The zero-divisor graph of a ring is built from its algebraic pathology. Vertices are the zero-divisors — nonzero elements whose product with some other nonzero element is zero. Edges connect pairs whose product is zero. The graph encodes the multiplicative failure of the ring: where multiplication collapses to zero.
For quaternion rings — rings built from the quaternion algebra over a base ring — the zero-divisor graph has a block structure that pins down its eigenvalues (arXiv: 2603.20947). The adjacency matrix decomposes into blocks corresponding to algebraically meaningful subsets of the ring. Each block contributes a calculable piece of the spectrum.
The spectrum of a graph — the set of eigenvalues of its adjacency matrix — captures global structural information. Eigenvalues determine walks, connectivity, expansion properties, and mixing times. Knowing the spectrum of the zero-divisor graph means knowing the structural geometry of the ring's multiplicative failures.
The through-claim: algebra determines geometry through the spectrum. The quaternion ring's algebraic structure (which elements multiply to zero) creates a graph, and the graph's spectral structure (eigenvalues of the adjacency matrix) is completely determined by the algebra. The spectrum is a geometric object derived from an algebraic one, and the block structure provides the dictionary. You can read the ring's pathology in the graph's frequencies.
2603.20947. Algebra / graph theory / zero-divisor graphs / quaternion rings / spectral graph theory.