On orthogonality graphs of Okubo algebras

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.

💡 Research Summary

The paper investigates the orthogonality graph Γ_O of an Okubo algebra O equipped with an isotropic norm over an arbitrary field F. The authors first recall the general framework of relation graphs associated with algebras: the directed zero‑divisor graph Γ_Z, whose vertices are the one‑dimensional subspaces generated by non‑zero zero‑divisors, and the undirected orthogonality graph Γ_O, whose vertices are the same but two vertices