L (2,1) Labeling On Graph

Abstract

L(2,1)-labeling is a specialized graph labeling technique that assigns integer values to vertices while ensuring strict separation constraints to optimize network performance, resource allocation, and interference reduction. This study explores the fundamental concepts, definitions, and theoretical foundations of L(2,1)-labeling, along with its applications in cloud computing, wireless communication, social networks, and distributed systems. A comprehensive literature review highlights existing research contributions, while identified gaps point toward the need for scalability, algorithmic efficiency, and real-world implementation. The study also discusses relevant theorems and optimization strategies that contribute to enhancing labeling performance in various graph structures. Future research directions suggest integrating L(2,1)-labeling with emerging technologies such as AI-driven networks and blockchain-based systems for improved computational efficiency.