A Spectral Adjacency Matrix Based Cryptographic Framework via Structural Graph Invariants

Abstract

This paper introduces a graph-theoretic cryptographic framework derived from structural and spectral properties of simple connected graphs. Unlike conventional graph-based encryption techniques that depend on labeling or coloring mechanisms, the proposed approach utilizes adjacency matrices and their associated spectral invariants to construct encryption transformations. A parametric graph family is employed to generate transformation matrices whose invertibility governs deterministic decryption. The encryption process is realized through modular matrix-vector multiplication, while security strength is analysed using spectral radius, determinant conditions, and combinatorial growth of graph structures. The model is entirely algebraic in nature and does not rely on classical symmetric encryption standards. Theoretical results concerning invertibility, structural uniqueness, computational complexity, and key space expansion are established. The framework provides a mathematically rigorous alternative for graph-based cryptographic design.