Advanced higher-order degree irregularity measures in Pythagorean fuzzy graph under threshold conditions
*R. ShahinCorresponding authorshahinshariff2015@gmail.comDepartment of MathematicsJamal Mohamed College (Autonomous)Affiliated to Bharathidasan UniversityTiruchirappalli, Tamil Nadu, 620020, IndiaView full profile → , M. Mohammed Jabarullam.md.jabarulla@gmail.comDepartment of MathematicsJamal Mohamed College (Autonomous)Affiliated to Bharathidasan UniversityTiruchirappalli, Tamil Nadu, 620020, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Jan 2026
- Published Online:
- 29 Aug 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2623
- Pages:
- 1–15
Abstract
This paper introduces advanced irregularity measures for Pythagorean fuzzy graphs under threshold conditions. Two types of indices are proposed. The first type focuses on edge-based irregularity, including a higher-order edge degree irregularity index and a correlation-based index that captures relationships between the degrees of adjacent vertices. The second type focuses on degree-based irregularity, introducing an entropy-based index that measures uncertainty in the degree distribution. For each measure, we establish fundamental properties, monotonicity behavior under increasing thresholds, and sharp upper and lower bounds. Extremal cases are also characterized to identify graphs that attain equality in these bounds. Numerical examples are provided to illustrate the validity and effectiveness of the proposed measures. The results offer a refined perspective on degree irregularity and extend the theoretical framework of Pythagorean fuzzy graphs under threshold conditions.
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References
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