Abstract:
Commuting graphs of Okubo algebras are considered, and the problem of their connectivity is studied. The commuting graph of a pseudo-octonion algebra $P_8(\mathbb{F})$ over a field $F$, $\mathrm{char}\, \mathbb{F} \neq 3$, that contains a primitive cubic root of unity is shown to be isomorphic to the commuting graph of the matrix algebra $M_3(\mathbb{F})$. As a consequence, if the field $\mathbb{F}$ is algebraically closed, then the diameter of the commuting graph for the unique Okubo algebra over $\mathbb{F}$ equals $4$. It is shown that the commuting graph of the real division Okubo algebra is connected and its diameter also equals $4$. The proof of this result relies on the fact that, given any two idempotents in an arbitrary Okubo algebra, the intersection of their centralizers is always nonzero.
Key words and phrases:Okubo algebras, composition algebras, pseudo-octonions, relation graphs, commuting graph.
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