A Graph-Theoretic Model for Battery Monitoring Systems Using Weighted Friendship Graphs and Harary Index Evaluation

¹ Assistant Professor, Department of Mathematics, Sri ChandrasekharendraSaraswathiViswaMahavidyalaya
² Assistant Professor, Department of Electrical and Electronics Engineering, Sri ChandrasekharendraSaraswathiViswaMahavidyalaya

Published online: 16 April 2026

Abstract

This study presents a graph-theoretic approach to model the communication architecture of a Battery Monitoring System (BMS) using an undirected, edge-weighted Friendship Graph. In this representation, each vertex corresponds to a battery module or sensor, while a central vertex symbolizes the controller system that coordinates overall communication. The edges denote communication links, and their associated weights capture two critical parameters, they are the physical distance between nodes and the effective distance determined by the infrared (IR) sensor’s radiation range. To assess the efficiency of communication within this architecture, the Harary Index is adopted and extended to incorporate these edge weights. The Harary Index, a well-known topological invariant in graph theory, reflects the closeness of vertices and thus serves as a measure of communication robustness and fault tolerance. By calculating the Harary Index for both physical distances and radiation distances, the study derives an average index that quantifies the monitoring efficiency of the system. This combined measure aids in predicting faults, improving communication reliability, and minimizing the risk of damage in the BMS. The findings emphasize the determination of an optimal placement strategy for sensors and the controller unit, thereby offering practical guidelines for designing efficient BMS communication networks.