Super Mean Graph Labeling: A Novel Cryptographic Framework Using Five-Star Graphs with Applications in Secure Communications and Epidemic Modeling

Abstract

This paper introduces an innovative cryptographic framework based on super mean labeling of five-star graphs K1,V1 ∪ K1,V2 ∪ K1,V3 ∪ K1,V4 ∪ K1,V5, where V1 ≤ V2 ≤ V3 ≤ V4 ≤ V5. We present a comprehensive mathematical foundation for super mean labeling and develop systematic methodologies for encoding messages through carefully constructed graph structures. Three distinct implementation approaches are demonstrated, incorporating computational techniques through C programming for alphabetical mapping, including subtraction-based and division-based numbering schemes. Each approach is thoroughly illustrated with complete message encoding examples, visual cryptography representations, and detailed security analysis. Beyond cryptographic applications, we extend the framework to epidemic modeling, demonstrating how super mean labeling can represent complex disease transmission dynamics in multi-population systems. The epidemic modeling application includes complete mathematical formulations of multi-population SIR models, transmission matrix encoding through graph labeling, and computational implementations for disease surveillance. Our results establish that five-star graphs with super mean labeling provide optimal balance between encoding capacity and structural complexity, offering robust security through multiple layers of mathematical obfuscation. The integration of graph theory with computational algorithms creates a versatile framework applicable to both secure communications and public health informatics.