Introduction:
This presentation offers a structured exploration of graph theory and its critical role in advancing the field of network science. We begin with formal mathematical definitions of graphs and progressively introduce foundational metrics—degree, centrality, clustering—and their relevance to biological, chemical, and engineered systems.
The session emphasizes computational applications, from protein-protein interaction networks and brain connectivity mapping to the quantification of phase transitions in complex fluids. We also explore recent advancements in node-based multifractal analysis (NMFA) to characterize self-similarity and heterogeneity within large-scale networks.
Through case studies ranging from gene regulatory architectures to AI-driven neuronal interaction networks, we illustrate how graph-theoretic tools serve not only as abstract mathematical models but also as practical descriptors of structure, function, and emergent behavior in complex systems.