A graph built from selected zero divisors can capture enough structure to classify a higher-dimensional family of real Cayley-Dickson algebras, according to a mathematical preprint. For algebras of at least 16 dimensions, the study's abstract states that two algebras are isomorphic, meaning they have the same structure in the study's terms, if and only if their graphs are isomorphic.
The result concerns a restricted graph, not the entire orthogonality graph. Its vertices are lines through selected pure zero divisors whose two components are standard basis elements. Two distinct vertices are joined when multiplication in both orders gives zero, written as , where a and b are algebra elements.
Turning algebra into a graph
The paper constructs this restricted graph inductively for an arbitrary real Cayley-Dickson algebra. It also provides an algorithm for retrieving the algebra's Cayley-Dickson parameters from the graph.
Those parameters are not all recovered in the same way. For n at least 3, the unlabeled graph determines every parameter from the third through the nth. The first three parameters are determined only up to isomorphism of the algebra called A3.
For n at least 4, the unlabeled graph also determines two auxiliary graphs defined at the preceding level. Those graphs support a recursive recovery of earlier parameters.
A distinctive pattern of components
The graph develops a regular component pattern from n at least 3. The study reports two possible component counts, with one exactly one fewer than the other, according to whether the final parameter is 1 or -1. Every component has diameter three, so no two vertices in the same component are more than three graph links apart.
Component sizes also have two reported alternatives, depending on a quantity the paper denotes as chi of C. The supplied analysis flags uncertainty in the extracted superscript formatting of those size expressions.
What the graph can and cannot identify
The study states both directions of the classification result for n at least 3. Isomorphic analyzed graphs imply isomorphic parameterized Cayley-Dickson algebras, and isomorphic parameterized algebras imply isomorphic analyzed graphs. Together, those statements provide the claimed equivalence in that range.
The lowest indices are less decisive. For n no greater than 2, the converse from algebra isomorphism to graph isomorphism is not generally true for this restricted graph: the paper gives isomorphic algebras whose graphs are not isomorphic. When both index values are between 1 and 3, however, isomorphic nonempty analyzed graphs still imply isomorphic associated algebras.
For n between 1 and 3, the component forms listed by the study include empty sets, disconnected vertices, hexagons, double hexagons, bundles of hexagons, and an almost complete bipartite graph with seven vertices on each side. The form depends on chi of C and on the parameter pattern.
This is a formal methods result, not an empirical study: the objects are algebraic structures and graph components, and no statistical analysis is reported. The parameter-recovery procedure is described formally, but the supplied review reports no implementation, runtime, or computational validation. The conclusions apply to the selected pure-zero-divisor subgraph and do not establish that the full orthogonality graph has the same classification property.
The document is identified in the supplied metadata as preprint arXiv:2608.28163v1, version 1, dated 28 Aug 2026. The work was supported by the Russian Science Foundation under project No. 17-11-01124.
Paper data and sources
Original title: Orthogonality graphs of real Cayley-Dickson algebras. Part II: The subgraph on pairs of basis elements
Authors: Svetlana Zhilina
Journal/Repository: S. Zhilina, Orthogonality graphs of real Cayley-Dickson algebras. Part II: The subgraph on pairs of basis elements, Int. J. Algebra Comput. 31(4) (2021) 691-725
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: 10.1142/s0218196721500338
Original paper · Full text