Abstract
Fuzzy graph theory, with its ability to handle uncertainty and varying relationship strengths, offers a powerful tool for modeling and solving complex problems across diverse fields like medical, social network, biological networks, etc. The topological index (TI) is the most useful tool in this field. It characterizes and predicts the properties of chemical compounds and other network systems. One of the degree-based TI is the Inverse Sum Indeg Index (ISI). This index is utilized in a number of ways in crisp graphs. So in this paper, ISI for fuzzy graph (FG) is defined. In this paper, we establish several key relationships between the ISI values of two isomorphic fuzzy graphs, as well as between a FG and its subgraphs, and a connected fuzzy graph and its corresponding spanning tree. Additionally, we derive upper bounds for the ISI values of various well-known fuzzy graph structures, including the fuzzy star, broom graph, complete fuzzy graph, fuzzy cycle, and complete bipartite graph. To demonstrate the practical significance of our findings, we present a real-world application in the medical field, specifically in the treatment of cancer patients in a hospital setting. By analyzing patient conditions through a fuzzy graph framework, we provide insights into personalized treatment strategies, offering a systematic and data-driven approach to patient care. Our results suggest a novel method for prioritizing and responding to cancer patients based on their health parameters and treatment needs. Furthermore, we introduce a comprehensive flowchart that outlines an advanced treatment planning strategy, serving as a decision-support tool for health-care professionals. This structured approach enhances treatment efficiency and precision, potentially leading to improved patient outcomes.