This study investigates the properties and applications of Eulerian and Hamiltonian graphs within complex networks, with a particular focus on their roles in transportation, communication, and biological systems. The primary objective is to develop a deeper mathematical understanding of these graph structures and their implications for optimizing network performance. Key methodologies employed include the implementation and analysis of Hierholzer’s and Fleury’s algorithms for detecting Eulerian circuits, as well as dynamic programming, backtracking, and approximation algorithms for Hamiltonian cycles. Our results reveal significant enhancements in network optimization, showcasing improvements in traversal strategies and connectivity while effectively minimizing operational costs. Furthermore, the study identifies computational challenges associated with large-scale networks and proposes heuristic methods to address these issues. The findings provide valuable insights and recommendations for designing more efficient and resilient network infrastructures, demonstrating the practical applicability of graph theory in solving real-world problems in various domains.