<![CDATA[Graph theory as a branch of discrete mathematics has experienced significant development in its application to modern complex network systems, particularly in digital social networks and transportation systems. This research aims to analyze fundamental concepts of graph theory, examine characteristics of cycle detection algorithms along with their computational complexity, investigate their application in digital social network analysis, and explore their implementation in digital transportation system optimization. The research method employs a qualitative approach with library research focusing on scientific literature from 2020-2025 period from accredited academic databases such as Scopus, Web of Science, and IEEE Xplore, utilizing thematic analysis techniques to identify meaningful patterns from the examined literature. Research findings indicate that fundamental graph theory concepts including vertices, edges, and graph classifications form the foundation for relational structure modeling. Cycle detection algorithms such as Depth-First Search, Union-Find, and Tarjan demonstrate effectiveness with O(V+E) complexity for large-scale graphs. Applications in digital social networks facilitate community identification through Multi-View Clustering, centrality analysis for influencer detection, and understanding viral information dissemination patterns. Implementation in digital transportation systems demonstrates route planning optimization using Dijkstra and Bellman-Ford algorithms, vulnerability analysis through articulation point and bridge identification, and bottleneck detection with betweenness centrality. The research concludes that integration of graph theory in discrete mathematics education enhances critical thinking skills and real-world application understanding, with recommendations for algorithm development for massive dynamic graphs and machine learning integration in graph algorithm optimization.]]>
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