Picture Fuzzy Threshold Graphs with Application in Medicine Replenishment
Sankar Das1,2, Ganesh Ghorai1, Qin Xin3
1Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India.
This study introduces picture fuzzy threshold graphs (PFTGs), demonstrating their unique properties and relationship to split graphs. These findings offer novel applications in medicine resource control.
Area of Science:
- Graph Theory
- Fuzzy Mathematics
Background:
- Picture fuzzy graphs (PFGs) are an extension of fuzzy graphs.
- Understanding structural properties of PFGs is crucial for applications.
Purpose of the Study:
- Introduce and define the novel concept of picture fuzzy threshold graphs (PFTGs).
- Explore fundamental properties of PFTGs and their relationship with existing graph structures.
- Define and analyze new parameters: picture fuzzy threshold dimension (TD) and PF partition number (PN).
Main Methods:
- Introduce PFTGs by adding dominating or isolated nodes.
- Analyze the underlying crisp graph (UCG) of PFTGs, identifying it as a split graph (SG).
- Investigate the unique decomposition of PFTGs into fuzzy threshold graphs (FTGs).
Main Results:
- PFTGs are free from alternating 4-cycles.
- Every PFG is equivalent to a PFTG under specific conditions.
- The UCG of a PFTG is an SG, and vice versa.
- PFTGs uniquely decompose into three distinct FTGs.
- Properties of PF TD and PF PN are established.
Conclusions:
- The study establishes PFTGs as a significant structure within fuzzy graph theory.
- The defined parameters (PF TD and PF PN) offer new analytical tools for PFGs.
- The research provides a framework for applying these graph concepts to medicine resource control.
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