# A Study on Neutrosophic Cubic Graphs with Real Life Applications in Industries

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (i)
- $({\mu}_{1}\times \phantom{\rule{4pt}{0ex}}{\mu}_{2})({x}_{1},{x}_{2})=min({\mu}_{1}({x}_{1}),{\mu}_{2}({x}_{2})),$ for all $({x}_{1},{x}_{2})\in V.$
- (ii)
- $({\nu}_{1}\times \phantom{\rule{4pt}{0ex}}{\nu}_{2})((x,{x}_{2})(x,{y}_{2}))=min({\mu}_{1}(x),{\nu}_{2}({x}_{2}{y}_{2})),$ for all $x\in {V}_{1}$, for all ${x}_{2}{y}_{2}\in {E}_{2}$.
- (iii)
- $({\nu}_{1}\times {\nu}_{2})(({x}_{1},z)({y}_{1},z))=min({\nu}_{1}({x}_{1}{y}_{1}),{\mu}_{2}(z)),$ for all $z\in {V}_{2}$, for all ${x}_{1}{y}_{1}\in {E}_{1}.$

**Definition**

**5.**

- (i)
- $({\mu}_{1}\circ {\mu}_{2})({x}_{1},{x}_{2})=min({\mu}_{1}({x}_{1}),{\mu}_{2}({x}_{2})),$ for all $({x}_{1},{x}_{2})\in V$.
- (ii)
- $({\nu}_{1}\circ {\nu}_{2})((x,{x}_{2})(x,{y}_{2}))=min({\mu}_{1}(x),{\nu}_{2}({x}_{2}{y}_{2})),$ for all $x\in {V}_{1},$ for all ${x}_{2}{y}_{2}\in {E}_{2}.$
- (iii)
- $({\nu}_{1}\circ {\nu}_{2})(({x}_{1},z)({y}_{1},z))=min({\nu}_{1}({x}_{1}{y}_{1}),{\mu}_{2}(z)),$ for all $z\in {V}_{2},$ for all ${x}_{1}{y}_{1}\in {E}_{1}.$
- (iv)
- $({\nu}_{1}\circ {\nu}_{2})(({x}_{1},{x}_{2})({y}_{1},{y}_{2}))=min({\mu}_{2}({x}_{2}),{\mu}_{2}({y}_{2}),{\nu}_{1}({x}_{1}{y}_{1})),$ for all $z\in {V}_{2},$ for all$({x}_{1},{x}_{2})({y}_{1},{y}_{2})\in \phantom{\rule{3.33333pt}{0ex}}{E}^{0}-E.$

**Definition**

**6.**

- (i)
- $({\mu}_{1}\cup {\mu}_{2})(x)={\mu}_{1}(x)$ if $x\in {V}_{1}\cap {V}_{2},$
- (ii)
- $({\mu}_{1}\cup {\mu}_{2})(x)={\mu}_{2}(x)$ if $x\in {V}_{2}\cap {V}_{1},$
- (iii)
- $({\mu}_{1}\cup {\mu}_{2})(x)=max({\mu}_{1}(x),{\mu}_{2}(x))$ if $x\in {V}_{1}\cap {V}_{2},$
- (iv)
- $({\nu}_{1}\cup {\nu}_{2})(xy)={\nu}_{1}(xy)$ if $xy\in {E}_{1}\cap {E}_{2},$
- (v)
- $({\nu}_{1}\cup {\nu}_{2})(xy)={\nu}_{2}(xy)$ if $xy\in {E}_{2}\cap {E}_{1},$
- (vi)
- (${\nu}_{1}\cup {\nu}_{2})(xy)=max($${\nu}_{1}(xy),$${\nu}_{2}(xy))$ if $xy\in {E}_{1}\cap {E}_{2}$.

**Definition**

**7.**

- (i)
- $({\mu}_{1}+{\mu}_{2})(x)=({\mu}_{1}\cup {\mu}_{2})(x)$ if $x\in {V}_{1}\cup {V}_{2},$
- (ii)
- $({\nu}_{1}+{\nu}_{2})(xy)=({\nu}_{1}\cup {\nu}_{2})(xy)={\nu}_{1}(xy)$ if $xy\in {E}_{1}\cup {E}_{2},$
- (iii)
- $({\nu}_{1}+{\nu}_{2})(xy)=min({\mu}_{1}(x),{\mu}_{2}(y))$ if $xy\in {E}^{\prime}$.

**Definition**

**8.**

## 3. Neutrosophic Cubic Graphs

**Definition**

**9.**

- (i)
- $\left({\tilde{T}}_{C}({u}_{i}{v}_{i})\u2aafrmin\{{\tilde{T}}_{A}({u}_{i}),{\tilde{T}}_{A}({v}_{i})\},{T}_{D}({u}_{i}{v}_{i})\le max\{{T}_{B}({u}_{i}),{T}_{B}({v}_{i})\}\right),$
- (ii)
- $\left({\tilde{I}}_{C}({u}_{i}{v}_{i})\u2aafrmin\{{\tilde{I}}_{A}({u}_{i}),{\tilde{I}}_{A}({v}_{i})\},{I}_{D}({u}_{i}{v}_{i})\le max\{{I}_{B}({u}_{i}),{I}_{B}({v}_{i})\}\right),$
- (iii)
- $\left({\tilde{F}}_{C}({u}_{i}{v}_{i})\u2aafrmax\{{\tilde{F}}_{A}({u}_{i}),{\tilde{F}}_{A}({v}_{i})\},{F}_{D}({u}_{i}{v}_{i})\le min\{{F}_{B}({u}_{i}),{F}_{B}({v}_{i})\}\right).$

**Example**

**1.**

**Remark**

**1.**

- 1.
- If $n\ge 3$ in the vertex set and $n\ge 3$ in the set of edges then the graphs is a neutrosophic cubic polygon only when we join each vertex to the corresponding vertex through an edge.
- 2.
- If we have infinite elements in the vertex set and by joining the each and every edge with each other we get a neutrosophic cubic curve.

**Definition**

**10.**

**Example**

**2.**

**Definition**

**11.**

- (i)
- $\left({\tilde{T}}_{{A}_{1}\times {A}_{2}}(x,y)=rmin({\tilde{T}}_{{A}_{1}}(x),{\tilde{T}}_{{A}_{2}}(y)),{T}_{{B}_{1}\times {B}_{2}}(x,y)=max({T}_{{B}_{1}}(x),{T}_{{B}_{2}}(y))\right),$
- (ii)
- $\left({\tilde{I}}_{{A}_{1}\times {A}_{2}}(x,y)=rmin({\tilde{I}}_{{A}_{1}}(x),{\tilde{I}}_{{A}_{2}}(y)),{I}_{{B}_{1}\times {B}_{2}}(x,y)=max({I}_{{B}_{1}}(x),{I}_{{B}_{2}}(y))\right),$
- (iii)
- $\left({\tilde{F}}_{{A}_{1}\times {A}_{2}}(x,y)=rmax({\tilde{F}}_{{A}_{1}}(x),{\tilde{F}}_{{A}_{2}}(y)),{F}_{{B}_{1}\times {B}_{2}}(x,y)=min({F}_{{B}_{1}}(x),{F}_{{B}_{2}}(y))\right),$
- (iv)
- $\left(\begin{array}{c}{\tilde{T}}_{{C}_{1}\times {C}_{2}}((x,{y}_{1})(x,{y}_{2}))=rmin({\tilde{T}}_{{A}_{1}}(x),{\tilde{T}}_{{C}_{2}}({y}_{1}{y}_{2})),\\ {T}_{{D}_{1}\times {D}_{2}}((x,{y}_{1})(x,{y}_{2}))=max({T}_{{B}_{1}}(x),{T}_{{D}_{2}}({y}_{1}{y}_{2}))\end{array}\right),$
- (v)
- $\left(\begin{array}{c}{\tilde{I}}_{{C}_{1}\times {C}_{2}}((x,{y}_{1})(x,{y}_{2}))=rmin({\tilde{I}}_{{A}_{1}}(x),{\tilde{I}}_{{C}_{2}}({y}_{1}{y}_{2})),\\ {I}_{{D}_{1}\times {D}_{2}}((x,{y}_{1})(x,{y}_{2}))=max({I}_{{B}_{1}}(x),{I}_{{D}_{2}}({y}_{1}{y}_{2}))\end{array}\right),$
- (vi)
- $\left(\begin{array}{c}{\tilde{F}}_{{C}_{1}\times {C}_{2}}((x,{y}_{1})(x,{y}_{2}))=rmax({\tilde{F}}_{{A}_{1}}(x),{\tilde{F}}_{{C}_{2}}({y}_{1}{y}_{2})),\\ {F}_{{D}_{1}\times {D}_{2}}((x,{y}_{1})(x,{y}_{2}))=min({F}_{{B}_{1}}(x),{F}_{{D}_{2}}({y}_{1}{y}_{2}))\end{array}\right),$
- (vii)
- $\left(\begin{array}{c}{\tilde{T}}_{{C}_{1}\times {C}_{2}}(({x}_{1},y)({x}_{2},y))=rmin({\tilde{T}}_{{C}_{1}}({x}_{1}{x}_{2}),{\tilde{T}}_{{A}_{2}}(y)),\\ {T}_{{D}_{1}\times {D}_{2}}(({x}_{1},y)({x}_{2},y))=max({T}_{{D}_{1}}({x}_{1}{x}_{2}),{T}_{{B}_{2}}(y))\end{array}\right),$
- (viii)
- $\left(\begin{array}{c}{\tilde{I}}_{{C}_{1}\times {C}_{2}}(({x}_{1},y)({x}_{2},y))=rmin({\tilde{I}}_{{C}_{1}}({x}_{1}{x}_{2}),{\tilde{I}}_{{A}_{2}}(y)),\\ {I}_{{D}_{1}\times {D}_{2}}(({x}_{1},y)({x}_{2},y))=max({I}_{{D}_{1}}({x}_{1}{x}_{2}),{I}_{{B}_{2}}(y))\end{array}\right),$
- (ix)
- $\left(\begin{array}{c}{\tilde{F}}_{{C}_{1}\times {C}_{2}}(({x}_{1},y)({x}_{2},y))=rmax({\tilde{F}}_{{C}_{1}}({x}_{1}{x}_{2}),{\tilde{F}}_{{A}_{2}}(y)),\\ {F}_{{D}_{1}\times {D}_{2}}(({x}_{1},y)({x}_{2},y))=min({F}_{{D}_{1}}({x}_{1}{x}_{2}),{F}_{{B}_{2}}(y))\end{array}\right),$∀$(x,y)\in ({V}_{1},{V}_{2})=V$ for $(i)-(iii),\forall x\in {V}_{1}$ and ${y}_{1}{y}_{2}\in {E}_{2}$ for $(iv)-(vi),\forall y\in {V}_{2}$ and ${x}_{1}{x}_{2}\in {E}_{1}$ for $(vi)-(ix)$.

**Example**

**3.**

**Proposition**

**1.**

**Proof.**

**Definition**

**12.**

**Example**

**4.**

**Definition**

**13.**

- (i)
- $\forall (x,y)\in ({V}_{1},{V}_{2})=V,$

- (ii)
- $\forall x\in {V}_{1}$ and ${y}_{1}{y}_{2}\in E$

- (iii)
- $\forall y\in {V}_{2}$ and ${x}_{1}{x}_{2}\in {E}_{1}$

- (iv)
- $\forall ({x}_{1},{y}_{1})({x}_{2},{y}_{2})\in {E}^{0}-E$

**Example**

**5.**

**Proposition**

**2.**

**Definition**

**14.**

**Example**

**6.**

**Proposition**

**3.**

**Remark**

**2.**

**Definition**

**15.**