# Vertex Degrees and Isomorphic Properties in Complement of an m-Polar Fuzzy Graph.

1. Introduction

Akram [1] introduced the notion of bipolar fuzzy graphs describing various methods of their construction as well as investigating some of their important properties. Bhutani [2] discussed automorphism of fuzzy graphs. Chen et al. [3] generalized the concept of bipolar fuzzy set to obtain the notion of an m-polar fuzzy set. The notion of an m-polar fuzzy set is more advanced than fuzzy set and eliminates ambiguity more absolutely. Ghorai and Pal [4] studied some operations and properties of an m-polar fuzzy graph. Mordeson and Peng [5] defined join, union, Cartesian product, and composition of two fuzzy graphs. Rashmanlou et al. [6] discussed some properties of bipolar fuzzy graphs and their results. Sunitha and Vijaya Kumar [7] defined the complement of a fuzzy graph in another way which gives a better understanding about that concept. We have studied product m-polar fuzzy graph, product m-polar fuzzy intersection graph, and product m-polar fuzzy line graph [8].

In this article, we study the Cartesian product and composition of two m-polar fuzzy graphs and compute the degrees of the vertices in these graphs. The notions of normal product and tensor product of m-polar fuzzy graphs are introduced and some properties are studied. Also in the present work, we introduce the concept of complement, complement of an m-polar fuzzy graph, and some properties are discussed.

These concepts strengthen the decision-making in critical situations. Some applications to decision-making are also studied.

In this article, unless and otherwise specified, all graphs considered are m-polar fuzzy graphs.

2. Preliminaries

Definition 1. The m-polar fuzzy graph of a graph [G.sup.*] = (V, E) is a pair G = (W,F), where W : V [right arrow] [0,1]m is an m-polar fuzzy set in V and F: V [??] V [right arrow] [[0,1].sup.m] is an m-polar fuzzy set in V [??] V such that F(qr) [less than or equal to] min{W(q), W(r)} for all qr [member of] V [??] V and F(qr) = 0 for all edges qr [member of] ([[??].sup.2] - E) (0 = 0,0,..., 0) is the smallest element in [[0,1].sup.m]. W is called the m-polar fuzzy vertex set of G and F is called m-polar fuzzy edge set of G.

Sometimes we denote the graph G = (W, F) by G = (V, W, F) also.

Definition 2. Given two graphs [G.sub.1], [G.sub.2] their Cartesian product, [G.sub.1] x [G.sub.2] = ([V.sub.1] x [V.sub.2], [W.sub.1] x [W.sub.2], [F.sub.1] x [F.sub.2]), is defined as follows:

for i = 1, 2, ..., m, we have

(i) [p.sub.i] [omicron] ([W.sub.1] x [W.sub.2])([q.sub.1], [q.sub.2]) = min{[p.sub.i] [omicron] [W.sub.1]([q.sub.1]), [p.sub.i] [W.sub.2]([q.sub.2])} for all ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2],

(ii) [p.sub.i] [omicron] ([F.sub.1] x [F.sub.2])((q,[q.sub.2])(q,[r.sub.2])) = min{[p.sub.i] [omicron] [W.sub.1](q), [p.sub.i] [omicron] [F.sub.2]([q.sub.2][r.sub.2])} for all q [member of] [V.sub.1], [q.sub.2][r.sub.2] [member of] [E.sub.2],

(iii) [p.sub.i] [omicron] ([F.sub.1] x [F.sub.2])(([q.sub.1], m)([r.sub.1], m)) = min{[p.sub.i] [omicron] [F.sub.1]([q.sub.1][r.sub.1]), [p.sub.i] [omicron] [W.sub.2](m)} for all m [member of] [V.sub.2], [q.sub.1][r.sub.1] [member of] [E.sub.1],

(iv) [p.sub.i] [omicron] ([F.sub.1] x [F.sub.2])(([q.sub.1],[q.sub.2])([r.sub.1], [r.sub.2])) = 0 for all ([q.sub.1], [q.sub.2])([r.sub.1], [r.sub.2]) [member of] [([V.sub.1] x [V.sub.2]).sup.2] - E.

Definition 3. Given two graphs [G.sub.1], [G.sub.2] their composition, [G.sub.1] [[G.sub.2]] = ([V.sub.1] x [V.sub.2], [W.sub.1] [[W.sub.2]], [F.sub.1] [[F.sub.2]]), is defined as follows: for i = 1,2, ...,m,we have

(i) [p.sub.i] [omicron] ([W.sub.1][[W.sub.2]])([q.sub.1], [q.sub.2]) = min{[p.sub.i] [omicron] [W.sub.1]([q.sub.1]), [p.sub.i] [omicron] [W.sub.2]([q.sub.2])} for all ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2],

(ii) [p.sub.i] [omicron] ([F.sub.1][[F.sub.2]])((q,[q.sub.2])(q,[r.sub.2])) = min{[p.sub.i] [omicron] [W.sub.1](q), [p.sub.i] [omicron] [F.sub.2] ([q.sub.2][r.sub.2])} for all q [member of] [V.sub.1], [q.sub.2][r.sub.2] [member of] [E.sub.2],

(iii) [p.sub.i] [omicron] ([F.sub.1][[F.sub.2]])(([q.sub.1],m)([r.sub.1],m)) = min{[p.sub.i] [omicron] [F.sub.1]([q.sub.1][r.sub.1]), [p.sub.i] [omicron] [W.sub.2](m)} for all m [member of] [V.sub.2], [q.sub.1][r.sub.1] [member of] [E.sub.1],

(iv) [mathematical expression not reproducible].

Definition 4. The normal product of [G.sub.1] and [G.sub.2] is defined as the m-polar fuzzy graph [G.sub.1] x [G.sub.2] = ([V.sub.1] x [V.sub.2], [W.sub.1] x [W.sub.2], [F.sub.1] x [F.sub.2]) on [G.sup.*] = (V,E), where E = {((q,[q.sub.2])(q,[r.sub.2])) | q [member of] [V.sub.1], [q.sub.2][r.sub.2] [member of] [E.sub.2]} [union] {([q.sub.1],m)([r.sub.1],m) | [q.sub.1][r.sub.1] [member of] [E.sub.1], m [member of] [V.sub.2]} [union] {([q.sub.1], [q.sub.2])([r.sub.1],[r.sub.2]) | [q.sub.1][r.sub.1] [member of] [E.sub.1], [q.sub.2][r.sub.2] [member of] [E.sub.2]} such that for i = 1,2, ...,m, we have

(i) [p.sub.i] [omicron] ([W.sub.1] x [W.sub.2])([q.sub.1],[q.sub.2]) = min{[p.sub.i] [omicron] [W.sub.1]([q.sub.1]), [p.sub.i] [omicron] [W.sub.2]([q.sub.2])} for all ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2],

(ii) [p.sub.i] [omicron] ([F.sub.1] x [F.sub.2])((q,[q.sub.2])(q,[r.sub.2])) = min{[p.sub.i] [omicron] [W.sub.1](q),[p.sub.i] [omicron] [F.sub.2]([q.sub.2][r.sub.2])} for all q [member of] [V.sub.1], [q.sub.2][r.sub.2] [member of] [E.sub.2],

(iii) [p.sub.i] [omicron] ([F.sub.1] x [F.sub.2])(([q.sub.1],m)([r.sub.1],m)) = min{[p.sub.i] [omicron] [F.sub.1]([q.sub.1][r.sub.1]), [p.sub.i] [omicron] [W.sub.2](m)} for all m [member of] [V.sub.2], [q.sub.1][r.sub.1] [member of] [E.sub.1],

(iv) [p.sub.i] [omicron] ([F.sub.1] x [F.sub.2])(([q.sub.1],[q.sub.2])([r.sub.1],[r.sub.2])) = min{[p.sub.i] [omicron] [F.sub.1]([q.sub.1][r.sub.1]), [p.sub.i] [omicron] [F.sub.2]([q.sub.2][r.sub.2])} for all [q.sub.1][q.sub.2] [member of] [E.sub.1] and [r.sub.1][r.sub.2] [member of] [E.sub.2].

Definition 5. The tensor product of [G.sub.1] and [G.sub.2] is defined as [G.sub.1] [cross product] [G.sub.2] = ([V.sub.1] x [V.sub.2], [W.sub.1] [cross product] [W.sub.2], [F.sub.1] [cross product] [F.sub.2]) on [G.sup.*] = (V, E), where E = {(([q.sub.1], [q.sub.2])([r.sub.1], [r.sub.2])) | [q.sub.1][r.sub.1] [member of] [E.sub.1], [q.sub.2][r.sub.2] [member of] [E.sub.2]} such that for i = 1, 2, ..., m, we have

(i) [p.sub.i] [omicron] ([W.sub.1] [cross product] [W.sub.2])([q.sub.1], [q.sub.2]) = min{[p.sub.i] [omicron] [W.sub.1]([q.sub.1]), [p.sub.i] [omicron] [W.sub.2]([q.sub.2])} for all ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2],

(ii) [p.sub.i] [omicron] ([F.sub.1] [cross product] [F.sub.2])(([q.sub.1],[q.sub.2])([r.sub.1],[r.sub.2])) = min{[p.sub.i] [omicron] [F.sub.1]([q.sub.1][r.sub.1]), [p.sub.i] [omicron] [F.sub.2]([q.sub.2][r.sub.2])} for all [q.sub.1][q.sub.2] [member of] [E.sub.1] and [r.sub.1][r.sub.2] [member of] [E.sub.2].

Definition 6. The union [G.sub.1] [union] [G.sub.2] = ([V.sub.1] [union] [V.sub.2], [W.sub.1] [union] [W.sub.2], [F.sub.1] [union] [F.sub.2]) of the graphs [G.sub.1] = ([V.sub.1], [W.sub.1], [F.sub.1]) and [G.sub.2] = ([V.sub.2], [W.sub.2], [F.sub.2]) of [G.sub.1.sup.*] and [G.sub.2.sup.*], respectively, is defined as follows: for i = 1,2,...,m, we have

(i) [mathematical expression not reproducible] (1)

(ii) [mathematical expression not reproducible] (2)

(iii) [p.sub.i] [omicron] ([F.sub.1] [union] [F.sub.2]) (qr) = 0 if qr [member of] [([V.sub.1] [??] [V.sub.2]).sup.2] - [E.sub.1] [union] [E.sub.2]. (3)

Definition 7. The join of the graphs [G.sub.1] = ([V.sub.1], [W.sub.1], [F.sub.1]) and [G.sub.2] = ([V.sub.2], [W.sub.2], [F.sub.2]) of [G.sub.1.sup.*] and [G.sub.2.sup.*], respectively, is defined as [G.sub.1] + [G.sub.2] = ([V.sub.1] [union] [V.sub.2], [W.sub.1] + [W.sub.2], [F.sub.1] + [F.sub.2]) such that for ii = 1, 2, ..., m, we have the following:

(i) [p.sub.i] [omicron] ([W.sub.1] + [W.sub.2])(q) = [p.sub.i] [omicron] ([W.sub.1] [union] [W.sub.2])(q) if q [member of] [V.sub.1] [union] [V.sub.2].

(ii) [p.sub.i] [omicron] ([F.sub.1] + [F.sub.2])(qr) = [p.sub.i] [omicron] ([F.sub.1] [union] [F.sub.2])(qr) if qr [member of] [E.sub.1] [upsilon] [E.sub.2].

(iii) [p.sub.i] [omicron] ([F.sub.1] + [F.sub.2])(qr) = min{[p.sub.i] [omicron] [W.sub.1] (q), [p.sub.i] [omicron] [W.sub.2](r)} if qr [member of] E', where E' denotes the set of all edges joining the vertices of [V.sub.1] and [V.sub.2].

(iv) [p.sub.i] [omicron] ([F.sub.1] + [F.sub.2])(qr) = 0 if qr [member of] [([V.sub.1] [??] [V.sub.2]).sup.2] - [E.sub.1] [union] [E.sub.2] [union] E.

3. Degree of Vertices in m-Polar Fuzzy Graph

Definition 8. Let G = (V, W, F) be an m-polar fuzzy graph. Then the degree of a vertex q in G is defined as [d.sub.G](q) = <[summation over (qr[member of]E q[not equal to]r)][p.sub.1] [omicron] F(qr), [summation over (qr[member of]E q[not equal to]r)][p.sub.2] [omicron] F(qr), [summation over (qr[member of]E q[not equal to]r)][p.sub.3] [omicron] F(qr),..., [summation over (qr[member of]E q[not equal to]r)][p.sub.m] [omicron] F(qr)>.

Further, a highly irregular m-polar fuzzy graph is defined as an m-polar fuzzy graph G = (V,W,F) in which every vertex of G = (V, W, F) is adjacent to vertices with distinct degrees.

Example 9. Consider the graph G = (V, W, F) of [G.sup.*] = (V, E), where V = {K, L, M, N}, E = {KL, LM, MN, NK}, W = {<0.2,0.5, 0.6>/K, <0.3, 0.5, 0.7>/L, <0.3, 0.6, 0.7>/M, <0.4, 0.7,0.8>/N}, and F = {<0.2, 0.4, 0.5>/KL, <0.3, 0.4, 0.6>/LM, <0.3, 0.5, 0.6>/MN, <0.2, 0.3, 0.6>/NK} as in Figure 1.

In this graph, we have dG(K) = <0.2, 0.4, 0.5> + <0.2, 0.3, 0.6> = <0.4, 0.7, 1.1>, [d.sub.G](L) = <0.2, 0.4, 0.5> + <0.3, 0.4, 0.6> = <0.5, 0.8, 1.1>, [d.sub.G](M) = <0.3, 0.4, 0.6> + <0.3, 0.5, 0.6> = <0.6, 0.9, 1.2>, [d.sub.G](N) = <0.3, 0.5, 0.6> + <0.2, 0.3, 0.6> = <0.5, 0.8, 1.2>.

4. Cartesian Product and Vertex Degree

From the definition of Cartesian product graphs, for every vertex ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2], we have

[mathematical expression not reproducible] (4)

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

Theorem 10. Let [G.sub.1] = ([V.sub.1], [W.sub.1], [F.sub.1]) and [G.sub.2] = ([V.sub.2], [W.sub.2], [F.sub.2]) be two graphs. If [p.sub.i] [omicron] [W.sub.1] [greater than or equal to] [p.sub.i] [omicron] [F.sub.2] and [p.sub.i] [omicron] [W.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1] then [mathematical expression not reproducible].

Proof. By definition, we get

[mathematical expression not reproducible] (5)

Example 11. Consider two graphs 1 , 2 and their Cartesian product, [G.sub.1] x [G.sub.2], as shown in Figure 2.

Since [p.sub.i] [omicron] [W.sub.1] [greater than or equal to] [p.sub.i] [omicron] [F.sub.2] and [p.sub.i] [omicron] [W.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1], by Theorem 10, we have

[mathematical expression not reproducible] (6)

5. Composition Graph and Vertex Degree

From the definition of composition graphs, for every vertex ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2], we have

[mathematical expression not reproducible] (7)

Theorem 12. Let [G.sub.1] = ([V.sub.1], [W.sub.1], [F.sub.1]) and [G.sub.2] = ([V.sub.2], [W.sub.2], [F.sub.2]) be two graphs. If [p.sub.i] [omicron] [W.sub.1] [greater than or equal to] [p.sub.i] [omicron] [F.sub.2] and [p.sub.i] [omicron] [W.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1] then [mathematical expression not reproducible] for all ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2] for i = 1, 2, ..., m.

Proof. By definition, we get

[mathematical expression not reproducible] (8)

Example 13. Consider two graphs [G.sub.1], [G.sub.2] and their composition, [G.sub.1] [[G.sub.2]], as shown in Figure 3.

Since [p.sub.i] [omicron] [W.sub.1] [greater than or equal to] [p.sub.i] [omicron] [F.sub.2] and [p.sub.i] [omicron] [W.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1], by Theorem 12, we have

[mathematical expression not reproducible] (9)

6. Normal Product and Vertex Degree

From the definition of normal product graphs, for every vertex ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2], we have

[mathematical expression not reproducible] (10)

Theorem 14. Let [G.sub.1] = ([V.sub.1],[W.sub.1],[P.sub.1]) and [G.sub.2] = ([V.sub.2],[W.sub.2],[F.sub.2]) be two graphs. If [p.sub.i] [omicron] [W.sub.1] [greater than or equal to] [p.sub.i] [omicron] [F.sub.2], [p.sub.i] [omicron] [W.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1] and [p.sub.i] [omicron] [F.sub.1] [less than or equal to] [p.sub.i] [omicron] [F.sub.2], then [mathematical expression not reproducible].

Proof. By definition, we get

[mathematical expression not reproducible] (11)

Example 15. Consider two graphs [G.sub.1], [G.sub.2] and their normal product, [G.sub.1] x [G.sub.2], is shown in Figure 4.

Since [p.sub.i] [omicron] [W.sub.1] [greater than or equal to] [p.sub.i] [omicron] [F.sub.2] [p.sub.i] [omicron] [W.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1] and [p.sub.i] [omicron] [F.sub.1] [less than or equal to] [p.sub.i] [omicron] [F.sub.2], by Theorem 14, we have

[mathematical expression not reproducible] (12)

7. Tensor Product and Vertex Degree

From the definition of tensor product, for every vertex ([q.sub.1], [q.sub.2]) [member of] [V.sub.1] x [V.sub.2] for i = 1,2,...,m

[mathematical expression not reproducible] (13)

Theorem 16. Let [G.sub.1] = ([V.sub.1],[W.sub.1],[F.sub.1]) and [G.sub.2] = ([V.sub.2],[W.sub.2],[F.sub.2]) be two m-polar fuzzy graphs. If [p.sub.i] [omicron] [F.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1], then [mathematical expression not reproducible].

Proof. Let [p.sub.i] [omicron] [F.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1]. Then we get

[mathematical expression not reproducible] (14)

Let [p.sub.i] [omicron] [F.sub.1] [greater than or equal to] [p.sub.i] [omicron] [F.sub.2]. Then we get

[mathematical expression not reproducible] (15)

Example 17. Consider two graphs [G.sub.1], [G.sub.2] and their tensor product, [G.sub.1] [cross product] [G.sub.2], is shown in Figure 5.

Since [p.sub.i] [omicron] [F.sub.2] [greater than or equal to] [p.sub.i] [omicron] [F.sub.1], by Theorem 16, we have

[mathematical expression not reproducible] (16)

8. [mu]-Complement of an m-Polar Fuzzy Graph

Now for an m-polar fuzzy graph G, we introduce the notion of its [mu]-complement.

Definition 18. The complement of an m-polar fuzzy graph G = (V, W, F) is also an m-polar fuzzy graph [bar.G] = (V, [bar.W], [bar.F]), where [bar.W] = W and [bar.F] is defined as follows:

[p.sub.i] [omicron] [bar.F] (qr) = [conjunction]([p.sub.i] [omicron] W(q), [p.sub.i] [omicron] W(r)) - [p.sub.i] [omicron] F (qr) [for all]q, r [member of] V for i = 1, 2, ..., m. (17)

Definition 19. Let G = (V, W, F) be an m-polar fuzzy graph. If G is isomorphic to [bar.G] we say G is self-complementary. Similarly, if G is weak isomorphic to [bar.G] then we say G is self-weak complementary.

Definition 20. The [mu]-complement of G is defined as [G.sup.[mu]] = (V, [W.sup.[mu]], [F.sup.[mu]]), where [W.sup.[mu]] = W and [F.sup.[mu]] is given by

[mathematical expression not reproducible] (18)

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

Theorem 21. Let G = (V, W, F) be a self-weak complementary and highly irregular graph. Then, we have [summation over (q[not equal to]r)][p.sub.i] [omicron] F(qr) [less than or equal to] 0.5 [summation over (q[not equal to]r)] [conjunction] {[p.sub.i] [omicron] W(q),[p.sub.i] [omicron] W(r)}.

Proof. Let G = (V, W, F) be a self-weak complementary, highly irregular graph of [G.sup.*] = (V, E). Then, there exists a weak isomorphism g from G to [bar.G] such that for all q,r [member of] V, we get [p.sub.i] [omicron] W(q) = [p.sub.i] [omicron] [bar.W](g(q)) = [p.sub.i] [omicron] W(g(q)), [p.sub.i] [omicron] F(qr) [less than or equal to] [p.sub.i] [omicron] [bar.F](g(q)g(r)). From the definition of complement, from the above inequality, for all q,r [member of] V, we get that [p.sub.i] [omicron] F(qr) [less than or equal to] [p.sub.i] [omicron] [bar.F](g(q)g(r)) = [conjunction]([p.sub.i] [omicron] W(q), [p.sub.i] [omicron] W(r)) - [p.sub.i] [omicron] F(g(q)g(r)) and [p.sub.i] [omicron] F(qr) + [p.sub.i] [omicron] F(g(q)g(r)) [less than or equal to] [conjunction]([p.sub.i] [omicron] W(q), [p.sub.i] [omicron] W(r)). Hence [summation over (q[not equal to]r)] [p.sub.i] [omicron] F(qr) + [summation over (q[not equal to]r)] [p.sub.i] [omicron] F(g(q)g(r)) [less than or equal to] [summation over (q[not equal to]r)]([p.sub.i] [omicron] W(q), [p.sub.i] [omicron] W(r)). Thus, we get

[mathematical expression not reproducible] (19)

Theorem 22. Let G = (V, W, F) be a graph which is self-complementary and highly irregular. Then, we have [summation over (q[not equal to]r)] [p.sub.i] [omicron] F(qr) = 0.5 [summation over (q[not equal to]r)] [conjunction]{[p.sub.i] [omicron] W(q), [p.sub.i] [omicron] W(r)},for all qr [member of] [[??].sup.2], i = 1, 2, ..., m.

Proof. Let G = (V, W, F) be a self-complementary, highly irregular graph of [G.sup.*] = (V,E). Then, there exists an isomorphism g from G to [bar.G] such that [p.sub.i] [omicron] W(q) = [p.sub.i] [omicron] [bar.W](q) for all q [member of] V and [p.sub.i] [omicron] F(qr) = [p.sub.i] [omicron] [bar.F](g(q)g(r)) for all qr [member of] [[??].sup.2].

Let qr [member of] [[??].sup.2]. Then, for i = 1, 2, ..., m, we have

[p.sub.i] [omicron] [bar.F](g(q)g(r)) = [conjunction] ([p.sub.i] [omicron] W(g (q)), [p.sub.i] [omicron] W(g (r))) - [p.sub.i] [omicron] F(g (q)g(r)); (20)

that is, [p.sub.i] [omicron] F(qr) = [conjunction]([p.sub.i] [omicron] W(g(q)), [p.sub.i] [omicron] W(g(r))) - [p.sub.i] [omicron] F(g(q)g(r)). Therefore,

[mathematical expression not reproducible] (21)

Theorem 23. Let [G.sub.1] = ([V.sub.1], [W.sub.1],[F.sub.1]) and [G.sub.2] = ([V.sub.2], [W.sub.2], [F.sub.2]) be two graphs. If [G.sub.1] and [G.sub.2] are isomorphic, then their complements, [G.sub.1.sup.[mu]] and [G.sub.2.sup.[mu]], are also isomorphic.

Proof. Let [G.sub.1] [congruent to] [G.sub.2] and let g be an isomorphism from [G.sub.1] to G2. Then for i = 1, 2, ..., m

[p.sub.i] [omicron] [W.sub.1](q) = [p.sub.i] [omicron] [W.sub.2](g(q)), [for all]q [member of] [V.sub.1], [p.sub.i] [omicron] [F.sub.1](qr) = [p.sub.i] [omicron] [F.sub.2](g(q)g(r)), [for all]qr [member of] [E.sub.1]. (22)

If [p.sub.i] [omicron] [F.sub.1](qr) > 0, then [p.sub.i] [omicron] [F.sub.2](g(q)g(r)) > 0, and [p.sub.i] [omicron] [F.sub.1.sup.[mu]] (qr) = min {[p.sub.i] [omicron] [W.sub.1] (q), [p.sub.i] [omicron] [W.sub.1] (r)} - [p.sub.i] [omicron] [F.sub.1] (qr) = min {[p.sub.i] [omicron] [W.sub.2] (g (q)), [p.sub.i] [omicron] [W.sub.2] (g (r))} - [p.sub.i] [omicron] [p.sub.2] (g (q)g(r)) = [p.sub.i] [omicron] [F.sub.2.sup.] (g(q)g(r)). (23)

If [p.sub.i] [omicron] [F.sub.1] (qr) = 0, then [p.sub.i] [omicron] [F.sub.2](g(q)g(r)) = 0, and [p.sub.i] [omicron] [F.sub.1.sup.[mu]](qr) = 0 = [p.sub.i] [omicron] [F.sub.2.sup.[mu]](g(q)g(r)).

Thus, [p.sub.i] [omicron] [F.sub.1.sup.[mu]] (qr) = [p.sub.i] [omicron] [F.sub.2.sup.[mu]] (g(q)g(r)) for all qr [member of] [E.sub.1].

Therefore, g from [G.sub.1.sup.[mu]] to [G.sub.2.sup.[mu]] is an isomorphism; that is, [G.sub.1.sub.[mu]] [congruent to] [G.sub.2.sup.[mu]].

Theorem 24. Let [G.sub.1] = ([V.sub.1], [W.sub.1], [F.sub.1]) and [G.sub.2] = ([V.sub.2], [W.sub.2], [F.sub.2]) be two graphs such that [V.sub.1] [intersection] [V.sub.2] = [phi]. Then [([G.sub.1] + [G.sub.2]).sup.[mu]] [congruent to] [G.sub.1.sup.[mu]] [union] [G.sub.2.sup.[mu]].

Proof. Let I: [V.sub.1] [union] [V.sub.2] [right arrow] [V.sub.1] [union] [V.sub.2] be the identity map. To show that [([G.sub.1] + [G.sub.2]).sup.[mu]] [congruent to] [G.sub.1.sup.[mu]] [union] [G.sub.2.sub.[mu]], it is enough to prove that [([p.sub.i] [omicron] [W.sub.1] + [p.sub.i] [omicron] [W.sub.2]).sup.[mu]](q) = [p.sub.i] [omicron] [W.sub.1.sup.[mu]] (q) [union] [p.sub.i] [omicron] [W.sub.2.sup.[mu]](q) and [([p.sub.i] [omicron] [F.sub.1] + [p.sub.i] [omicron] [F.sub.2]).sup.[mu]] (qr) = [p.sub.i] [omicron] [F.sub.1.sup.[mu]](qr) [union] [p.sub.i] [omicron] [F.sub.2.sup.[mu]] (qr) for all q, r [member of] V and for i = 1, 2, ..., m. Then for any q, r [member of] V, we have

[mathematical expression not reproducible] (24)

Moreover,

[mathematical expression not reproducible] (25)

Hence, the result is proved.

Theorem 25. Let [G.sub.1] = ([V.sub.1], [W.sub.1],[F.sub.1]) and [G.sub.2] = ([V.sub.2], [W.sub.2], [F.sub.2]) be two graphs such that [V.sub.1] [intersection] [V.sub.2] = [phi]. Then [([G.sub.1] [union] [G.sub.2]).sup.[mu]] [congruent to] [G.sub.1.sup.[mu]] [union] [G.sub.2.sup.[mu]].

Proof. We shall prove that the identity map I : [V.sub.1] [union] [V.sub.2] [right arrow] [V.sub.1] [union] [V.sub.2] is the required isomorphism. Then for any i, 1 [less than or equal to] i [less than or equal to] m we have the following:

[mathematical expression not reproducible].

Hence, the result is proved.

9. m-Polar Fuzzy Graphs and Their Applications

One of the important applications of -polar fuzzy graphs is guidance in decision-making. For example, let us consider a scenario where a college wants to select a principal. Let W = {Kasim, Leo, Modi, Nandhu} be the set of doctorates shortlisted and let F = {Krishna, Srinu, Mano, Ramesh} be the set of college management members. The management has to make a selection taking into account the set of qualities = {management skills, attitude, patents, and research}. Each member from set F can assign a value between 0 and 1 to each element listed in the set for each candidate mentioned in set W, such as W(K) = <0.3, 0.4, 0.6, 0.2>, W(L) = <0.9, 0.5, 0.7, 0.3>, W(M) = <0.4, 0.7, 0.3, 0.5>, and W(N) = <0.3, 0.6, 0.7, 0.6>. Then we have a 4-polar fuzzy graph, in which each member gives his opinion to w that belongs to W on the basis of his qualities. So W(K), W(L), W(M), and W(N) denote the degree of management skills, attitude, patents, and research of each person given by the members of college management and edges denote the identical qualities of two persons (see Figure 6).

The membership degree of the edge indicates that Kasim is 30% suitable, Leo is 40% suitable, Modi is 60% suitable, and Nandhu is 20% suitable. Thus, the edge specifies that Modi is the suitable person for the post of principal. In the same way, LM specifies Leo, MN specifies Leo, and also specifies Leo. Hence, according to all, Leo has a higher value of desirability among the shortlisted candidates and should be appointed as the principal.

10. Conclusions

Numerous uses can be harnessed from the theory of a fuzzy graph in the areas of number theory, algebra, topology, operation research, and so on. Not only researchers but also the common man can benefit from m-polar fuzzy data. It solves the day to day problems that are faced in the society related to probabilistic data. The novel approach discussed here enables decision makers to formulate better- preferred option with the help of unique case pattern than the conventional fuzzy graph approaches. The concepts complement, [mu]-complement of an m-polar fuzzy graph, Cartesian product, composition, tensor product, and normal product are applied to m-polar fuzzy graphs and many results have been obtained.

https://doi.org/10.1155/2017/3817469

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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[7] M. S. Sunitha and A. Vijaya Kumar, "Complement of a fuzzy graph," Indian Journal of Pure and Applied Mathematics, vol. 33, no. 9, pp. 1451-1464, 2002.

[8] Ch. Ramprasad, P. L. N. Varma, N. Srinivasarao, and S. Satyanarayana, "Regular product m-polar fuzzy graphs and product m-polar fuzzy line graphs," Ponte, vol. 73, pp. 264-282, 2017.

Ch. Ramprasad, (1,2) P. L. N. Varma, (2) S. Satyanarayana, (3) N. Srinivasarao (4)

(1) Department of Mathematics, Vasireddy Venkatadri Institute of Technology, Nambur 522 508, India

(2) Department of Mathematics, VFSTR University, Vadlamudi 522 237, India

(3) Department of CSE, KL University, Vaddeswaram 522502, India

(4) Department of Mathematics, Tirumala Engineering College, Narasaraopet 522034, India

Correspondence should be addressed to S. Satyanarayana; s.satyans1@gmail.com

Received 22 May 2017; Accepted 5 July 2017; Published 22 August 2017

Caption: Figure 1: Highly irregular 3-polar fuzzy graph ??.

Caption: Figure 2: Cartesian product of two 3-polar fuzzy graphs [G.sub.1] and [G.sub.2].

Caption: Figure 3: Composition of two 3-polar fuzzy graphs [G.sub.1] and [G.sub.2]

Caption: Figure 4: Normal product of two 3-polar fuzzy graphs [G.sub.1] and [G.sub.2].

Caption: Figure 5: Tensor product of two 3-polar fuzzy graphs [G.sub.1] and [G.sub.2].

Caption: Figure 6: 4-polar fuzzy graph in decision-making.