Fuzzy node rupture degree of some fuzzy graph families

Department of Mathematics, Manisa Celal Bayar University, Sehit Prof. Dr. Ilhan Varank Campus, 45140 Manisa, Turkey

^* Corresponding author: goksen.turan@cbu.edu.tr; goksenbacak@gmail.com

Received: 14 August 2024
Accepted: 6 March 2025

Abstract

In the event of the malfunction of nodes and/or links connecting nodes, the vulnerability parameters defined in graph theory may be employed as a metric of the quality of service received via a network. In graph theory, numerous different vulnerability parameters have been defined, including toughness, rupture degree, tenacity, integrity, connectivity and others. The modelling of real-world problems is made more effective by the use of fuzzy graphs as a specific type of graph. These issues can be more accurately addressed through the use of membership values as they better represent the inherent uncertainties involved. However, despite this advantage, there has been limited research conducted on the vulnerability parameters in fuzzy graphs. In this paper, we consider the importance of fuzzy graphs and the limitations of the existing literature on vulnerability parameters for these graphs. We propose a definition of the fuzzy node rupture degree, a commonly used concept in graph theory, for application in fuzzy graph theory. Furthermore, this study examines the theoretical aspects of fuzzy wheel graphs, fuzzy cycle graphs and fuzzy star graphs, with the objective of providing general formulas that can be applied in practice, and also an algorithm for calculating the fuzzy node rupture degree of a given fuzzy graph is presented.