Structural Cryptanalysis of a Graph-Based Adjacency-Matrix Transformation Derived from the Middle Graph of a Star Graph

This paper provides a structural cryptanalysis of a graph-based matrix transformation obtained from the normalized adjacency matrix of the middle graph of a star. The construction is specified with canonical indexed vertices and edges, exact rational arithmetic, a deterministic auxiliary sequence, and an invertible integer key matrix. For every valid block of length n, the graph matrix has rank 2n rather than 2n+1. This invariant row dependence survives right multiplication by the secret matrix: a single ciphertext reveals all auxiliary values yi and n complete rows of the key. If the same key matrix is reused for two known plaintext/ciphertext pairs whose recovered y-vectors differ, the complete matrix key is uniquely recovered from a stacked linear system. A seeded numerical experiment confirms this recovery over multiple block lengths. We further show that fuzzy normalization is only public reversible scaling and therefore adds no cryptographic hiding. The results establish the construction as a reproducible negative case study in graph-based cryptography rather than a stand-alone secure cipher.