Optimal Network Analysis through Vertex Order Coloring of Intuitionistic Fuzzy Graph Operations

Intuitionistic fuzzy graphs IFGs are a powerful tool for modeling uncertainty and complex relationships. They offer versatile frameworks for addressing real-world challenges. In this research, we have introduced intuitionistic fuzzy vertex order coloring IFVOC and analyzed the alpha-strong (α˜str), beta-strong (β˜str), and gamma-strong (γ˜str) vertices through their degree. We explored important theorems based on the types of strong vertices, broadening the scope of our study. We analyzed multiple IFG products to determine the most optimal network based on some important metrics, including the weight and total number of α˜str vertices, the chromatic number, and the weight of the graph’s minimum spanning tree. We systematically evaluate different types of IFG products, such as conormal, modular, residue, and maximal, and consider their implications for fuzzy graph structure and connectivity. We have investigated new metrics to measure the presence and importance of α˜str vertices in the graph, assessing both their frequency and overall impact. Additionally, we have investigated the impact of product operations on the significance of α˜str nodes and the resultant minimum spanning tree, providing insights into the overall robustness and efficiency of each product. We have explored how variations in product operations impact the distribution of α˜str vertices and the characteristics of the minimum spanning tree and chromatic number, elucidating the trade-offs between different product options. We have modeled the product graph as a network to represent complex systems composed of interconnected entities. The outcomes of this research contribute to the advancement of IFG theory and its use in various fields, such as network design, strategic planning, and system analysis.