E1 and v1v2 e E2. Now, G1 and G2 being strong, we have for each i = 1,2,...,m

p, + (B1 • B2)((u1, v1)(u2, v2)) = pi°B1(u1u2)Ap,oB2(v1v2)

= p, oA1 (u1 )Ap, 0A1 (u2)Ap, 0A2 (v1 )Ap, 0A2 (v2)

= p, + (A1 • A2)(u1, v1 )Ap, o (A1 • A2)(u2, v2).

Hence, G1»G2 is a strong m-polar fuzzy graph. The strong product between m-polar fuzzy graphs is an important construction of m-polar fuzzy graphs, which is defined below.

Definition 4.9. The strong product of two m-polar fuzzy graphs G1 = (A1,B1) of G1 = (V1,E1) and G2 = (A2B2) ofG'2 = (V2,E2) such that V1 nV2 = 0, is defined to be the m-polar fuzzy graph G1 5 G2 = (A1 5A2, B1 5B2) of G* = (V1 x V2,E), where

Fig. 4. Semi-strong Product of G1 and G2

E = {(u, v1)(u, v2)|ueV1, v1v2eE2}u{(u1, w)(u2, w)|weV2,

u1u2eE1}u{(u1, v1)(u2, v2)|u1u2eE1, v^eE2 x V2

such that the following condition holds: for each i = 1,2, ...,m p,o(A15A2)(u, v) = p, °A1 (u)Ap, o A2 (v) for all (u,v) e V1 x V2, p,o(B1 5B2)((u,v1)(u,v2)) = p,+A1 (u)Ap,0B2(v1 v2)forallu e V1 and v1v2 e E2,

p, 0 (B1 5B2)((u, w)(u2, w)) = p, 0B1 (u1 u2)Ap, 0A2 (w) for all w e V2 and u1u2 e E1,

p,0(B15B2)((u1,v1)(u2,v2))= p,0B1(u1u2)Ap,0B2(v1v2) for all u1u2 e E1 and v1v2 e E2,

and p, ◦ (B1 5 B2)((w,x)(y,z)) = 0for all (w, x)(y, z) e (V1 x Vf - E). We now give an example that shows that the strong product of m-polar fuzzy graphs is again an m-polar fuzzy graph.

Example 4.10 Consider the 3-polar fuzzy graphs G1 = (A1,B1) and G2 = (A2,B2) as in Example 4.2. Also consider the strong product G1 5 G2, which is shown in Fig. 5.

It is easily checked that G1 5 G2 is a 3-polar fuzzy graph.

Theorem 4.11. The strong product G1 5 G2 of two m-polar fuzzy graphs is an m-polar fuzzy graph.

Theorem 4.12. If G1 = (A1,B1) and G2 = (A2,B2) are complete m-polar fuzzy graphs, then G1 5 G2 is complete.

Proof: By Theorem 4.11, we have that the strong product of m-polar fuzzy graphs is an m-polar fuzzy graph. Because G1 and G2 are complete, every pair of vertices are adjacent in the graph G1 5 G2 and

E = V1 x V|.

Let (u,v1)(u,v2) e E. Because G2 is complete, we have for each i = 1,2, ...,m

p,0(B15B2)((u, v1)(u, v2)) = p, 0A1(u) A p,0B2(v1v2)

= p, 0 A1 (u)Ap, 0 A2 (v1 )Ap, 0A2 (v2)

= p, 0 (A1 5A2)(u, v1)Ap,0 (A1 5A2)(u, v2).

Let (u1,w)(u2,w) e E. Because G1 is complete, we have for each i = 1,2, ...,m

p, 0(B15B2)((u1, w)(u2, w)) = p,0B1 (u1u2 )Ap, 0A2 (w) = p, 0 A1 (u1 )Ap, 0A1 (u2)Ap, 0 A2 (w)

= p,0 (A1 5A2 )(u1, w)Ap,0 (A1 5A2)(u2, w).

Finally, let (u1,v1)(u2,v2) e E. Then, because G1 and G2 are complete, we have for each i = 1,2,...,m

Pi = (Bj®B2)((Ui, Vj)(U2; V2))

= Pi + Bj (UjU2)APi°B2(vjV2)

= Pi °Aj (Uj )APi + Aj (U2)APi + A2 (vi )APi + A2 (V2)

= p,0(A15A2)(u1, v1)Ap,0(A15A2)(u2, v2).

Hence, G1 5 G2 is complete.

Theorem 4.13. If G1 = (A1,B1) and G2 = (A2,B2) are m-polar fuzzy graphs such that G1 nG2 is strong, then at least G1 or G2 must be strong.

Proof: Let us assume that both G1 and G2 are not strong m-polar fuzzy graphs. Then, there exists at least one u1v1 e E1 and u2v2 e E2 such that for each i = 1,2,.. ,,m p, 0 B1 (u1 v1) < p, 0 A1 (u1) Ap, 0 A1 (v1) and

p, 0 B2 (u2 v2) < p, 0 A2 (u2)Ap, 0A2 (v2).

Now, for each i = 1,2,...,mwe have p,0(B1nB2)((u1, v1), (u2, v2))

= p,0B1 (u1u2)Ap,0B2(v1v2)

<pi0A1(u1)Api0A1(u2)Api0A2(v1 )Ap,0A2(v2) (from the above assumption)

= p, 0(A1nA2)(u1, v1)Ap,0(A1nA2)(u2, v2).

This shows that, G1nG2 is not strong, which is a contradiction. Therefore, our assumption is wrong. This means one of G1 or G2 is strong.

The following result follows from the preceding theorem.

Theorem 4.14. If G1 = (A1,B1) and G2 = (A2,B2) are m-polar fuzzy graphs such that G1»G2 or G1 5 G2 is strong, then at least G1 or G2 must be strong.

5. Balanced m-polar fuzzy graphs

This section begins by defining the density of an m-polar fuzzy graph and balanced m-polar fuzzy graphs. Then it is proved that any complete m-polar fuzzy graph is balanced, but the converse is not always true.

Definition 5.1. The density of an m-polar fuzzy graph G = (A,B) of G* = (V,E) is D(G) = (p10D(G), p20D(G),..., pm°D(G)), whereforeach i = 1,2,...,m

2( £ Pi + B(Uv))

Pi + D(G)

£ (Pi°A(u)APi +A(v))-

U;V2 V

G is said to be balanced if for each i = 1,2, ...,m

Fig. 5. Strong product of Gj and G2

pi°D(H) < pi°D(G) for all non-empty subgraphs H of G. Example 5.2 Consider the 3-polar fuzzy graph G = (A,B) of G* = (V,E) where V = {a,b,c}, E = {ab,bc,ca},

a_i< 0.3,0.4,0.5 > < 0.3,0.4,0.5 > < 0.3,0.4,0.5 >

A = <-a-,-b-,-c-

B __ I < 0.1,0.2,0.2 > < 0.1,0.2,0.2 > < 0.1,0.2,0.2 > B j ab , bc , ca

We have,

Theorem 5.4. Every self-complementary strong m-polarfuzzy graph has density equal to 1 = (1,1,...,1).

Proof: Let G = (A,B) be a self-complementary strong m-polar fuzzy graph of G* = (V,E). Then, by Proposition 6.12 of [8], we have for each

i = 1,2,...,m and xyeV2, Xpi°B(xy) = 1 X (pi°A(x)Api°A(y)).

p D(n=_2(p1°B(ab)+p1°B(bc)+p1°B(ca))_

p1° 1 j (p1=A(a)Ap1=A(b)+p1=A(b)Ap1=A(c)+p1=A(c)Ap1=A(a)) 2(0.1 + 0.1 + 0.1) n6_ 0.3 + 0.3 + 0.3

Similarly, p2-D(G) = 1 and p3°D(G) = 0.8. Hence, D(G) = (0.67,1,0.8).

The non-empty subgraphs of G are H1 = {a,b}, H2 = {b,c} and H3 = {c,a}. Then,

DH ) = (2^ ■ T^TT2) = (0.67, ,,0.8),

D(H2 ) = (2^ ■ T^TT2) = (0-67,10.8)

and D(H3) = (2X01, 2X42, nxy2) = (0 67,1,0.8).

We see that, D(H1) = D(H2) = D(H3) = D(G) = (0.67,1,0.8). Hence, G is a balanced 3-polar fuzzy graph (see Fig. 6).

Theorem 5.3. Any complete m-polar fuzzy graph is balanced.

Proof: Let G = (A,B) be a complete m-polar fuzzy graph and H be a non-empty subgraph of G. Then, for each i = 1,2,...,m

p;°D(G) =

2( £ p; + B(uv)) 2( £ pt +A(u)Ap;+A(v))

U,v2V U,v2V

" J2 (p;+A(u)Ap;+A(v)) Z (p;oA(u)Ap;+A(v))

U,v2 V U,v2 V

Hence, p;+D(G) =

2( £ pi+B(uv))

u.vaV_

Y, CPi'0A(u)APi'oA(v))

= 1 (by the above) for each

i = 1,2,...,m. Thus, D(G) = 1.

The converse of this theorem is not true in general. For example, the 3-polar fuzzy graph in Fig. 7 has density equal to (1,1,1), but it is not self-complementary strong.

Here, we see that D(G) = (1,1,1) but G^G.

Theorem 5.5. Let G = (A,B) be a strong m-polar fuzzy graph such

that for each i = 1,2,...,m and uveV2, pi°B(uv) = 2 (pi°A(u)Api°A(v)).

Then, D(G) = 1 = (1,1_____1).

Proof: Because G = (A,B) is a strong m-polar fuzzy graph such that

for each i = 1,2,...,m and uveV2, pi°B(uv) =2(pi°A(u)Api°A(v)); therefore, by Proposition 6.13 of [8], we have G is self-complementary. Hence, by Theorem 5.4, it follows that D(G) = 1. Next, necessary and sufficient conditions are established for the direct product, semi-strong product and strong product of two m-polar fuzzy graphs to be balanced.

Theorem 5.6 Let G1 = (A1,B1) and G2 = (A2,B2) be two m-polar fuzzy graphs of G1 = (V1,E1) and G2 = (V2 ,E2), respectively. Then, D(Gk) < D(G1nG2) for k = 1,2 if and only if D(G1)=D(G2) = D(GmG2).

Proof: Let D(Gk) < D(G1 nG2) fork = 1,2. Then fori = 1,2,...,m

p;+D(H)

2( £ p;oB(uv)j

\u , v2V (H)_/_

£ (p;+A(u)ap; +A(v))

u , veV (H)

2( £ pi°A(u)Api°A(v))

Vu ,veV(H)_7 =2

" £ (pi°A(u)Api°A(v))

u , veV (H)

(where V(H) represents the vertex of H). This shows that G is balanced.

The converse of the above theorem is not always true. For example, the 3-polar fuzzy graph in Fig. 6 is balanced but not complete.

Below, we discuss two types of m-polar fuzzy graphs, each with density equal to 1 = (1,1,...,1).

p; + D(G1)

2( £ ^(u^))

_u1 ,u2 2 V1_

P (p;+A1(u1)Ap; +A1(u2))

u1, u2 2 V1

Fig. 6. 3-polar balanced fuzzy graph G

u ,V 2 V

2( £ Pi"Bi (UiU2)M2(V1)AA2(V2))

Ui ,U2 e V,

Vi ,vj e V2

£ (Pi "Ai (Ui ^Pi "Ai (U2)aA2 (Vi )aA2 (V2)) Ui,U2eVi

Vi ,vj e V2

2( £ Pi"Bl(UlU2)лpi"B2 (V1V2)) Ui,U2eVi

vi ,V2 e V2

£ (Pi "Ai (Ui ^Pi "Ai (U2)aA2 (Vi )aA2 (V2))

Ui ,U2 e Vi

vi ,V2 e V2

2( £ Pi°(BlПB2)((UlU2)(VlV2)) Ui ,U2 e Vi

V1 ;V2 e V2

£ Pi "(Ai nA2)((Ui)U2)(ViiV2)

Ui ,U2 e Vi vi ,V2 e V2

Hence, p,0D(G1) >p,0D(G1nG2) for each fori = 1,2,...,m, i.e., D(G1) > D(G1nG2). Similarly, D(G2) > D(G1nG2). Therefore, D(G1) = D(G2) = D(G1nG2).

Theorem 5.7. Let G1 = (A1,B1) and G2 = (A2,B2) be two balanced m-polar fuzzy graphs. Then, G1nG2 is balanced if and only if D(GO=D(G2)=D(GmG2).

Proof: Suppose D(G1nG2) is balanced. Then, D(Gk) < D(G1 nG2) for k = 1,2 and by Theorem 5.6, D(G1) = D(G2) = D(G1nG2).

Conversely, letD(G1) = D(G2) = D(G1nG2) and H be a non-empty subgraph of G1nG2. Then there exist subgraphs H1 ofG1 and H2 of G2.

Let pt 0D(G1) = p 0D(G2) = q, p, 0DH) = fand p, 0D(H2) = fr i = 1,2,...,mand ai,bi,qi,ri,si,ti e R.

Because G1 and G2 are balanced, therefore for i = 1,2,...,m p,0D(H1 )= s- < pi0D(G1)=qqiandpi 0D(H2)=b < p,°D(G2)= | Thus, s,r, + a-r,- < t,q, + b,-q,- i.e., siftgi < q-fori = 1,2,...,m. Hence, p,0D(H) < | = p,-0D(G1nG2) fori = 1,2,_,m.

Therefore, G1 nG2 is balanced. Similarly, we have the following results.

Theorem 5.8. Let G1 = (A1,B1) and G2 = (A2,B2) be two balanced m-polar fuzzy graphs. Then (i) G1$G2 is balanced if and only if D(G1) = D(G2) = D(G1$G2).

(ii) G1 5 G2 is balanced if and only ifD(G1) = D(G2) = D(G1 5 G2).

6. Product m-polar fuzzy graphs

In this section, a new type of m-polar fuzzy graph, known as product m-polar fuzzy graphs, is defined.

Definition 6.1. A product m-polar fuzzy graph of a graph G* = (V,E) is a pair G = (A,B) where A:V / [0,1]m is an m-polar fuzzy set in Vand

B : V2/[0,1]m is an m-polar fuzzy set in V2 such that for each

i = 1,2,...,m; p,0B(xy) <p,0A(x) xp,0A(y) forallxyeV2.

Remark 6.2 Because for each i = 1,2,...,m; p,°A(x) and p,°A(y) are less than or equal to 1, it follows thatp, 0B(xy) < p, 0A(x) x p, 0A(y) <

pi0A(x)Ap,0A(y) for all xyeV2. Hence, every product m-polar fuzzy graph is an m-polar fuzzy graph.

Definition 6.3. A product m-polar fuzzy graph G = (A,B) is said to be complete if for each i = 1,2,...,m and x,y e V,

p,0B(xy)=p,0A(x) xp,0A(y).

Definition 6.4. The complement of the product m-polar fuzzy graph G = (A,B) is an m-polar fuzzy graph G = (A, B) where A = A and B is defined by

p,0B(xy) =p,0A(x) xp,0A(y) -p,0B(xy)foreachi = 1,2,.,.,mand xyeV2.

Remark 6.5 Because for all xyeV2, i = 1,2, ...,m; p,0B(xy)=p,0A(x) x p,0A(y) -p,0B(xy) < p,0A(x) x p,0A(y),

Therefore, G is a product m-polar fuzzy graph.

Definition 6.6. The union G1 uG2 = (A1 uA2, B1 uB2) of two product m-polar fuzzy graphs G1 = (A1,B1) of G\ = (V1, E1) and G2 = (A2,B2) of G'2 = (V2, E2) is defined as follows: for each i = 1,2...,m

Pi" (A,uA2)(X) =

Pi°(BlUB2)(xy)

Pi°Al(x) if xeVi - V2

Pi " A2 (x) if xe V2 - Vi Pi°Al(x)vpi°A2(x) if xeVinV2.

Pi°Bl(xy) if xyeEi - E2

Pi ° B2 (xy) if xyeE2 - El , Pi ° Bi (xy)vp, ° B2 (xy) if xy e El nE2.

Proposition 6.7. The direct product Gi nG2 of two product m-polar fuzzy graphs G1 = (A1,B1) and G2 = (A2,B2) is a product m-polar fuzzy graph.

Proof: Let (u1,v1)(u2,v2) e E. Then u1u2 e E1 and v1v2 e E2. Now, for each i = 1,2,...,m we have

Pi "(B^X^i, Vi), (U2, V2))

Fig. 7. G is a 3-polar fuzzy graph with density (1,1,1) but is not self-complementary and strong.

Fig. 8. G1 and G2 are complete product 3-polar fuzzy graphs, but G1 nG2 is not complete.

= minfpi °B1(u1u2), pi°B2 (V1V2)}

< min{pi°A1(u1) X pi °A1(u2),pi°A2(V1) X pi °A2(V2)}

= minfpi°A1(u1),pi°A2(v1)} X minfpi°A1(u2),pi°A2(v2)}

= pi ° (A1 n^2 )(u1, V1) X pi ° (A1 nA2 )(u2, V2) . Hence, the result.

Remark 6.8 Let G1 = (A1,B1) and G2 = (A2,B2) be two complete product m-polar fuzzy graphs of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively. Then, G1 nG2 may not be complete.

For example, let us take product 3-polar fuzzy graphs G1 and G2, which are complete, but G1 nG2 is not complete (see Fig. 8)

Definition 6.9. Let G1 = (A1,B1) and G2 = (A2,B2) be the product m-polar fuzzy graphs of the graphs G1 = (V1,E1) and G2 = (V2,E2), respectively. Then, the ring sum of G1 and G2 is denoted by G = G1 © G2 = (A1 © A2,B1 © B2) and defined as follows: for each i = 1,2...,m

pi ° (A1 ©A2 )(u) = pi ° (A1 uA2 )(u) for all ue V1 uV2 and

< pi°B1 (uv) if uveE1 - E2 pi°(B1©B2)(uv) = : pi°B2(uv) if uveE2 - E1 I 0 otherwise.

Proposition 6.10. Let G1 = (A1,B1) and G2 = (A2,B2) be the product m-polar fuzzy graphs of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively. Then, the ring sum G = G1 © G2 = (A1 © A2,B1 © B2) is a product m-polar fuzzy graph.

Proof: We show the following: for each i = 1,2,...,m pi°(B1 ©B2)(uv) < pio(A1©A2)(u) X pi o(A1©A2)(v) for all uveE1uE2.

Case (i): Let uv e E1 - E2 and u,v e V1 - V2. Then, for each i = 1,2,...,m

pi °(B1©B2 )(uv)

= pi°B1 (uv) <pi°A1(u) x pi °A1(v)

= pi°(A1 ©A2)(u) x pi°(A1©A2)(v).

Case (ii): Let uv e E1 - E2 and u e V1-V2, ve V1nV2. Then, for each i = 1,2,...,m

pi °(B1©B2 )(uv)

= pi °B1(uv)

< pi°A1 (u) x max {pi °A1(v), pi °A2(v)}

< pi ° (A1UA2 )(u) x pi ° (A1UA2 )(v)

= pi ° (A1 ©A2)(u) xpi °(A1©A2)(v).

Case (iii): Letuv e E1 - E2 and u, veV1nV2. Then, for each i = 1,2,...,m

pi°(B1©B2)(uv) = pi °B1(uv)

< maxfpi°A1(u),pi°A2(u)} x max{pi°A1(v),pi°A2(v)}

< pi ° (A1UA2 )(u) x pi ° (A1UA2 )(v)

= pi ° (A1 ©A2)(u) xpi °(A1©A2)(v).

Similarly, we can show that if uv e E2-E1, then for each i = 1,2, ...,m

pi°(B1©B2)(uv) < pi°(A1©A2)(u) x pi°(A1©A2)(v). Hence, the result.

7. Conclusions

The theory of graphs is an extremely useful tool in solving combinatorial problems in different areas, including algebra, number theory, geometry, topology, operation research, optimization, and computer science. Because research and models of real world problems often involve multi-agent, multi-attribute, multi-object, multi-index, and multi-polar information and uncertainty, the study of m-polar fuzzy graphs is significant. The m-polar fuzzy models gives increasing precision, flexibility, and comparability compared to the classical, fuzzy and bipolar fuzzy models. Therefore, we studied some properties of m-polar fuzzy graphs. In this study, we defined three new operations, density and balanced m-polar fuzzy graphs, but the degree of the resultant graphs was not determined. We are currently working on this topic. The results of this paper will help to find new algorithms and more important results. Our next plan is to extend our research work to regular and irregular m-polar fuzzy graphs, m-polar fuzzy intersection graphs, m-polar fuzzy interval graphs, and m-polar fuzzy hypergraphs.

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 thankfully acknowledged. We are thankful to the Editor Dr. Zhi Zhou and all of the reviewers for their valuable comments and suggestions to improve the paper.

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