# Picture Fuzzy Threshold Graphs with Application in Medicine Replenishment

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## Abstract

**:**

## 1. Introduction

- (1)
- The TG is an advance studied topic and it was discussed in several fields except in PF field.
- (2)
- The crisp TG, FTG and intuitionistic FTG models do not recognize all real life systems having an indeterminate information.
- (3)
- The PFTG models are more helpful to control the flow of information than the other existing models. But, till now it was not formulate in PF environment.

- (1)
- Is it possible to formulate a new TG model can help to solve resources allocation problems in PF surroundings?
- (2)
- Is it possible to handle the systems having an indeterminate information by using PFTG models?

- (1)
- To extend the concept of FTG and intuitionistic FTG to PFTG to model systems having an indeterminate information.
- (2)
- To fill the research gap, we propose TG models under PF environment.
- (3)
- To compare the obtained results derived from the proposed model with the existing models.
- (4)
- To control medicine resources using the proposed models.

- (1)
- We formulate PFTG along with its many interesting properties and then initiate the notions of picture fuzzy alternating 4-cycle (PFA4-C), TD and PN of PFGs.
- (2)
- We introduce the relation of threshold values (TV) and vertex cardinality of a PFTG. We can decompose a PFTG in a unique way and it generates 3 distinct FTGs.
- (3)
- We establish a relation between picture fuzzy TD and PN, and illustrates many important properties on decomposed FTG.
- (4)
- We present the comparison of proposed TGs with the existing TGs.
- (5)
- Finally, an application of PFTG is present in medicine replenishment problem. In this way, the research gap mentioned above can be filled.

- (1)
- It is capable to recognize all real life systems having an indeterminate information.
- (2)
- It is more efficient and effective than in other field.
- (3)
- It is based to control resources rather than in other existing TG models.

## 2. Related Works

## 3. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Example**

**1.**

**Definition**

**6.**

## 4. Picture Fuzzy Threshold Graph

**Definition**

**7.**

**Example**

**2.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

- (1)
- a PF square ${C}_{4}$ graph if $((p,s),{\mu}_{B},{\eta}_{B},{\nu}_{B})\ne (0,0,0)$ and $((q,r),{\mu}_{B},{\eta}_{B},{\nu}_{B})\ne (0,0,0)$.
- (2)
- a PF path ${P}_{4}$ graph if $((p,s),{\mu}_{B},{\eta}_{B},{\nu}_{B})\ne (0,0,0)$ and $((q,r),{\mu}_{B},{\eta}_{B},{\nu}_{B})=(0,0,0)$;or, $((p,s),{\mu}_{B},{\eta}_{B},{\nu}_{B})=(0,0,0)$ and $((q,r),{\mu}_{B},{\eta}_{B},{\nu}_{B})\ne (0,0,0)$.
- (3)
- a PF matching $2{K}_{2}$ graph if $((p,s),{\mu}_{B},{\eta}_{B},{\nu}_{B})=(0,0,0)$ and $((q,r),{\mu}_{B},{\eta}_{B},{\nu}_{B})=(0,0,0)$.

**Example**

**3.**

**Definition**

**12.**

**Definition**

**13.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Definition**

**14.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Example**

**4.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Definition**

**15.**

**Example**

**5.**

**Theorem**

**11.**

**Proof.**

**Definition**

**16.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Theorem**

**14.**

**Example**

**6.**

**Theorem**

**15.**

**Proof.**

**Example**

**7.**

**Definition**

**17.**

**Theorem**

**16.**

**Proof.**

**Theorem**

**17.**

**Proof.**

## 5. An Application of Picture Fuzzy Threshold Graphs in Medicine Replenishment Problem

#### 5.1. Model Construction

#### 5.2. Decision Making

## 6. Comparative Study with Existing Methods

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GT | Graph theory |

IF | Intuitionistic fuzzy |

PF | Picture fuzzy |

PFS | Picture fuzzy set |

PFG | Picture fuzzy graph |

PFA4-C | Picture fuzzy alternating 4-cycle |

FG | Fuzzy graph |

SS | Stable set |

ES | Edges set |

TG | Threshold graph |

SG | Split graph |

TV | Threshold value |

TD | Threshold dimension |

PN | Partition number |

MV | Membership value |

UCG | Underlying crisp graph |

FTG | Fuzzy threshold graph |

PFTG | Picture fuzzy threshold graph |

PFSG | Picture fuzzy split graph |

PFTSG | Picture fuzzy threshold-subgraph |

TMS, AMS and FMS | Truth, abstinence and false membership value, respectively. |

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**MDPI and ACS Style**

Das, S.; Ghorai, G.; Xin, Q.
Picture Fuzzy Threshold Graphs with Application in Medicine Replenishment. *Entropy* **2022**, *24*, 658.
https://doi.org/10.3390/e24050658

**AMA Style**

Das S, Ghorai G, Xin Q.
Picture Fuzzy Threshold Graphs with Application in Medicine Replenishment. *Entropy*. 2022; 24(5):658.
https://doi.org/10.3390/e24050658

**Chicago/Turabian Style**

Das, Sankar, Ganesh Ghorai, and Qin Xin.
2022. "Picture Fuzzy Threshold Graphs with Application in Medicine Replenishment" *Entropy* 24, no. 5: 658.
https://doi.org/10.3390/e24050658