Resolving Parameters in Generalized Sierpi ́nski3 Networks Over Cycle of Length Five

The fascinating family of fractal-based networks known as generalized Sierpi\'nski networks has garnered significant interest across various research domains and practical applications. These graphs exhibit self-similar structures, making them particularly valuable in areas such as antenna structures, interconnection networks and the porous materials. Their recursive nature and hierarchical organization enhance their relevance in modeling complex systems and network structures. A key parameter for analyzing these graphs is the metric dimension, which represents the smallest set of reference vertices (or landmarks) needed to uniquely identify the distances between all other vertices in the graph. This parameter is crucial for network localization, efficient routing, and information retrieval. Beyond the standard metric dimension, other variations play important roles in different applications. The fault-tolerant metric dimension is essential in robust network design, ensuring that localization remains possible even if certain reference points fail. The edge metric dimension is widely used in network security and surveillance, where monitoring specific connections is more relevant than individual nodes. Meanwhile, the fault-tolerant edge metric dimension has applications in resilient communication networks, guaranteeing reliable identification of edges even under failure conditions. In this study, we specifically examine the metric, fault-tolerant metric, edge metric, and fault-tolerant edge metric dimensions of generalized Sierpi\'nski networks over \ (C₅ \). These findings provide deeper insights into their structural properties and distinguish them from traditional cycle networks, highlighting their potential in real-world applications.