This chapter introduces quantum planar graphs, which build on classical graph theory by adding key quantum features like superposition and entanglement. The focus is on how these quantum graphs have dynamic vertices and possibly entangled edges. By combining entangled edges and changing weights, quantum graphs become a better tool for studying quantum behavior. The study also explains the how quantum graphs can be used is in quantum planar graphs. Through theoretical analysis and practical application examples, this chapter demonstrates how quantum planar graphs can effectively identify critical nodes, manage data flow, and reduce congestion. Our findings reveal that this quantum-based approach has significant potential to enhance network resilience, optimize resource utilization, and support real-time adaptability in fast-evolving communication environments. These insights highlight the promise of quantum planar graphs as a next-generation framework for advanced communication network design and management.