Exploring Symmetry Structures in Integrity-Based Vulnerability Analysis Using Bipolar Fuzzy Graph Theory

The integrity parameter in vulnerability refers to a set of removed vertices and the maximum number of connected components that remain functional. A bipolar fuzzy graph (BFG) assigns membership values to both positive and negative attributes. A new parameter, integrity, is defined and discussed using an example of a BFG. The integrity value of a special type of graph is determined, and the node strength sequence (NSS) for BFG is introduced. Specific NSS values are used to discuss the integrity values of paths and cycles. The integrity of the union, join, and Cartesian product of two BFGs is presented. This parameter is then applied to a road network with both positive and negative attributes, and the findings are discussed with a conclusion.