Connectivity Concepts in Intuitionistic Fuzzy Incidence Graphs with Application

Area of Science:

• Graph Theory
• Fuzzy Mathematics
• Decision Making

Background:

• Fuzzy incidence graphs (FIGs) lack non-membership values, limiting their applicability.
• Intuitionistic fuzzy graphs (IFGs) address this by including both membership and non-membership values.
• A need exists for a graph structure that combines incidence concepts with intuitionistic fuzzy sets.

Purpose of the Study:

• To introduce and define intuitionistic fuzzy incidence graphs (IFIGs) as a generalization of FIGs.
• To explore connectivity concepts within IFIGs, including intuitionistic fuzzy cut-vertices and bridges.
• To develop the concepts of intuitionistic incidence gain and loss for paths and vertex pairs.

Main Methods:

• Formal definition of IFIGs and their properties.
• Extension of existing graph connectivity concepts (cut-vertices, bridges) to the intuitionistic fuzzy incidence context.
• Introduction of new metrics: intuitionistic incidence gain and loss.

Main Results:

• IFIGs are established as a more comprehensive structure than FIGs due to the inclusion of non-membership values.
• Novel concepts analogous to intuitionistic fuzzy cut-vertices, intuitionistic incidence cut-vertices, intuitionistic fuzzy bridges, and intuitionistic incidence bridges are defined.
• The notions of intuitionistic incidence gain and loss are introduced for enhanced path analysis.

Conclusions:

• IFIGs offer a more reliable and valuable framework compared to FIGs for complex modeling.
• The developed concepts are essential for analyzing connectivity and relationships in systems where uncertainty is represented by both membership and non-membership.
• IFIGs provide a practical tool for applications, such as investment selection, that require non-membership information.