(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Signed Fuzzy Graphs (SFGs), a mathematical structure having nodes associated with membership values and edges consisting of positive and negative signs, give a multi-faceted mathematical model to deal with uncertainty and polarity in graph-based networks. Exploring the theories of SFG and its applications helps to understand the complex networks and their characteristics. This study intends to examine the key theorems pertaining to vertex and edge degrees and their membership values by highlighting their real-time applications in Wireless Sensor Networks (WSNs) and Environmental Monitoring Systems (EMSs), where sensor nodes must maintain reliable communication. The main focus of this paper shifts from the exploration of theorems about vertex and edge degree and their membership values to the introduction of new domination metrics tailored for SFGs, the domination sets and minimum domination sets and their relevance to efficient placement and the operation of WSNs in EMSs. SFGs facilitate effective data collection by ensuring network coverage. This study makes a significant contribution to the field of Fuzzy Mathematics with a comprehensive understanding of complex networks and their characteristics.