# A Novel Study of Graphs Based onm-Polar Cubic Structures

Authors: G. Muhiuddin; A. M. Alanazi; A. Mahboob; A. H. Alkhaldi; Wejdan Alatwai
Journal: Journal of Function Spaces
 (2022)

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## Abstract

By combining the notions of interval-valued m-polar fuzzy graphs and m-polar fuzzy graphs, the notion of m-polar cubic graphs is first introduced. Then, the degree of a vertex in m-polar cubic graphs and complete m-polar cubic graphs is defined. After that, the concepts of direct product and strong product of m-polar cubic graphs are given. Moreover, weak isomorphism and co-weak isomorphism are defined, and examples are given to prove that weak isomorphism and co-weak isomorphism are not an isomorphism. Finally, the notion of complement m-polar cubic graphs and weak complement m-polar cubic graphs is presented.


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## Body

## 1. Introduction

Fuzzy set is a set that comprises components with just a partial degree of membership and without crisp. The significant advances in the interesting area of fuzzy sets began with the work of Zadeh, who pioneered new directions and concepts. A study of the fuzzy graph theory has started in the pioneering paper of Rosenfeld in 1975 [1]. Following that, the concept of fuzzy graph theory was extensively studied by several authors, particularly Mordeson and Nair [2, 3], Bhutani et al. [4, 5], Al-Masarwah and Qamar [6, 7], and Akram et al. [8, 9].Akram et al. [10, 11] initiated the study of bipolar fuzzy graphs. Rashmanlou et al. [12–14] studied bipolar fuzzy graphs with categorical properties, bipolar fuzzy graph products, and their degrees. Chen et al. (2014) [15] initially established the concept of m-polar fuzzy graphs by generalizing the concept of bipolar fuzzy graphs. After that, Ghorai and Pal [16] provided properties of generalized m-polar fuzzy graphs and described various operations with density of m-polar fuzzy graphs [17]. They also introduced the notions of m-polar fuzzy planar graphs, faces, and dual of m-polar fuzzy planar graph in [18]. With examples, Akram and Younas [19] described specific types of irregular m-polar fuzzy graphs and shown m-polar fuzzy graph applications in decision making and social networks. Regular m-polar fuzzy graphs, totally regular m-polar fuzzy graphs, complete m-polar fuzzy graphs, and m-polar fuzzy line graphs have been introduced and investigated by Akram et al. [20]. Ramprasad et al. [21] defined the notion of m-polar h-morphism of m-polar fuzzy graphs. For more concepts related to this work, we refer readers to [22–27].The concept of IVFG is defined by Hongmei and Lianhua [28] in 2009. In 2011, Akram and Dudek [9] defined the notion of interval-valued fuzzy complete graphs and considered some vital properties of self-complementary and self-weak complementary interval-valued fuzzy complete graphs. Product of IVFGs and degree of IVFGs based on strong product, tensor product, and lexicographic product is investigated by Rashmanlou et al. [29]. Pramanick and Pal [30] defined interval-valued planar graph and presented several properties. By combining the notions of m-polar fuzzy graph (m-PFG) and interval-valued fuzzy graph (IVFG), the notion of m-polar interval-valued fuzzy graph (m-PIVFG) was introduced by Sanchari and Pal [31]. After that, the notions of palner graph and strong edges based on IVmPF set theory were introduced and investigated by Mahapatra et al. [32].The present paper is divided into four sections in which the first section is the introduction. In Section2, we first review some fundamental concepts that will be used in the sequel. In Section 3, the notion of m-polar cubic graphs is introduced, and the degree of a vertex in m-polar cubic graphs and complete m-polar cubic graphs is defined. Moreover, the concepts of direct product and strong product of m-polar cubic graphs are given. In Section 4, weak isomorphism and co-weak isomorphism are defined, and the relation between these notions is considered. Also, the notions of complement m-polar cubic graphs and weak complement m-polar cubic graphs are presented.


## 2. Preliminaries

In this section, we collect the basic notions that will be used throughout this article.A mappingη:X⟶0,1 is called a fuzzy set of X. For any two fuzzy subsets η1 and η2 of X, η1≤η2 means that, for all ℏ∈X, η1ℏ≤η2ℏ. The intersection η1∩η2 and union η1∪η2 are defined as
(1)η1∩η2ℏ=minη1ℏ,η2ℏandη1∪η2ℏ=maxη1ℏ,η2ℏ,for all ℏ∈X.By an interval numberc~, we mean an interval, denoted by c−,c+, where 0≤c−≤c+≤1. The set of all interval numbers is denoted by D0,1. In whatever follows, the interval c,c is identified by the number c∈0,1. For the interval numbers c~i=ci−,ci+, d~i=di−,di+∈D0,1,i∈I, we define
(a)
rminc~i,d~i=minci−,di−,minci+,di+;(b)
rmaxc~i,d~i=maxci−,di−,maxci+,di+;(c)
c~1≤c~2⇔c1−≤c2− and c1+≤c2+;(d)
c~1=c~2⇔c1−=c2− and c1+=c2+.LetX be a nonempty set. A mapping P~:X⟶D0,1 is called an interval-valued fuzzy set in X, where P~ℏ=P~−ℏ,P~+ℏ for all ℏ∈X, P~− and P~+ are fuzzy sets of X with P~−ℏ≤P~+ℏ for all ℏ∈X.Definition 1.
LetX be a nonempty set. A cubic set A is a structure
(2)A=ℏ,P~ℏ,ηℏ∣ℏ∈Xwhich is denoted by A=P~,η, where P~ is an interval-valued fuzzy set and η is a fuzzy set in X.Definition 2.
A cubic graph is a tripletG=G∗,A,B, where G∗=V,E is a graph, A=PA~,ηA is a cubic set on V, and B=PB~,ηB is a cubic set on V×V such that
(3)PB~ℏϑ≤rminPA~ℏ,PA~ϑ,ηBℏϑ≥maxηAℏ,ηAϑ.Definition 3.
A mappingη:X⟶0,1m is called an m-polar fuzzy set (briefly, mpF set) of X and is defined as
(4)ηυ=π1∘ηυ,π2∘ηυ,⋯,πm∘ηυ,where πi∘ηυ denotes the ith degree of membership for an element υ.
Define an order “≤” on 0,1m as pointwise, i.e.,
(5)ℏ≤ϑ⇔πiℏ≤πiϑ∀i∈1,2,⋯,m.
πi:0,1m⟶0,1 is the ith projection mapping. For an element ℓ~∈D0,1m, we mean that ℓ,ℓ,⋯,ℓ. Clearly, the elements 0~ and 1~ are the smallest and largest elements in 0,1m.Definition 4.
A mappingP~:X⟶D0,1m is called an interval-valued m-polar fuzzy set (briefly, IVmPF set) of X and is defined as
(6)P~υ=π1~∘P~υ,π2~∘P~υ,⋯,πm~∘P~υ,where π~i∘P~υ denoted the ith degree of membership of element υ.
That is(7)P~υ=P1−υ,P1+υ,P2−υ,P2+υ,⋯,Pm−υ,Pm+υ.For all υ∈X, Pi− and Pi+ are fuzzy sets of X with Pi−υ≤Pi+υ for all υ∈X and i∈1,2,⋯,m.
We define an order onD0,1m as pointwise, i.e.,
(8)ℏ≤ϑ⇔π~iℏ≤π~iϑ∀i∈1,2,⋯,m.
πi~:D0,1m⟶D0,1 is the i-th projection mapping. For an element α,β~∈D0,1m, we mean that α,β,α,β,.⋯,α,β. Clearly, the elements 0,0~ and 1,1~ are the smallest and largest elements in D0,1m.


## 3.m-Polar Cubic Graphs

In this section, we define m-polar fuzzy graphs and the degree of a vertex in m-polar cubic graphs and complete m-polar cubic graphs. In addition, the notions of direct product and strong product of m-polar cubic graphs are presented.LetX be a nonempty set. An m-polar cubic set A is a structure
(9)A=ℏ,P~ℏ,ηℏ∣ℏ∈X,which is denoted by A=P~,η, where P~ is an IVmPF set and η is an mpF set in X.Definition 5.
An m-polar cubic graph (briefly, mpC graph) is a tripletG=G∗,A,B, where G∗=V,E is a graph, A=PA~,ηA is an m-polar cubic set on V and B=PB~,ηB is an m-polar cubic set on V×V such that for each ι∈1,2,⋯,m, π~i∘PB~ℏy≤rminπ~i∘PA~ℏ,π~i∘PA~ϑ, and πi∘ηBℏϑ≥maxπi∘ηAℏ,πi∘ηAϑ. Also, PB~ℏϑ=0~ and ηBℏϑ=0^ for all ℏϑ∈V×V−E, where 0~=0,0,0,0,⋯,0,0 and 0^=0,0,⋯,0 are the smallest elements of D0,1m and 0,1m.Example 1.
Take a graphG∗=V,E, where V=ℏ,ϑ,κ,ω and E=ℏϑ,ϑκ,κω,ℏω,ϑω. Define P=PA~,ηA a cubic set on V and Q=PB~,ηB a cubic set on E as shown in Tables 1 and 2.Table 1VℏϑκωηA0.3,0.4,0.50.4,0.5,0.60.5,0.6,0.70.2,0.3,0.4EℏϑϑκκωℏωηB0.5,0.6,0.70.6,0.7,0.80.7,0.8,0.90.4,0.5,0.6Table 2VP~EP~ℏ([0.7,0.6],[0.8,0.5],[0.4,0.2])ℏϑ([0.3,0.2],[0.5,0.3],[0.2,0.1])ϑ([0.6,0.5],[0.5,0.4],[0.4,0.3])ϑκ([0.2,0.1],[0.4,0.2],[0.3,0.2])ω([0.3,0.2],[0.4,0.3],[0.2,0.1])κω([0.2,0.1],[0.3,0.1],[0.2,0.1])κ([0.3,0.2],[0.4,0.2],[0.5,0.4])ℏω([0.3,0.2],[0.2,0.1],[0.2,0.1])It is a routine to verify that graphG=G∗,A,B is an 3pC graph as shown in Figure 1.Definition 6.
The degree of a vertexa in an mpC graph G=G∗,A,B is defined as
(10)da=d1Pa,d2Pa,⋯,dmPad1ηa,,d2ηa⋯,dmηa,where diPa=∑a≠ba,b∈Vπ~i∘PB~ab and diηa=∑a≠ba,b∈Vπi∘ηBab.Example 2.
In Figure1, we have
(11)dℏ=0.6,0.4,0.7,0.4,0.4,0.2,0.9,0.9,0.9.
Similarly, we can computedv,dw, and dℏ.Definition 7.
An mpC graphG=G∗,A,B is called complete if π~i∘PB~ℏϑ=rminπ~i∘PA~ℏ,π~i∘PA~ϑ and πi∘ηBℏϑ=maxπi∘ηAℏ,πi∘ηAϑ, for all ℏ,ϑ∈V.Example 3.
Consider the graphG∗=V,E, where V=ℏ,ϑ,κ,ω and E=ℏϑ,ϑκ,κω,ℏω,ϑω. Define P=PA~,ηA a 3pC set on V and Q=PB~,ηB a 3pC set on E as shown in Tables 3 and 4.Figure 1
3pC graph G.Table 3VηAEηBℏ(0.3,0.4,0.5)ℏϑ(0.5,0.6,0.7)ϑ(0.4,0.5,0.6)ϑκ(0.6,0.7,0.8)κ(0.5,0.6,0.7)κω(0.7,0.8,0.9)ω(0.2,0.3,0.4)ℏω(0.4,0.5,0.6)ϑω(0.5,0.6,0.7)ℏκ(0.5,0.6,0.7)Table 4VP~EP~ℏ([0.6,0.7],[0.5,0.8],[0.2,0.4])ℏϑ([0.2,0.3],[0.1,0.4],[0.1,0.2])ϑ([0.5,0.6],[0.5,0.5],[0.3,0.4])ϑκ([0.2,0.3],[0.1,0.4],[0.1,0.2])κ([0.3,0.4],[0.1,0.2],[0.1,0.2])κω([0.2,0.3],[0.1,0.4],[0.1,0.2])ω([0.2,0.3],[0.2,0.3],[0.1,0.2])ℏω([0.2,0.3],[0.1,0.4],[0.1,0.2])ϑω([0.2,0.3],[0.1,0.4],[0.1,0.2])ℏκ([0.3,0.4],[0.1,0.2],[0.1,0.2])It is a routine to verify that graphG=G∗,A,B is a complete 3pC graph as shown in Figure 2:Definition 8.
An mpC graphG=G∗,A,B is called strong if π~i∘PB~ab=rminπ~i∘PA~a,πi~∘PA~b and πi∘ηBab=maxπi∘ηAa,πi∘ηAb for all ab∈E.Definition 9.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two mpC graphs, where G1∗=V1,E1 and G2∗=V2,E2. Then, direct product of G1 and G2 is defined by G1⨀G2=G1∗⨀G2∗,A1⨀A2,B1⨀B2, where G1∗⨀G2∗=V1×V2,E, E=a1,ϑ1a2,ϑ2∣a1,a2∈E1,ϑ1,ϑ2∈E2, and for all a,b,a1,ϑ1,a2,ϑ2∈V1×V2, we have
(12)πi~∘PA1~⨀PA2~a,b=rminπi~∘PA1~a,π~i∘PA2~b,πi∘ηA1⨀ηA2a,b=maxπi∘ηA1a,πi∘ηA2b,(13)π~i∘PB1~⨀PB2~ℏ1,ϑ1ℏ2,ϑ2=rminπ~i∘PB1~ℏ1ℏ2,π~i∘PB2~ϑ1ϑ2,πi∘ηB1⨀ηB2ℏ1,ϑ1ℏ2,ϑ2=maxπi∘ηB1ℏ1ℏ2,πi∘ηB2ϑ1ϑ2.Theorem 10.
The direct productG1⨀G2 of two mpC graphs G1 and G2 is an mpC graph.Proof.
The proof is straightforward.Example 4.
LetG1∗=V1,E1 and G2∗=V2,E2 be two crisp graphs such that V1=ℏ,ϑ, V2=κ,ω, E1=ℏϑ, and E2=κω. Consider 3pC graphs G1=G1∗,A1,B1 and G2=G2∗,A2,B2 as shown in Figure 3.Figure 2
Complete3pC graph.Figure 3
3pc graphs G1 and G2.The direct product ofG1 and G2 is G1⨀G2 as shown in Figure 4.Figure 4
3pc graph G=G1×G2.It is easy to show thatG1⨀G2 is a 3pC graph.Proposition 11.
IfG1=G1∗,A1,B1 and G2=G2∗,A2,B2 are two strong mpC graphs, then, G1⨀G2 is a strong mpC graph, too.Proof.
Ifℏ1,b1ℏ2,ϑ2∈E, since G1 and G2 are strong, then, we have
(14)π~i∘PB1~⨀PB2~ℏ1,ϑ1ℏ2,ϑ2=rminπ~i∘PB1~ℏ1ℏ2,π~i∘Pϑ2~ϑ1ϑ2=rminrminπ~i∘PA1~ℏ1,π~i∘PA1~ℏ2,rminπ~i∘PA2~ϑ1,π~i∘PA2~ϑ2=rminπ~i∘PA1~⨀PA2~ℏ1,ϑ1,π~i∘PA1~⨀PA2~ℏ2,ϑ2,(15)πi∘ηB1⨀ηB2ℏ1,ϑ1ℏ2,ϑ2=maxπi∘ηB1ℏ1ℏ2,πi∘ηB2ϑ1ϑ2=maxmaxπi∘ηA1ℏ1,πi∘ηA1ℏ2,maxπi∘ηA2ϑ1,πi∘ηA2ϑ2=maxπi∘ηA1⨀ηA2ℏ1,ϑ1,πi∘ηA1⨀ηA2ℏ2,ϑ2.
Hence,G1⨀G2 is a strong mpC graph.Remark 12.
IfG1=G1∗,A1,B1 and G2=G2∗,A2,B2 are two complete mpC graphs, then, G1⨀G2 may not be complete, as shown by the following illustration.Example 5.
Consider the3pC graphs G1=G1∗,A1,B1 and G2=G2∗,A2,B2 as shown in Figure 3. It is routine to verify that G1 and G2 are complete 3pC graphs, but their direct product is not a complete 3pC graph because π~i∘PB~a,db,c≠rminπ~i∘PA~a,d,π~i∘PA~b,c and πi∘ηBa,db,c≠maxπi∘ηAa,d,πi∘ηAb,c.Definition 13.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two mpC graphs, where G1∗=V1,E1 and G2∗=V2,E2. Then, strong product of G1andG2 is defined by G1⊗G2=G1∗⊗G2∗,A1⊗A2,B1⊗B2, where G1∗⊗G2∗=V1×V2,E, E=a,ϑ1a,ϑ2∣a∈V1,ϑ1,ϑ2∈E2∪ℏ1,wℏ2,w∣w∈V2,ℏ1,ℏ2∈E1∪ℏ1,ϑ1ℏ2,ϑ2∣ℏ1,ℏ2∈E1,ϑ1,ϑ2∈E2, and for all a,b,ℏ1,ℏ2,ϑ1,ϑ2,w∈V, we have
(1)
π~i∘PA1~⊗PA2~a,b=rminπ~i∘PA1~a,π~i∘PA2~b,πi∘ηA1⊗ηA2a,b=maxπi∘ηA1a,πi∘ηA2b,(2)
π~i∘PB1~⊗PB2~a,b1a,ϑ2=rminπ~i∘PA1~a,π~i∘PB2~ϑ1ϑ2,πi∘ηϑ1⊗ηB2a,ϑ1a,ϑ2=maxπi∘ηA1a,πi∘ηB2ϑ1ϑ2,(3)
π~i∘PB1~⊗PB2~ℏ1,ωℏ2,ω=rminπ~i∘PA2~ω,π~i∘PB1~ℏ1ℏ2,πi∘ηB1⊗ηB2ℏ1,ωℏ2,ω=maxπi∘ηA2w,πi∘ηB1ℏ1ℏ2,(4)
π~i∘PB1~⊗PB2~ℏ1,ϑ1ℏ2,ϑ2=rminπ~i∘PB1~ℏ1ℏ2,π~i∘PB2~ϑ1ϑ2,πi∘ηB1⊗ηB2ℏ1,ϑ1ℏ2,ϑ2=maxπi∘ηB1ℏ1ℏ2,πi∘ηB2ϑ1ϑ2.Example 6.
Consider3pC graphs G1=G1∗,A1,B1 and G2=G2∗,A2,B2 as follows by following Figure 5:
The direct product ofG1 and G2 is G1⨀G2 as shown in below by following Figure 6:Theorem 14.
The strong productG1⊗G2 of two mpC graphs G1 and G2 is an mpC graph.Proof.
The proof is straightforward.Proposition 15.
IfG1=G1∗,A1,B1 and G2=G2∗,A2,B2 are two complete mpC graphs, then, G1⊗G2 is also complete mpC graph.Proof.
For alla,ϑ1a,ϑ2∈E, since G2 is complete, we have
(16)π~i∘PB1~⊗PB2~a,ϑ1a,ϑ2=rminπ~i∘PA1~a,π~i∘PB2~ϑ1ϑ2=rminπ~i∘PA1~a,rminπ~i∘PA2~ϑ1,π~i∘PA2~ϑ2=rminπ~i∘PA1~⊗PA2~a,ϑ1,π~i∘PA1~⊗PA2~a,ϑ2,(17)πi∘ηB1⊗ηB2a,ϑ1a,ϑ2=maxπi∘ηA1a,πi∘ηB2ϑ1ϑ2=maxπi∘ηA1a,maxπi∘ηA2ϑ1,πi∘ηA2ϑ2=maxπi∘ηA1⊗ηA2a,ϑ1,πi∘ηA1⊗ηA2a,ϑ2.
Now, ifℏ1,bℏ2,b∈E, then,
(18)π~i∘PB1~⊗PB2~ℏ1,bℏ2,b=rminπ~i∘PA2~b,π~i∘PB1~ℏ1ℏ2=rminπ~i∘PA2~b,rminπ~i∘PA1~ℏ1,π~i∘PA1~ℏ2=rminπ~i∘PA1~⊗PA2~ℏ1,b,π~i∘PA1~⊗PA2~ℏ2,b.
Similarly,(19)πi∘ηB1⊗ηB2ℏ1,bℏ2,b=maxπi∘ηA1⊗ηA2ℏ1,b,πi∘ηA1⊗ηA2ℏ2,b.
Now, ifℏ1,ϑ1ℏ2,ϑ2∈E, then, since G1 and G2 are complete, we get
(20)π~i∘PB1~⊗PB2~ℏ1,ϑ1ℏ2,ϑ2=rminπ~i∘PB1~ℏ1ℏ2,π~i∘PB2~ϑ1ϑ2=rminrminπ~i∘PA1~ℏ1,π~i∘PA1~ℏ2,rminπ~i∘PA2~ℏ2,π~i∘PA2~ϑ2=rminπ~i∘PA1~⊗PA2~ℏ1,ϑ1,π~i∘PA1~⊗PA2~ℏ2,ϑ2,(21)πi∘ηB1⊗ηB2ℏ1,ϑ1ℏ2,ϑ2=maxπi∘ηB1ℏ1ℏ2,πi∘ηB2ϑ1ϑ2=maxmaxπi∘ηA1ℏ1,πi∘ηA1ℏ2,maxπi∘ηℏ2ϑ1,πi∘ηA2ϑ2=maxπi∘ηA1⊗ηA2ℏ1,ϑ1,πi∘ηA1⊗ηA2ℏ2,ϑ2.
Hence,G1⊗G2 is complete.Theorem 16.
IfG1=G1∗,A1,B1 and G2=G2∗,A2,B2 are two mpC graphs, such that G1⨀G2 is strong, then, at least one of G1 or G2 must be strong.Proof.
Suppose thatG1 and G2 are not strong. Then, there exist ℏ1,ϑ1∈E1 and ℏ2,ϑ2∈E2 such that
(22)π~i∘PB1~ℏ1ϑ1<rminπ~i∘PA1~ℏ1,π~i∘PA1~ϑ1,πi∘ηB1ℏ1ϑ1>maxπi∘ηA1ℏ1,πi∘ηA1ϑ1,(23)π~i∘PB2~ℏ2ϑ2<rminπ~i∘PA2~ℏ2,π~i∘PA2~ϑ2,πi∘ηB2ℏ2ϑ2>maxπi∘ηA2ℏ2,πi∘ηA2ϑ2.
Hence,(24)π~i∘PB1~⨀PB2~ℏ1,ϑ1ℏ2,ϑ2=rminπ~i∘PB1~ℏ1ℏ2,π~i∘PB2~ϑ1ϑ2<rminrminπ~i∘PA1~ℏ1,π~i∘PA1~ℏ2,rminπ~i∘PA2~ϑ1,π~i∘PA2~ϑ2,(25)πi∘ηB1⨀ηB2ℏ1,ϑ1ℏ2,ϑ2>maxmaxπi∘ηA1ℏ1,πi∘ηA1ℏ2,maxπi∘ηA2ϑ1,πi∘ηA2ϑ2.
Now, sinceπ~i∘PA1~⨀PA2~ℏ1,ϑ1=rminπ~i∘PA1~ℏ1,π~i∘PA2~ϑ1 and π~i∘PA1~⨀PA2~ℏ2,ϑ2=rminπ~i∘PA1~ℏ1,π~i∘PA2~ϑ2, thus,
(26)rminπ~i∘PA1~⨀PA2~ℏ1,ϑ1,π~i∘PA1~⨀PA2~ℏ2,ϑ2=rminrminπ~i∘PA1~ℏ1,πi~∘PA1~ℏ2,rminπ~i∘PA2~ϑ1,π~i∘PA2~ϑ2>π~i∘PB1~⨀PB2~ℏ1,ϑ1ℏ2,ϑ2.
Similarly,maxπi∘ηA1⨀ηA2ℏ1,ϑ1,πi∘ηA1⨀ηA2ℏ2,ϑ2<πi∘ηB1⨀ηB2ℏ1,ϑ1ℏ2,ϑ2. Therefore, G1eG2 is not strong, which is a contradiction.Remark 17.
The degree of a vertex in direct product can be calculated by following formula. By definition of direct product for anyℏ1,ℏ2∈V1×V2, we have dG1⨀G2a=π1~∘d1G1⨀G2Pa,π2~∘d2G1⨀G2Pa,⋯,πm~∘dmG1⨀G2Paπ1∘d1G1⨀G2ηa,π2∘d2G1⨀G2ηa⋯,πm∘dmG1⨀G2ηa, where
(27)π~i∘diG1⨀G2Pℏ1,ℏ2=∑π~i∘PB1~⨀PB2~ℏ1,ℏ2ϑ1,ϑ2=∑rminπ~i∘PB1~ℏ1ϑ1,π~i∘PB2~ℏ2ϑ2,(28)πi∘diG1⨀G2ηℏ1,ℏ2=∑πi∘ηB1⨀ηB2ℏ1,ℏ2ϑ1,ϑ2=∑maxπi∘ηB1ℏ1ϑ1,πi∘ηB2ℏ2ϑ2.Theorem 18.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two mpC graphs. If π~i∘PB2~≥π~j∘PB1~ and πi∘ηB2≤πj∘ηB1 for all i,j∈1,2,⋯,m, then, dG1⨀G2ℏ1,ℏ2=dG1ℏ1. Also, if π~i∘PB1~≥π~j∘PB2~ and πi∘ηB1≤πj∘ηB2 for all i,j∈1,2,⋯,m, then, dG1⨀G2ℏ1,ℏ2=dG2ℏ2.Proof.
Letπ~i∘PB2~≥πj∘PB1~, and π~i∘ηB2≤πj∘ηB1. Then,
(29)π~i∘dG1⨀G2Pℏ1,ℏ2=∑ℏ1ϑ1∈E1rminπ~i∘PB1~ℏ1ϑ1,π~i∘PB2~ℏ2ϑ2=∑π~i∘PB1~ℏ1ϑ1=π~i∘dG1Pℏ1,πi∘dG1⨀G2ηℏ1,ℏ2=∑ℏ1ϑ1∈E1maxπi∘ηB1ℏ1ϑ1,πi∘ηB2ℏ2ϑ2=∑πi∘ηB1ℏ1ϑ1=πi∘dG1ηℏ1.
Hence,dG1⨀G2ℏ1,ℏ2=dG1ℏ1. Similarly, if π~i∘PB1≥πj∘PB2 and πi∘ηB1≤πj∘ηB2, then, dG1⨀G2ℏ1,ℏ2=dG2ℏ2.Remark 19.
By definition of direct product for anyℏ1,ℏ2∈V1×V2, we have dG1⊗G2a=π1∘d1G1⊗G2Pa,π2∘d2G1⊗G2Pa,⋯,πm∘dmG1⊗G2Paπ1∘d1G1⊗G2ηa,π2∘d2G1⊗G2ηa⋯,πm∘dmG1⊗G2ηa, where
(30)π~i∘diG1⊗G2Pℏ1,ℏ2=∑ℏ1,ϑ1ℏ2,ϑ2∈Eπ~i∘PB1~⊗PB2~ℏ1,ϑ1ℏ2,ϑ2=∑ℏ1=ϑ1,ℏ2ϑ2∈E2rminπ~i∘PA1~ℏ1,π~i∘PB2~ℏ2ϑ2+∑ℏ2=ϑ2,ℏ1ϑ1∈E1rminπ~i∘PA2~ℏ2,π~i∘Pϑ1~ℏ1ϑ1+∑ℏ2ϑ2∈E2,ℏ1ϑ1∈E1rminπ~i∘PB2~ℏ2ϑ2,π~i∘PB1~ℏ1ϑ1,(31)πi∘diG1⊗G2ηℏ1,ℏ2=∑ℏ1,ϑ1ℏ2,ϑ2∈Eπi∘ηB1⊗ηB2ℏ1,ϑ1ℏ2,ϑ2=∑ℏ1=ϑ1,ℏ2ϑ2∈E2maxπi∘ηA1ℏ1,πi∘ηB2a2ϑ2+∑ℏ2=ϑ2,ℏ1ϑ1∈E1maxπi∘ηA2ℏ2,πi∘ηB1ℏ1ϑ1=∑ℏ2ϑ2∈E2,ℏ1ϑ1∈E1maxπi∘ηB2ℏ2ϑ2,πi∘ηB1ℏ1ϑ1.Theorem 20.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two mpC graphs. If PA1~≥PB2~, ηA1≤ηB2, PA2~≥PB1~, ηA2≤ηB1, PB1~≤PB2~, and ηB1≥ηB2, then, dG1⊗G2a1,a2=V2dG1a1+dG2a2.Proof.
LetPA1~≥PB2~, ηA1≤ηB2, PA2~≥PB1~, ηA2≤ηB1, PB1~≤PB2~, and ηB1≥ηB2. Then,
(32)dG1⊗G2Pℏ1,ℏ2=∑ℏ1,ϑ1ℏ2,ϑ2∈EPB1~⊗PB2~ℏ1,ϑ1ℏ2,ϑ2=∑ℏ1=ϑ1,ℏ2ϑ2∈E2rminPA1~ℏ1,PB2~ℏ2ϑ2+∑ℏ2=ϑ2,u1ϑ1∈E1rminPA2~u2,PB1~u1ϑ1+∑u2ϑ2∈E2,u1ϑ1∈E1rminPB2~u2ϑ2,PB1~u1ϑ1=∑u2ϑ2∈E2PB2~u2ϑ2+∑u2=ϑ2,u1ϑ1∈E1PB1~u1ϑ1+∑u1ϑ1∈E1PB1~u1ϑ1=dG2Pu2+V2dG1Pu1.
In the same way, we can show thatdG1⊗G2ηu1,u2=dG2ηu2+V2dG1ηu1. Hence, dG1⊗G2u1,u2=dG2u2+V2dG1u1.Figure 5
3pc graphs G1 and G2.Figure 6
3pc graph G1⊗G2.


## 4. Self-Complement ofmpC Graph

In this section, we define homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism between two cubic graphs and show that isomorphism between cubic graphs is an equivalence relation.Definition 21.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two mpC graphs. A homomorphism from G1 to G2 is a mapping ϕ:V1⟶V2 such that
(1)
π~i∘PA1~ℏ1≤π~i∘PA2~ϕℏ1,πi∘ηA1ℏ1≥πi∘ηA2ϕℏ1,(2)
π~i∘PB1~ℏ1ϑ1≤π~i∘PB2~ϕℏ1ϕϑ1,πi∘ηB1ℏ1ϑ1≥πi∘ηB2ϕℏ1ϕϑ1,for all ℏ1∈V1, ℏ1ϑ1∈E1 and i∈1,2,⋯,m.Definition 22.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two m-PCGs. An isomorphism ϕ:G1⟶G2 is a bijective mapping ϕ:V1⟶V2 such that
(1)
π~i∘PA1~ℏ1=π~i∘PA2~ϕℏ1,πi∘ηA1ℏ1=πi∘ηA2ϕℏ1,(2)
π~i∘PB1~ℏ1ϑ1=π~i∘PB2~ϕℏ1ϕϑ1,πi∘ηB1ℏ1ϑ1=πi∘ηB2ϕℏ1ϕϑ1,for all ℏ1∈V1 and ℏ1ϑ1∈E1 and i∈1,2,⋯,m. If ϕ:G1⟶G2 is an isomorphism, then, we say that G1 and G2 are isomorphic and we write G1≅G2.Definition 23.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two mpC graphs. A weak isomorphism ϕ:G1⟶G2 is a bijective mapping ϕ:V1⟶V2 such that
(1)
ϕ is homomorphism,(2)
π~i∘PA1~ℏ1=π~i∘PA2~ϕℏ1,πi∘ηA1ℏ1=πi∘ηA2ϕℏ1,for all ℏ1∈V1 and i∈1,2,⋯,m.Remark 24.
Every isomorphism is a weak isomorphism, but the converse implication is not true in general. We show it by the following illustration:Example 7.
Consider3pC graphs G1 and G2 as follows by following Figure 7.
Letϕ:V1⟶V2 is defined by ϕa=e,ϕb=g, and ϕc=f. Then, we see that
PA1~c=PA2~f, ηA1c=ηA2f, PA1~b=PA2~g, ηA1b=ηA2g, PA1~a=PA2~e, ηA1a=ηA2e, but PB1~ac≠PB2~ϕaϕc=PB2~ef, PB1~ab≠PB2~ϕaϕb=PB2~eg, ηB1ac≠ηB2ϕaϕc=ηB2ef and ηB1ab≠ηB2ϕaϕb=ηB2eg. Hence, the map is a weak isomorphism, but it is not an isomorphism.Definition 25.
LetG1 and G2 be two mpC graphs. A co-weak isomorphism ϕ:G1⟶G2 is a bijective mapping ϕ:V1⟶V2 which satisfies: 1ϕ is homomorphism, 2π~i∘PB1~ℏ1ϑ1=π~i∘PB2~ϕℏ1ϕϑ1, ηB1ℏ1ϑ1=ηB2ϕℏ1ϕϑ1 for all ℏ1,ϑ1∈V1.Example 8.
Consider3pC graphs G1 and G2 as follows by following Figure 8.
Letϕ:V1⟶V2 be defined by ϕb=d and ϕa=c. By routine computations, we can show that the map ϕ is a co-weak isomorphism, but it is not an isomorphism, since PA1~b≠PA2~c and ηA1b≠ηA2c.Definition 26.
The complement of mpC graphG=G∗,A,B, where G∗=V,E is an mpC graph G¯=G∗¯,A¯,B¯, where A¯=A=PA~,ηA, B¯=PB~¯,ηB¯, and PB~¯ℏϑ=rminPA~ℏ,PA~ϑ−PB~ℏϑ, ηB¯ℏϑ=ηBℏϑ−maxηAℏ,ηAϑ, for all ℏ,ϑ∈V.Theorem 27.
LetG1=G1∗,A1,B1 and G2=G2∗,A2,B2 be two mpC graphs. If G1≅G2, then, G1¯≅G2¯.Proof.
LetG1≅G2. Then, ∃ϕ:V1⟶V2 such that π~i∘PA1~ℏ1=π~i∘PA2~ϕℏ1,πi∘ηA1ℏ1=πi∘ηA2ϕℏ1, and π~i∘PB1~ℏ1ϑ1=π~i∘PB2~ϕℏ1ϕϑ1,πi∘ηB1ℏ1ϑ1=πi∘ηB2ϕℏ1ϕϑ1 for all ℏ1∈V1 and ℏ1ϑ1∈E1 and i∈1,2,⋯,m. Now for all ℏ∈V1, we have
(33)π~i∘PA1~¯ℏ=π~i∘PA1~ℏ=π~i∘PA2~ϕℏ=π~i∘PA2~¯ϕℏ,πi∘ηA1¯ℏ=πi∘ηA1ℏ=πi∘ηA2¯ϕℏ=πi∘ηA2ϕℏ.
Also for allℏ,ϑ∈V1, we have
(34)π~i∘PB1~¯ℏϑ=rminπ~i∘PA1~ℏ,π~i∘PA1~ϑ−π~i∘PB1~ℏϑ=rminπ~i∘PA2~ℏ,π~i∘PA2~ϑ−π~i∘PB2~ℏϑ=π~i∘PB2~¯ℏϑ,πi∘ηB1¯ℏϑ=πi∘ηB1ℏϑ−maxπi∘ηA1ℏ,πi∘ηA1ϑ=πi∘ηB2ℏϑ−maxπi∘ηA2ℏ,πi∘ηA2ϑ=πi∘ηB2¯ℏϑ.
Hence,G1≅G2.Definition 28.
An mpC graphG is said to be self complement if there exists an isomorphism from G onto G¯.Example 9.
Consider2pC graph G as follows by following Figure 9.
Consequently,G is self complementary, since ϕ:G⟶G¯ is an isomorphism as defined by b⟶b, a⟶c, c⟶a.Theorem 29.
LetG be a self complement mpC graph. Then,
(35)∑ℏ≠ϑπ~i∘PB~ℏϑ=12∑ℏ≠ϑrminπ~i∘PA~ℏ,π~i∘PA~ϑ,∑ℏ≠ϑπi∘ηBℏϑ=12∑ℏ≠ϑmaxπi∘ηAℏ,πi∘ηAϑ.Proof.
LetG=G∗,A,B be a self complement mpC graph of graph G∗=V,E. Then, there exists an isomorphism ϕ from G to G¯ such that for every ℏ,ϑ∈V, we have π~i∘PA~ℏ=πi~∘PA~¯ϕℏ, πi∘ηAℏ=πi∘ηA¯ϕℏ, for all ℏ∈V and π~i∘PB~ℏϑ=π~i∘PB~¯ϕℏϕϑ, πi∘ηBℏϑ=πi∘ηB¯ϕℏϕϑ, for all ℏ,ϑ∈V. Now by definition of G¯, for every ℏ,ϑ∈V, we have
(36)π~i∘PB~¯ϕℏϕϑ=rminπ~i∘PA~ϕℏπ~i∘PA~ϕϑ−π~i∘PB~ϕℏϕϑ,πi∘ηB¯ϕℏϕϑ=πi∘ηBϕℏϕϑ−maxπi∘ηAϕℏ,πi∘ηAϕϑ.
It follows thatπ~i∘PB~ℏϑ=rminπ~i∘PA~ℏ,π~i∘PA~ϑ−π~i∘PB~ϕℏϕϑ and πi∘ηBℏϑ=πi∘ηBϕℏϕϑ−maxπi∘ηAℏ,πi∘ηAϑ. Thus,
(37)∑rminπ~i∘PA~ℏ,πi~∘PA~ϑ=∑ℏ≠ϑπ~i∘PB~ℏϑ+∑π~i∘PB~ϕℏϕϑ=2∑πi∘ηBℏϑ,∑maxπi∘ηAℏ,πi∘ηAϑ=∑ℏ≠ϑπi∘ηBℏϑ+∑πi∘ηBϕℏϕϑ=2∑πi∘ηBℏϑ.
Hence,∑π~i∘PB~ℏϑ=1/2∑rminπ~i∘PA~ℏ,π~i∘PA~ϑ and ∑πi∘ηBℏϑ=1/2∑ℏ≠ϑmaxπi∘ηAℏ,πi∘ηAϑ.Figure 7Figure 8
3pC grphs G1 and G2.Figure 9


## 5. Conclusion

The main focus in this study is on connecting the theories of m-polar fuzzy graphs and m-polar interval-valued fuzzy graphs and developing a new model m-polar cubic graphs. The goal of this paper is to use an m-polar cubic set to deal with various types of graph theories. To begin, we defined m-polar cubic graphs. The concepts of direct product and strong product of m-polar cubic graphs are presented with illustrations. Also, the direct product and strong product of m-polar cubic graphs are studied by concerning the concepts of degree of a vertex and completeness of an m-polar cubic graph. Moreover, weak isomorphism and co-weak isomorphism are defined. Finally, the notion of complement m-polar cubic graphs and weak complement m-polar cubic graphs is presented.We believe that our research will have a substantial influence on other fuzzy algebraic studies and future research in this subject, pioneering the way for new areas of interest and technological advancement. To extend, one can further apply m-polar cubic set theory on different algebras such as hemirings,LA-semigroups, semihypergroups, semihyperrings, BCK/BCI-algebras, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, EQ-algebras, and lattice implication algebras.



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