abstract

Graph theory and topological data analysis provide complementary mathematical approaches for studying the structural organization of complex data. Graph theory represents relationships through vertices and edges, whereas topological data analysis extends this representation to higher-dimensional structures capable of identifying connected components, cycles, cavities, and persistent geometric patterns. The present paper examines the integration of graph-theoretic methods with persistent homology for multiscale topological data analysis. Particular attention is given to graph construction, simplicial complexes, Vietoris–Rips complexes, filtrations, homology groups, Betti numbers, persistence diagrams, barcodes, and the birth and death of topological features. The paper adopts a theoretical, descriptive, and analytical methodology based on mathematical concepts and developments reported up to 2025. It explains how graph structures form the one-dimensional foundation of simplicial complexes and how persistent homology extends conventional graph connectivity by investigating topological features across multiple spatial or functional scales. Applications in image analysis, biological networks, medical imaging, brain connectivity, point-cloud analysis, and machine learning are discussed. The analysis indicates that persistent homology provides a robust framework for distinguishing significant structural patterns from short-lived fluctuations or noise. The study concludes that combining graph theory with persistent homology provides an effective mathematical framework for multiscale analysis of complex data and offers substantial potential for computational topology, artificial intelligence, and scientific data analysis.