(untitled) (Journal of Mathematical Modeling and Fractional Calculus, 2026)
Abstract: Graph theory is like the backbone of our understanding of complex networks, whether they be social, computational, or biological. In this study, we dive into the fascinating world of the Tickysim Spiking Neural Network (TSNN) to explore its metric, edge metric, and fault-tolerant metric dimensions. We aim To unravel its structure’s intricacies and uncover its potential applications. At its core, the metric dimension tells us how many vertices we need to pinpoint the locations of all other vertices based solely on distance measurements. Similarly, the edge metric dimension reveals the minimal number of edges necessary to achieve the same goal. To add another layer of reliability, the fault-tolerant metric dimension ensures that the network can still be recognized even when some vertices or edges are out of action. Throughout our research, we discovered unique symmetries and structural features of TSNN, which enabled us to identify resolving sets for both vertex- and edge-based metrics. What’s exciting is that these resolving sets work consistently, regardless of how we label the network, allowing for reliable identification even in complex scenarios. Notably, we found that the fault-tolerant metric dimension of TSNN is 3, while its metric dimension stands at 2, highlighting the network’s impressive adaptability and resilience. By exploring these dimensions, our work sheds new light on the reliability and flexibility of TSNN, emphasizing its potential for groundbreaking advancements in areas like computational neuroscience and neural network modeling. We believe these insights not only enrich the theoretical landscape of graph theory but also pave the way for innovative applications in fields that thrive on robust and intricate network designs.