On orthogonality graphs of Okubo algebras

TL;DR

This paper studies the structure of orthogonality graphs derived from Okubo algebras over arbitrary fields, detailing their connected components, diameters, and shortest path properties.

Contribution

It characterizes the connected components and diameters of orthogonality graphs of Okubo algebras, and analyzes shortest path uniqueness conditions.

Findings

Connected components of the orthogonality graph are described.

Diameters of these components are computed.

Conditions for the uniqueness of shortest paths are established.

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.