On orthogonality graphs of Okubo algebras
Danil Pavlinov, Svetlana Zhilina
TL;DR
This work analyzes the orthogonality graphs of Okubo algebras with isotropic norm over arbitrary fields. It leverages the O_{α,β} framework and the pseudo-octonion model to classify zero divisors, idempotents, and the resulting graph components, deriving exact diameters and shortest-path behavior. Key findings show the orthogonality graph is typically geodetic or bigeodetic, with diameters depending on field characteristics, the presence of primitive cube roots of unity, and whether the algebra is split; in the split-char≠3, ω∈𝔽 case there are up to two shortest paths for some pairs. The paper also connects these graphs to matrix rings when ω∈𝔽, providing alternative proofs and linking to known results on nilpotent matrices, and includes a computational appendix validating the constructions.
Abstract
The orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field $\mathbb{F}$ is considered. Its connected components are described, and their diameters are computed. It is shown that there exist at most two shortest paths between any pair of vertices, and the conditions under which the shortest path is unique are determined.