Quantum computing endangers classical cryptographic systems by efficiently solving integer factorization and discrete logarithm problems foundational to secure communications. This survey explores how graph theory provides quantum-resistant primitives—such as graph coloring, Hamiltonian paths, and isomorphism—absent efficient quantum algorithms, enabling streamlined constructions in isogeny, multivariate, lattice, and code-based post-quantum schemes. Graph structures further bolster network security through attack/trust graphs and graph neural networks for threat detection, while quantum graph-theoretic encryption emerges as a promising paradigm fusing quantum states with graph topologies. Addressing persistent challenges in complexity, scalability, and integration, this work illuminates critical research directions at the graph theory–post-quantum cryptography intersection, essential for resilient digital infrastructures.