Some isomorphic properties of mpolar fuzzy graphs with applications
 Ganesh Ghorai^{1}Email author and
 Madhumangal Pal^{1}
Received: 14 July 2016
Accepted: 1 December 2016
Published: 20 December 2016
Abstract
The theory of graphs are very useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we present a frame work to handle mpolar fuzzy information by combining the theory of mpolar fuzzy sets with graphs. We introduce the notion of weak self complement mpolar fuzzy graphs and establish a necessary condition for mpolar fuzzy graph to be weak self complement. Some properties of self complement and weak self complement mpolar fuzzy graphs are discussed. The order, size, busy vertices and free vertices of an mpolar fuzzy graphs are also defined and proved that isomorphic mpolar fuzzy graphs have same order, size and degree. Also, we have presented some results of busy vertices in isomorphic and weak isomorphic mpolar fuzzy graphs. Finally, a relative study of complement and operations on mpolar fuzzy graphs have been made. Applications of mpolar fuzzy graph are also given at the end.
Keywords
Background
After the introduction of fuzzy sets by Zadeh (1965), fuzzy set theory have been included in many research fields. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, management sciences, social sciences engineering, statistic, graph theory, artificial intelligence, signal processing, multi agent systems, decision making and automata theory. In a fuzzy set, each element is associated with a membership value selected from the interval [0, 1]. Zhang (1994, 1998) introduced the concept of bipolar fuzzy sets. Instead of using particular membership value as in fuzzy sets, mpolar fuzzy set can be used to represent uncertainty of a set more perfectly. Chen et al. (2014) introduced the notion of mpolar fuzzy set as a generalization of fuzzy set theory. The membership value in mpolar fuzzy set is more expressive in capturing uncertainty of data.
An mpolar fuzzy set on a nonvoid set X is a mapping \(\mu :X\rightarrow [0,1]^m\). The idea behind this is that “multipolar information” exists because data of real world problems are sometimes come from multiple agents. mpolar fuzzy sets allow more graphical representation of vague data, which facilitates significantly better analysis in data relationships, incompleteness, and similarity measures. Graph theory besides being a well developed branch of Mathematics, it is an important tool for mathematical modeling. Realizing the importance, Rosenfeld (1975) introduced the concept of fuzzy graphs, Mordeson and Nair (2000) discussed about the properties of fuzzy graphs and hypergraphs. After that, the operation of union, join, Cartesian product and composition on two fuzzy graphs was defined by Mordeson and Peng (1994). Sunitha and Vijayakumar (2002) further studied the other properties of fuzzy graphs. The concept of weak isomorphism, coweak isomorphism and isomorphism between fuzzy graphs was introduced by Bhutani (1989). Later many researchers have worked on fuzzy graphs like in Bhutani et al. (2004); AlHawary (2011); Koczy (1992); Leekwang and Lee (1995); Nagoorgani and Radha (2008), Samanta and Pal (2011a, b, 2013, 2014, 2015). Akram (2011, 2013) introduced and defined different operations on bipolar fuzzy graphs. Again, Rashmanlou et al. (2015a, 2015b, 2016) studied bipolar fuzzy graphs with categorical properties, product of bipolar fuzzy graphs and their degrees, etc. Using these concepts many research is going on till date on bipolar fuzzy graphs such as Ghorai and Pal (2015b), Samanta and Pal (2012a, b, 2014), Yang et al. (2013). Chen et al. (2014) first introduced the concept of mpolar fuzzy graphs. Then Ghorai and Pal (2016a) presented properties of generalized mpolar fuzzy graphs, defined many operations and density of mpolar fuzzy graphs (2015a), introduced the concept of mpolar fuzzy planar graphs (2016b) and defined faces and dual of mpolar fuzzy planar graphs (2016c). Akram and Younas (2015), Akram et al. (2016) introduced irregular mpolar fuzzy graphs and metrics in mpolar fuzzy graphs. In this paper, weak self complement mpolar fuzzy graphs is defined and a necessary condition is mentioned for an mpolar fuzzy graph to be weak self complement. Some properties of self complement and weak self complement mpolar fuzzy graphs are discussed. The order, size, busy vertices and free vertices of an mpolar fuzzy graphs are also defined and proved that isomorphic mpolar fuzzy graphs have same order, size and degree. Also, we have proved some results of busy vertices in isomorphic and weak isomorphic mpolar fuzzy graphs. Finally, a relative study of complement and operations on mpolar fuzzy graphs have been made.
Preliminaries
First of all we give the definitions of mpolar fuzzy sets, mpolar fuzzy graphs and other related definitions from the references (AlHarary 1972; Lee 2000).
Throughout the paper, \([0,1]^m\) (mpower of [0, 1]) is considered to be a poset with pointwise order \(\le \), where m is a natural number. \(\le \) is defined by \(x\le y \Leftrightarrow \) for each \(i=1,2,\ldots ,m\), \(p_i(x)\le p_i(y)\) where \(x, y\in [0,1]^m\) and \(p_i:[0,1]^m \rightarrow [0,1]\) is the ith projection mapping.
As a generalization of bipolar fuzzy sets, Chen et al. (2014) defined the mpolar fuzzy sets in 2014.
Definition 1
(Chen et al. 2014) Let X be a nonvoid set. An mpolar fuzzy set on X is defined as a mapping \(\mu :X\rightarrow [0,1]^m\).
The mpolar fuzzy relation is defined below.
Definition 2
(Ghorai and Pal 2016a) Let A be an mpolar fuzzy set on a set X. An mpolar fuzzy relation on A is an mpolar fuzzy set B of \(X\times X\) such that \(p_i\circ B(x,y)\le min\{p_i\circ A(x),p_i\circ A(y)\}\) for all \(x,y\in X\), \(i=1,2,\ldots ,m\). B is called symmetric if \(B(x,y)=B(y,x)\) for all \(x,y\in X\).
We define an equivalence relation \(\sim \) on \(V\times V\{(x,x): x\in V\}\) as follows:
We say \((x_1,y_1)\sim (x_2,y_2)\) if and only if either \((x_1,y_1)=(x_2,y_2)\) or \(x_1=y_2\) and \(y_1=x_2\). Then we obtain an quotient set denoted by \(\widetilde{V^2}\). The equivalence class containing the element (x, y) will be denoted as xy or yx.
We assume that \(G^*=(V,E)\) is a crisp graph and \(G=(V,A,B)\) is an mpolar fuzzy graph of \(G^*\) throughout this paper.
Chen et al. (2014) first introduced mpolar fuzzy graph. We have modified their definition and introduce generalized mpolar fuzzy graph as follows.
Definition 3
(Chen et al. 2014; Ghorai and Pal 2016a) An mpolar fuzzy graph (or generalized mpolar fuzzy graph) of \(G^*=(V,E)\) is a pair \(G=(V,A,B)\) where \(A: V\rightarrow [0,1]^m\) is an mpolar fuzzy set in V and \(B: \widetilde{V^2}\rightarrow [0,1]^m\) is an mpolar fuzzy set in \(\widetilde{V^2}\) such that \(p_i\circ B(xy)\le min\{p_i\circ A(x),p_i\circ A(y)\}\) for all \(xy\in \widetilde{V^2}\), \(i=1,2,\ldots ,m\) and \(B(xy)={\mathbf{0}} \) for all \(xy\in \widetilde{V^2}E\), \(\big ({\mathbf{0}} =(0,0,\ldots ,0)\) is the smallest element in \([0,1]^m\big )\). We call A as the mpolar fuzzy vertex set of G and B as the mpolar fuzzy edge set of G.
Example 4
Let \(G^*=(V, E)\) be a crisp graph where \(V=\{u_1, u_2, u_3, u_4\}\) and \(E=\{u_1u_2, u_2u_3, u_3u_4, u_4u_1\}\). Then, \(G=(V, A, B)\) be a 3polar fuzzy graph of \(G^*\) where \(A=\left\{ \frac{\langle 0.5, 0.7, 0.8\rangle }{u_1}, \frac{\langle 0.4, 0.7, 0.8\rangle }{u_2}, \frac{\langle0.7, 0.6, 0.8\rangle}{u_3}, \frac{\langle0.3, 0.6, 0.9\rangle}{u_4}\right\} \) and \(B=\left\{ \frac{\langle 0.4, 0.6, 0.7\rangle }{u_1u_2}, \frac{\langle 0.3, 0.6, 0.5\rangle }{u_2u_3}, \frac{\langle 0.2, 0.5, 0.6\rangle }{u_3u_4}, \frac{\langle 0.2, 0.4, 0.8\rangle }{u_4u_1}, \frac{\langle 0, 0, 0\rangle }{u_1u_3}, \frac{\langle 0, 0, 0\rangle }{u_4u_2}\right\} \).
Ghorai and Pal (2016a) introduced many operations on mpolar fuzzy graphs such as Cartesian product, composition, union and join which are given below.
Definition 5
 (i)
\(p_i\circ (A_1\times A_2)(x_1,x_2)= min\{p_i\circ A_1(x_1),p_i\circ A_2(x_2)\}\) for all \((x_1,x_2)\in V_1\times V_2\).
 (ii)
\(p_i\circ (B_1\times B_2)((x,x_2)(x,y_2))= min\{p_i\circ A_1(x),p_i\circ B_2(x_2y_2)\}\) for all \(x\in V_1\), \(x_2y_2\in E_2\).
 (iii)
\(p_i\circ (B_1\times B_2)((x_1,z)(y_1,z))= min\{p_i\circ B_1(x_1y_1),p_i\circ A_2(z)\}\) for all \(z\in V_2\), \(x_1y_1\in E_1\).
 (iv)
\(p_i\circ (B_1\times B_2)((x_1,x_2)(y_1,y_2))=0\) for all \((x_1,x_2)(y_1,y_2)\in \widetilde{(V_1\times V_2)^2}E\).
Definition 6
 (i)
\(p_i\circ (A_1\circ A_2)(x_1,x_2)= min\{p_i\circ A_1(x_1),p_i\circ A_2(x_2)\}\) for all \((x_1,x_2)\in V_1\times V_2\).
 (ii)
\(p_i\circ (B_1\circ B_2)((x,x_2)(x,y_2))= min\{p_i\circ A_1(x),p_i\circ B_2(x_2y_2)\}\) for all \(x\in V_1\), \(x_2y_2\in E_2\).
 (iii)
\(p_i\circ (B_1\circ B_2)((x_1,z)(y_1,z))= min\{p_i\circ B_1(x_1y_1),p_i\circ A_2(z)\}\) for all \(z\in V_2\), \(x_1y_1\in E_1\).
 (iv)
\(p_i\circ (B_1\circ B_2)((x_1,x_2)(y_1,y_2)) =min\{p_i\circ A_2(x_2),p_i\circ A_2(y_2),p_i\circ B_1(x_1y_1)\}\) for all \((x_1,x_2)(y_1,y_2)\in E^0E\).
 (v)
\(p_i\circ (B_1\circ B_2)((x_1,x_2)(y_1,y_2))=0\) for all \((x_1,x_2)(y_1,y_2)\in \widetilde{(V_1\times V_2)^2}E^0\).
Definition 7
 (i)
\(p_i\circ (A_1\cup A_2)(x)=\left\{ \begin{array}{ll} p_i\circ A_1(x) &{}\quad {\text {if}}\; x\in V_1V_2\\ p_i\circ A_2(x) &{}\quad {\text {if}}\; x\in V_2V_1\\ max\{p_i\circ A_1(x),p_i\circ A_2(x)\} &{}\quad {\text {if}}\; x\in V_1\cap V_2. \end{array}\right. \)
 (ii)
\(p_i\circ (B_1\cup B_2)(xy)=\left\{ \begin{array}{ll} p_i\circ B_1(xy) &{}\quad {\text {if}}\; xy\in E_1E_2\\ p_i\circ B_2(xy) &{}\quad {\text {if}}\; xy\in E_2E_1\\ max\{p_i\circ B_1(xy),p_i\circ B_2(xy)\} &{}\quad {\text {if}}\; xy\in E_1\cap E_2. \end{array}\right. \)
 (iii)
\(p_i\circ (B_1\cup B_2)(xy)=0\) if \(xy\in \widetilde{(V_1\times V_2)^2}E_1\cup E_2\).
Definition 8
 (i)
\(p_i\circ (A_1+A_2)(x)=p_i\circ (A_1\cup A_2)(x)\) if \(x\in V_1\cup V_2\).
 (ii)
\(p_i\circ (B_1+B_2)(xy)=p_i\circ (B_1\cup B_2)(xy)\) if \(xy\in E_1\cup E_2\).
 (iii)
\(p_i\circ (B_1+B_2)(xy)=min\{p_i\circ A_1(x),p_i\circ A_2(y)\}\) if \(xy\in E^\prime \), where \(E^\prime \) denotes the set of all edges joining the vertices of \(V_1\) and \(V_2\).
 (iv)
\(p_i\circ (B_1+B_2)(xy)=0\) if \(xy\in \widetilde{(V_1\times V_2)^2}E_1\cup E_2\cup E^\prime \).
Remark 9
Later on, Akram et al. (2016) applied the concept of mpolar fuzzy sets on graph structure and also defined the above operations on them.
Different types of morphism are defined on mpolar fuzzy graphs by Ghorai and Pal (2016a).
Definition 10
 (i)
\(p_i\circ A_1(x_1)\le p_i\circ A_2(\phi (x_1))\) for all \(x_1\in V_1\),
 (ii)
\(p_i\circ B_1(x_1y_1)\le p_i\circ B_2(\phi (x_1)\phi (y_1))\) for all \(x_1y_1\in \widetilde{V^2_1}\).
 (i)
\(p_i\circ A_1(x_1)= p_i\circ A_2(\phi (x_1))\) for all \(x_1\in V_1\),
 (ii)
\(p_i\circ B_1(x_1y_1)= p_i\circ B_2(\phi (x_1)\phi (y_1))\) for all \(x_1y_1\in \widetilde{V^2_1}\).
Definition 11
 (i)
\(\phi \) is a homomorphism,
 (ii)
\(p_i\circ A_1(x_1)= p_i\circ A_2(\phi (x_1))\) for all \(x_1\in V_1\), for each \(i=1,2,\ldots ,m\).
Definition 12
(Ghorai and Pal 2016a) \(G=(V,A,B)\) is called strong if \(p_i\circ B(xy)=min\{p_i\circ A(x),p_i\circ A(y)\}\) for all \(xy\in E\), \(i=1,2,\ldots ,m\).
A strong mpolar fuzzy graph G is called self complementary if \(G\cong \overline{G}\).
Degree of a vertex in an mpolar fuzzy graph is defined as below.
Definition 13
(Akram and Younas 2015) The neighborhood degree of a vertex v in the mpolar fuzzy graph G is denoted as \(deg(v)=\big (p_1\circ deg(v), p_2\circ deg(v), \ldots , p_m\circ deg(v)\big )\) where \(p_i\circ deg(v)=\sum \nolimits _{\begin{array}{c} u\ne v\\ uv\in E \end{array}}p_i\circ {B}(uv)\), \(i=1,2,\ldots ,m\).
Remark 14
If \(G_1=(V_1, A_1, B_1)\) and \(G_2=(V_2, A_2, B_2)\) are two mpolar fuzzy graphs. Then the canonical projection maps \(\pi _1: V_1\times V_2\rightarrow V_1\) and \(\pi _2: V_1\times V_2\rightarrow V_2\) are indeed homomorphisms from \(G_1\times G_2\) to \(G_1\) and \(G_1\times G_2\) to \(G_2\) respectively. This can be seen as follows:
\(p_i\circ (A_1\times A_2)(x_1, x_2)=min\{p_i\circ A_1(x_1), p_i\circ A_2(x_2)\}\le p_i\circ A_1(x_1)=p_i\circ A_1(\pi _1(x_1, x_2))\) for all \((x_1, x_2)\in V_1\times V_2\) and \(p_i\circ (B_1\times B_2)((x_1, z)(y_1, z))=min\{p_i\circ B_1(x_1y_1), p_i\circ A_2(z)\}\le p_i\circ B_1(x_1y_1)=p_i\circ B_1(\pi _1(x_1, z)\pi _1(y_1, z))\) for all \(z\in V_2\) and \(x_1y_1\in E_1\). In a similar way we can check the other conditions also. This shows that the canonical projection maps \(\pi _1: V_1\times V_2\rightarrow V_1\) is a homomorphism from \(G_1\times G_2\) to \(G_1\).
Weak self complement mpolar fuzzy graphs
Self complement mpolar fuzzy graphs have many important significant in the theory of mpolar fuzzy graphs. If an mpolar fuzzy graph is not self complement then also we can say that it is self complement in some weaker sense. Simultaneously we can establish some results with this graph. This motivates to define weak self complement mpolar fuzzy graphs.
Definition 15
Let \(G=(V,A,B)\) be an mpolar fuzzy graph of the crisp graph \(G^*=(V,E)\). The complement of G is an mpolar fuzzy graph \(\overline{G}=(V,\overline{A},\overline{B})\) of \(\overline{G^*}=(V,\widetilde{V^2})\) such that \(\overline{A}=A\) and \(\overline{B}\) is defined by \(p_i\circ \overline{B}(xy)=min\{p_i\circ A(x), p_i\circ A(y)\} p_i\circ B(xy)\) for \(xy\in \widetilde{V^2}\), \(i=1,2,\ldots ,m\).
Example 16
Let \(G=(V,A,B)\) be a 3polar fuzzy graph of the graph \(G^*=(V,E)\) where \(V=\{u,v,w,x\}\), \(E=\{uv,vw,wu,ux\}\), \(A=\left\{ \frac{\langle 0.2,0.3,0.5\rangle }{u},\frac{\langle 0.5,0.6,0.3\rangle }{v},\frac{\langle 0.7,0.2,0.3\rangle }{w},\,\frac{\langle 0.2,0.5,0.7\rangle }{x} \right\} \), \(B=\left\{ \frac{\langle 0.2,0.3,0.3\rangle }{uv},\frac{\langle 0.4,0.1,0.1\rangle }{vw},\frac{\langle 0.1,0.1,0.1\rangle }{wu}, \frac{\langle 0.1,0.2,0.4\rangle }{xu},\frac{\langle 0,0,0\rangle }{xv},\frac{\langle 0,0,0\rangle }{wx}\right\} \). Then by Definition 15, we have constructed the complement \(\overline{G}\) of G which is shown in Fig. 1.
Remark 17
Definition 18
 (i)
\(p_i\circ A(u)= p_i\circ \overline{A}(\phi (u))\) for all \(u\in V\),
 (ii)
\(p_i\circ B(uv)\le p_i\circ \overline{B}(\phi (u)\phi (v))\) for all \(uv\in \widetilde{V^2}\).
Example 19
Let \(G=(V,A,B)\) be a 3polar fuzzy graph of the graph \({G^*}=(V,E)\) where \(V=\{u,v,w\}\), \(E=\{uv,vw\}\), \(A=\left\{ \frac{\langle 0.3,0.4,0.4\rangle }{u},\frac{\langle 0.2,0.5,0.7\rangle }{v},\frac{\langle 0.3,0.6,0.7\rangle }{w}\right\} \), \(B=\left\{ \frac{\langle 0.1,0.1,0.2\rangle }{uv},\frac{\langle 0.1,0.2,0.2\rangle }{vw},\frac{\langle 0,0,0\rangle }{wu}\right\} \). Then \(\overline{G}=(V,\overline{A},\overline{B})\) is also a 3polar fuzzy graph where \(\overline{A}=A\) and \(\overline{B}=\left\{ \frac{\langle 0.1,0.3,0.2\rangle }{uv},\frac{\langle 0.1,0.3,0.5\rangle }{vw},\frac{\langle 0.3,0.4,0.4\rangle }{wu}\right\} \). We can easily verify that, the identity map is an weak isomorphism from G onto \(\overline{G}\)(see Fig. 2). Hence G is weak self complement.
Example 20
Now suppose an mpolar fuzzy graph \(G=(V,A,B)\) is a weak self complement. Then the following inequality holds.
Theorem 21
Proof
Since G is weak self complement, therefore there exists a weak isomorphism \(\phi : V \rightarrow V\) such that \(p_i\circ A(x)= p_i\circ \overline{A}(\phi (x))\) for all \(x\in V\) and \(p_i\circ B(xy)\le p_i\circ \overline{B}(\phi (x)\phi (y))\) for all \(xy\in \widetilde{V^2}\), \(i=1,2,\ldots ,m\).
Remark 22
The converse of the above theorem is not true in general. For example, consider the 3polar fuzzy graph of Fig. 3. We see that for the 3polar fuzzy graph G, the condition of Theorem 21 is satisfied. But, G is not weak self complementary as there is no weak isomorphism from G onto \(\overline{G}\).
Theorem 23
If \(p_i\circ B(xy)\le \frac{1}{2} min\{p_i\circ A(x),p_i\circ A(y)\}\) for all \(xy\in \widetilde{V^2}\), \(i=1,2,\ldots ,m\) then G is a weak self complement mpolar fuzzy graph.
Proof
Let \(\overline{G}=(V,\overline{A},\overline{B})\) be the complement of G where \(\overline{A}(x)=A(x)\) for all \(x\in V\) and \(p_i\circ \overline{B}(xy)=min\{p_i\circ A(x), p_i\circ A(y)\} p_i\circ B(xy)\) for \(xy\in \widetilde{V^2}\), \(i=1,2,\ldots ,m\).
Example 24
Consider the 3polar fuzzy graph \(G=(V,A,B)\) of \({G^*}=(V,E)\) where \(V=\{u,v,w\}\), \(E=\{uv,vw,wu\}\), \(A=\left\{ \frac{\langle 0.2,0.3,0.4\rangle }{u},\frac{\langle 0.4,0.5,0.6\rangle }{v},\frac{\langle 0.5,0.7,0.9\rangle }{w}\right\} \), \(B=\left\{ \frac{\langle 0.1,0.1,0.2\rangle }{uv},\frac{\langle 0.2,0.2,0.3\rangle }{vw},\frac{\langle 0.1,0.1,0.2\rangle }{wu}\right\} \). We see that for each \(i=1,2,3\) and \(xy\in \widetilde{V^2}\), \( p_i\circ B(xy)\le \frac{1}{2} min\{p_i\circ A(x),p_i\circ A(y)\}\) .
Also, consider the complement of G of Fig. 4. Let us now consider the identity mapping \(I: G\rightarrow \overline{G}\) such that \(I(u)=u\) for all \(u\in V\). Then, I is the required weak isomorphism from G onto \(\overline{G}\). Hence, G is weak self complementary.
Order, size and busy value of vertices of mpolar fuzzy graphs
In this section, the order, size, busy value of vertices of an mpolar fuzzy graph is defined.
Definition 25
Theorem 26
Two isomorphic mpolar fuzzy graphs \(G_1=(V_1,A_1,B_1)\) and \(G_2=(V_2,A_2,B_2)\) of the graphs \(G^*_1=(V_1,E_1)\) and \(G^*_2=(V_2,E_2)\) have same order and size.
Proof
Let \(\phi \) be an isomorphism from \(G_1\) onto \(G_2\). Then \(A_1(x)=A_2(\phi (x))\) for all \(x\in V_1\) and \(p_i\circ B_1(xy)=p_i\circ B_2(\phi (x)\phi (y))\) for \(i=1,2,\ldots ,m\), \(xy\in \widetilde{V^2_1}\).
Definition 27
The busy value of a vertex u of an mpolar fuzzy graph G is denoted as \(D(u)=(p_1\circ D(u),p_2\circ D(u),\ldots ,p_m\circ D(u))\) where \(p_i\circ D(u)=\sum \limits _{k}min\{p_i\circ A(u),p_i\circ A(u_k)\}\); \(u_k\) are the neighbors of u. The busy value of G is denoted as D(G) where \(D(G)=\sum \limits _{k}D(u_k)\), \(u_k\in V\).
Example 28
Definition 29
If \(p_i\circ A(u)\le p_i\circ deg(u)\) for \(i=1,2,\ldots ,m\), then the vertex u of G is called a busy vertex. Otherwise it is a free vertex.
Definition 30
If \(p_i\circ B(u_1v_1)=min\{p_i\circ A(u_1),p_i\circ A(v_1)\}\), \(i=1,2,\ldots ,m\) for \(u_1v_1\in E\), then it is called an effective edge of G.
Definition 31
 (i)
u is called a partial free vertex if it is a free vertex of G and \(\overline{G}\).
 (ii)
u is called a fully free vertex if it is a free vertex of G and it is a busy vertex of \(\overline{G}\).
 (iii)
u is called a partial busy vertex if it is a busy vertex of G and \(\overline{G}\).
 (iv)
u is called a fully busy vertex if it is a busy vertex in G and it is a free vertex of \(\overline{G}\).
Theorem 32
Let \(\phi \) be an isomorphism from \(G_1=(V_1,A_1,B_1)\) onto \(G_2=(V_2,A_2,B_2)\). Then \(deg(u)=deg(\phi (u))\) for all \(u\in V_1\).
Proof
Since \(\phi \) is an isomorphism between \(G_1\) and \(G_2\), we have \(p_i\circ A_1(u)=p_i\circ A_2(\phi (u))\) for all \(u\in V_1\) and \(p_i\circ B_1(x_1y_1)=p_i\circ B_2(\phi (x_1)\phi (y_1))\) for all \(x_1y_1\in \widetilde{V_1^2}\), \(i=1,2,\ldots ,m\).
Hence, \(p_i\circ deg(u)=\sum \nolimits _{\begin{array}{c} u\ne v\\ uv\in E_1 \end{array}} p_i\circ B_1(uv) =\sum \nolimits _{\begin{array}{c} \phi (u)\ne \phi (v)\\ \phi (u)\phi (v)\in E_2 \end{array}} p_i\circ B_2(\phi (u)\phi (v)) =p_i\circ deg(\phi (u))\) for \(u\in V_1\), \(i=1,2,\ldots ,m\). So, \(deg(u)=deg(\phi (u))\) for all \(u\in V_1\). \(\square \)
Theorem 33
If \(\phi \) is an isomorphism from \(G_1\) onto \(G_2\) and u is a busy vertex of \(G_1\), then \(\phi (u)\) is a busy vertex of \(G_2\).
Proof
Since \(\phi \) is an isomorphism between we have, \(p_i\circ A_1(u)=p_i\circ A_2(\phi (u))\) \(u\in V_1\) and \(p_i\circ B_1(x_1y_1)=p_i\circ B_2(\phi (x_1)\phi (y_1))\) for \(x_1y_1\in \widetilde{V_1^2}\), \(i=1,2,\ldots ,m\).
If u is a busy vertex of \(G_1\), then \(p_i\circ A_1(u)\le p_i\circ deg(u)\) for \(i=1,2,\ldots ,m\). Then by the above and Theorem 32, \(p_i\circ A_2(\phi (u))=p_i\circ A_1(u)\le p_i\circ deg(u)=p_i\circ deg(\phi (u))\) for \(i=1,2,\ldots ,m\). Hence, \(\phi (u)\) is a busy vertex in \(G_2\). \(\square \)
Theorem 34
Let the two mpolar fuzzy graphs \(G_1\) and \(G_2\) be weak isomorphic. If \(u\in V_1\) is a busy vertex of \(G_1\), then the image of u under the weak isomorphism is also busy in \(G_2\).
Proof
Let \(\phi :V_1\rightarrow V_2\) be a weak isomorphism between \(G_1\) and \(G_2\).
Then, \(p_i\circ A_1(x)=p_i\circ A_2(\phi (x))\) for all \(x\in V_1\) and \(p_i\circ B_1(x_1y_1)\le p_i\circ B_2(\phi (x_1)\phi (y_1))\) for all \(x_1y_1\in \widetilde{V_1^2}\), \(i=1,2,\ldots ,m\).
Let \(u\in V_1\) be a busy vertex. Then, for \(i=1,2,\ldots ,m\), \(p_i\circ A_1(u)\le p_i\circ deg(u)\).
Complement and isomorphism in mpolar fuzzy graphs
In this section some important properties of isomorphism, weak isomorphism, co weak isomorphism related with complement are discussed.
Theorem 35
Let \(G_1=(V_1,A_1,B_1)\) and \(G_2=(V_2,A_2,B_2)\) be two mpolar fuzzy graphs of the graphs \(G^*_1=(V_1,E_1)\) and \(G^*_2=(V_2,E_2)\). If \(G_1\cong G_2\) then \(\overline{G_1}\cong \overline{G_2}\).
Proof
Let \(G_1\cong G_2\). Then there exists an isomorphism \(\phi : V_1\rightarrow V_2\) such that \(A_1(x)=A_2(\phi (x))\) for all \(x\in V_1\) and \(p_i\circ B_1(xy)=p_i\circ B_2(\phi (x)\phi (y))\), for each \(i=1,2,\ldots ,m\) and \(xy\in \widetilde{V^2_1}\).
Now, \(\overline{A_1}(x)=A_1(x)=A_2(\phi (x))=\overline{A_2}(\phi (x))\) for all \(x\in V_1\).
Remark 36
Suppose there is a weak isomorphism between two mpolar fuzzy graphs \(G_1\) and \(G_2\). Then there may not be a weak isomorphism between \(\overline{G_1}\) and \(\overline{G_2}\).
For example, consider two 3polar fuzzy graphs \(G_1\) and \(G_2\) of Fig. 6. Let us now define a mapping \(\phi : V_1 \rightarrow V_2\) such that \(\phi (a)=u\), \(\phi (b)=v\), \(\phi (c)=w\). Then \(\phi \) is a weak isomorphism from \(G_1\) onto \(G_2\). But, there is no weak isomorphism from \(\overline{G_1}\) onto \(\overline{G_2}\) (see Fig. 7) because \(\overline{B_2}(uw=\phi (a)\phi (c))={\mathbf{0}} =(0,0,\ldots ,0)<\overline{B_1}(ac)=(0.1,0.1,0.05)\), and \(\overline{B_2}(vw=\phi (b)\phi (c))={\mathbf{0}} =(0,0,\ldots ,0)<\overline{B_1}(bc)=(0.1,0.1,0.1)\).
Remark 37
In a similar way, we can construct example to show that if there is a coweak isomorphism between two mpolar fuzzy graphs \(G_1\) and \(G_2\) then there may not be a coweak isomorphism between \(\overline{G_1}\) and \(\overline{G_2}\).
Theorem 38
Let \(G_1=(V_1,A_1,B_1)\) and \(G_2=(V_2,A_2,B_2)\) be two mpolar fuzzy graphs of the graphs \(G^*_1=(V_1,E_1)\) and \(G^*_2=(V_2,E_2)\) such that \(V_1\cap V_2= \emptyset \). Then \(\overline{G_1+G_2}\cong \overline{G_1}\cup \overline{G_2}\).
Proof
To show that \(\overline{G_1+G_2}\cong \overline{G_1}\cup \overline{G_2}\), we need to show that there exists an isomorphism between \(\overline{G_1+G_2}\) and \(\overline{G_1}\cup \overline{G_2}\).
We will show that the identity map \(I:V_1\cup V_2 \rightarrow V_1\cup V_2\) is the required isomorphism between them. For this, we will show the following:
for all \(x\in V_1\cup V_2\), \(\overline{(A_1+A_2)}(x)=(\overline{A_1}\cup \overline{A_2})(x)\),
and \(p_i\circ \overline{(B_1+B_2)}(xy)=p_i\circ (\overline{B_1}\cup \overline{B_2})(xy)\) for \(i=1,2,\ldots ,m\) and \(xy\in \widetilde{V_1\times V_2}^2\).
Let \(x\in V_1\cup V_2\).
Theorem 39
Let \(G_1=(V_1,A_1,B_1)\) and \(G_2=(V_2,A_2,B_2)\) be two mpolar fuzzy graphs of the graphs \(G^*_1=(V_1,E_1)\) and \(G^*_2=(V_2,E_2)\) such that \(V_1\cap V_2= \emptyset \). Then \(\overline{G_1\cup G_2}\cong \overline{G_1}+\overline{G_2}\).
Proof
Consider the identity map \(I:V_1\cup V_2\rightarrow V_1\cup V_2\). We will show that I is the required isomorphism between \(\overline{G_1\cup G_2}\) and \(\overline{G_1}+\overline{G_2}\).
For this, we will show the following:
for all \(x\in V_1\cup V_2\), \(\overline{(A_1\cup A_2)}(x)=(\overline{A_1}+\overline{A_2})(x)\),
and \(p_i\circ \overline{(B_1\cup B_2)}(xy)=p_i\circ (\overline{B_1}+\overline{B_2})(xy)\) for \(i=1,2,\ldots ,m\) and \(xy\in \widetilde{V_1\times V_2}^2\).
Let \(x\in V_1\cup V_2\).
Theorem 40
Let \(G_1=(V_1,A_1,B_1)\) and \(G_2=(V_2,A_2,B_2)\) be two strong mpolar fuzzy graphs of the graphs \(G^*_1=(V_1,E_1)\) and \(G^*_2=(V_2,E_2)\) respectively. Then \(\overline{G_1\circ G_2}\cong \overline{G_1}\circ \overline{G_2}\).
Proof
Let \(G_1\circ G_2=(V_1\times V_2,A_1\circ A_2,B_1\circ B_2)\) be an mpolar fuzzy graph of the graph \(G^*=(V,E)\) where \(V=V_1\times V_2\) and \(E=\{(x,x_2)(x,y_2): x\in V_1, x_2y_2\in E_2\}\cup \{(x_1,z)(y_1,z): z\in V_2, x_1y_1\in E_1\}\cup \{(x_1,x_2)(y_1,y_2): x_1y_1\in E_1, x_2\ne y_2\}\).
We show that the identity map I is the required isomorphism between the graphs \(\overline{G_1\circ G_2}\) and \(\overline{G_1}\circ \overline{G_2}\). Let us consider the identity map \(I: V_1\times V_2 \rightarrow V_1\times V_2\).

Case (i): Let \(e=(x,x_2)(x,y_2)\) where \(x\in V_1\), \(x_2y_2\in E_2\). Then \(e\in E\).
Since \(G_1\circ G_2\) is strong mpolar fuzzy graph, we have for each \(i=1,2,\ldots ,m\)(since \(G_2\) is strong and \(x_2y_2\in E_2\), therefore for each \(i=1,2,\ldots ,m\), \(p_i\circ \overline{B_2}(x_2y_2)=0\)).$$\begin{aligned}&p_i\circ \overline{(B_1\circ B_2)}(e)=0{\text { and}}\\&p_i\circ (\overline{B_1} \circ \overline{B_2})(e)=min\{p_i\circ A_1(x),p_i\circ \overline{B_2}(x_2y_2)\}=0 \end{aligned}$$ 
Case (ii): Let \(e=(x,x_2)(x,y_2)\) where \(x_2\ne y_2\), \(x_2y_2\notin E_2\). Then \(e\notin E\).
So for each \(i=1,2,\ldots ,m\), \(p_i\circ (B_1\circ B_2)(e)=0\) andAgain, since \(x_2y_2\in \overline{E_2}\), therefore for each \(i=1,2,\ldots ,m\),$$\begin{aligned} p_i\circ \overline{(B_1\circ B_2)}(e)&= {} min\{p_i\circ (A_1\circ A_2)(x,x_2),p_i\circ (A_1\circ A_2)(x,y_2)\}\\&= {} min\{p_i\circ A_1(x),p_i\circ A_2(x_2),p_i\circ A_2(y_2)\}. \end{aligned}$$$$\begin{aligned} p_i\circ (\overline{B_1} \circ \overline{B_2})(e)&= {} min\{p_i\circ A_1(x),p_i\circ \overline{B_2}(x_2y_2)\}\\&= {} min\{p_i\circ A_1(x),p_i\circ A_2(x_2),p_i\circ A_2(y_2)\}. \end{aligned}$$ 
Case (iii): Let \(e=(x_1,z)(y_1,z)\) where \(x_1y_1\in E_1\), \(z\in V_2\).
Then \(e\in E\). So for each \(i=1,2,\ldots ,m\), \(p_i\circ \overline{(B_1\circ B_2)}(e)=0\) as in Case (i).
Also, since \(x_1y_1\notin \overline{E_1}\), therefore for each \(i=1,2,\ldots ,m\), \(p_i\circ (\overline{B_1}\circ \overline{B_2})(e)=0\).

Case (iv): Let \(e=(x_1,z)(y_1,z)\) where \(x_1y_1\notin E_1\), \(z\in V_2\). Then \(e\notin E\).
Hence for each \(i=1,2,\ldots ,m\), \(p_i\circ (B_1\circ B_2)(e)=0\),$$\begin{aligned} p_i\circ \overline{(B_1\circ B_2)}(e)&= {} min\{p_i\circ (A_1\circ A_2)(x_1,z),p_i\circ (A_1\circ A_2)(y_1,z)\}\\&= {} min\{p_i\circ A_1(x_1),p_i\circ A_1(y_1),p_i\circ A_2(z)\}\text { and}\\ p_i\circ (\overline{B_1} \circ \overline{B_2})(e)&= {} min\{p_i\circ A_2(z),p_i\circ \overline{B_1}(x_1y_1)\}\\&= {} min\{p_i\circ A_1(x_1),p_i\circ A_1(y_1),p_i\circ A_2(z)\} \;(G_1{\text { being strong}}). \end{aligned}$$ 
Case (v): Let \(e=(x_1,x_2)(y_1,y_2)\) where \(x_1y_1\in E_1\), \(x_2\ne y_2\). Then \(e\in E\). So we have for each \(i=1,2,\ldots ,m\), \(p_i\circ \overline{(B_1\circ B_2)}(e)=0\) as in Case (i).
Also, since \(x_1y_1\in E_1\), we have for each \(i=1,2,\ldots ,m\), \(p_i\circ (\overline{B_1} \circ \overline{B_2})(e)=0\).

Case (vi): Let \(e=(x_1,x_2)(y_1,y_2)\) where \(x_1y_1\notin E_1\), \(x_2\ne y_2\). Then \(e\notin E\) and hence for each \(i=1,2,\ldots ,m\), \(p_i\circ (B_1\circ B_2)(e)=0\),and since \(x_1y_1\in \overline{E_1}\),$$\begin{aligned} p_i\circ \overline{(B_1\circ B_2)}(e)&= {} min\{p_i\circ (A_1\circ A_2)(x_1,x_2),p_i\circ (A_1\circ A_2)(y_1,y_2)\}\\&= {} min\{p_i\circ A_1(x_1),p_i\circ A_1(y_1),p_i\circ A_2(x_2),p_i\circ A_2(y_2)\} \end{aligned}$$$$\begin{aligned} p_i\circ (\overline{B_1} \circ \overline{B_2})(e)&= {} min\{p_i\circ A_2(x_2),p_i\circ A_2(y_2),p_i\circ \overline{B_1}(x_1y_1)\}\\&= {} min\{p_i\circ A_1(x_1),p_i\circ A_1(y_1),p_i\circ A_2(x_2),p_i\circ A_2(y_2)\}\, (\overline{G_1} \text { being strong by } [10]). \end{aligned}$$

Case (vii): Finally, let \(e=(x_1,x_2)(y_1,y_2)\) where \(x_1y_1\notin E_1\), \(x_2y_2\notin E_2\). Then \(e\notin E\) and hence for each \(i=1,2,\ldots ,m\), \(p_i\circ (B_1\circ B_2)(e)=0\),Now, \(x_1y_1\in \overline{E_1}\) and if \(x_2=y_2=z\), then we have the Case (iv).$$\begin{aligned} p_i\circ \overline{(B_1\circ B_2)}(e)=min\{p_i\circ (A_1\circ A_2)(x_1,x_2),p_i\circ (A_1\circ A_2)(y_1,y_2)\}. \end{aligned}$$
Again, if \(x_1y_1\in \overline{E_1}\) and if \(x_2\ne y_2\), then we have Case (vi).
Thus combining all the cases we have, for each \(i=1,2,\ldots ,m\), and \(xy\in \widetilde{V_1\times V_2}^2\),\(\square \)$$\begin{aligned} p_i\circ \overline{(B_1\circ B_2)}(xy)=p_i\circ (\overline{B_1} \circ \overline{B_2})(xy). \end{aligned}$$
Remark 41
If \(G_1\) and \(G_2\) are not strong, then \(\overline{G_1\circ G_2} \ncong \overline{G_1}\circ \overline{G_2}\) always. For example, consider the two 3polar fuzzy graphs \(G_1\) and \(G_2\) which are not strong (see Fig. 8). From Figs. 8 and 9, we see that, \(\overline{G_1\circ G_2}\ncong \overline{G_1}\circ \overline{G_2}\).
Applications
Now a days, fuzzy graphs and bipolar fuzzy graphs are most familiar graphs to us and they can also be thought of as 1polar and 2polar fuzzy graphs respectively. These graphs have many important application in social networks, medical diagnosis, computer networks, database theory, expert system, neural networks, artificial intelligence, signal processing, pattern recognition, engineering science, cluster analysis, etc. The concepts of bipolar fuzzy graphs can be generalized to mpolar fuzzy graphs. For example, consider the sorting of mangoes and guavas. Now the different characteristics of a given fruit can change the decision in sorting process more towards the decision mango or vice versa. There are two poles present in this case. One is \(100\%\) sure mango and the other is \(100\%\) sure guava. This shows that the situation is bipolar. This situation can be generalized further by adding a new fruit, for example sweet lemon into the sorting process.
Graphical representation of tug of war
Consider the another example of tug of war where two people pull the rope in opposite directions. Here, who uses the bigger force, the center of the rope will move in the respective direction of their pulling. The situation is symmetric in this case. We present an example where m people pull a special rope in m different directions. We use this example to represent it as an mpolar fuzzy graph. We assume that O is the origin and there are m straight paths leading from O. We also assume that there is a wall in between these paths. In this setting, we have the special rope with one node at O and m endings going out from this nodes—one end corresponding to each of the paths. Suppose on every path there is a man standing and pulling the rope in the direction of the path on which he is standing. This situation can be represented as an mpolar fuzzy graph by considering the nodes as mpolar fuzzy set and edges between them as mpolar fuzzy relations, which is shown in Fig. 10. In this context, one can ask the question what is the strength require in order to pull the node O from the center into one of the paths (assuming no friction)? The answer to this is that if the corresponding forces which are pulling the rope are \(F_k\), \(k=1,2,\ldots ,m\), then the node O will move to the \(j\hbox {th}\) path if \(F_j > \sum \nolimits _{\begin{array}{c} k=1,2,\ldots ,m\\ k\ne j \end{array}}{F_k}\).
Evaluation graph corresponding to the teacher’s evaluation by the students
In this section we present the model of mpolar fuzzy graph which is used in evaluating the teachers by the students of 4th semester of a department in an university during the session 2015–2016. Here the nodes represent the teachers of the corresponding department and edges represent the relationship between two teachers. Suppose the department has six teachers denoted as \(T=\{t_1,t_2,t_3,t_4,t_5,t_6\}\). The membership value of each node represents the corresponding teachers feedback response of the students depending on the following: {regularity of classes, style of presentation, quality of lectures, generation of interest and encouraging future reading among students, updated information}. Since all the above characteristics of a teacher according to the different students are uncertain in real life, therefore we consider 5polar fuzzy subset of the vertex set T (Fig. 11).
5Polar fuzzy set A of T
\(t_1\)  \(t_2\)  \(t_3\)  \(t_4\)  \(t_5\)  \(t_6\)  

\(p_1\circ A\)  0.6  0.7  0.8  0.8  0.8  0.7 
\(p_2\circ A\)  0.7  0.6  0.9  0.7  0.9  0.8 
\(p_3\circ A\)  0.8  0.7  0.7  0.8  0.7  0.9 
\(p_4\circ A\)  0.9  0.8  0.8  0.9  0.7  0.7 
\(p_5\circ A\)  0.9  0.8  0.9  0.8  0.8  0.8 
5Polar fuzzy relation B on A
\(t_1t_2\)  \(t_1t_5\)  \(t_1t_6\)  \(t_2t_3\)  \(t_2t_4\)  \(t_2t_5\)  \(t_3t_4\)  \(t_3t_5\)  \(t_4t_5\)  \(t_4t_6\)  \(t_5t_6\)  

\(p_1\circ A\)  0.6  0.5  0.6  0.6  0.8  0.7  0.8  0.7  0.8  0.6  0.7 
\(p_2\circ A\)  0.6  0.7  0.7  0.6  0.7  0.5  0.7  0.5  0.7  0.7  0.8 
\(p_3\circ A\)  0.7  0.7  0.8  0.6  0.6  0.6  0.7  0.7  0.7  0.7  0.6 
\(p_4\circ A\)  0.8  0.6  0.7  0.7  0.6  0.7  0.8  0.7  0.7  0.7  0.6 
\(p_5\circ A\)  0.7  0.8  0.8  0.8  0.8  0.6  0.8  0.8  0.8  0.7  0.7 
Average response score of the teachers
Teachers  

\(t_1\)  \(t_2\)  \(t_3\)  \(t_4\)  \(t_5\)  \(t_6\)  
Scores  0.78  0.72  0.82  0.8  0.78  0.78 
Edge membership values which represent the relationship between the teachers can be calculated by using the relation \(p_i\circ B(uv)\le min\{p_i\circ A(u), p_i\circ A(v)\}\) for all \(u,v\in T\), \(i=1,2,\ldots ,5\). These values are given in the Table 2.

Teacher’s average response score <60%, teacher’s performance according to the students is \(\mathbf{Average}\).

Teacher’s average response score ≥60% and <70%, teacher’s performance according to the students is \(\mathbf{Good}\).

Teacher’s average response score ≥70% and <80%, teacher’s performance according to the students is \(\mathbf {Very}\) \(\mathbf{Good}\).

Teacher’s average response score is ≥80%, teacher’s performance according to the students is \(\mathbf{Excellent}\).
Conclusions
The theory of fuzzy graphs play an important role in many fields including decision makings, computer networking and management sciences. An mpolar fuzzy graph can be used to represent real world problems which involve multiagent, multiattribute, multiobject, multiindex, multipolar information and uncertainty. In this research paper, we have studied the isomorphic properties of mpolar fuzzy graphs with some applications. We are extending our research work on mpolar fuzzy intersection graphs, mpolar fuzzy interval graphs, properties of mpolar fuzzy hypergraphs, degrees of vertices of mpolar fuzzy graphs and its application in decision making, etc.
Declarations
Authors' contributions
Both authors have significant contributions to this paper and the final form of this paper is approved by both of them. Both authors read and approved the final manuscript.
Acknowledgements
The authors would wish to express their sincere gratitude to the Editor in Chief and anonymous referees for their valuable comments and helpful suggestions.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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