Relation graphs of the sedenion algebra

A. E. Guterman ; S. A. Zhilina

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 61-86

Résumé

Let $\mathbb{S}$ denote the algebra of sedenions and let $\Gamma_O(\mathbb{S})$ denote its orthogonality graph. We observe that any pair of zero divisors in $\mathbb{S}$ produces a double hexagon in $\Gamma_O(\mathbb{S})$. The set of vertices of a double hexagon can be extended to a basis of $\mathbb{S}$ that has a convenient multiplication table. We explicitly describe the set of vertices of an arbitrary connected component of $\Gamma_O(\mathbb{S})$ and find its diameter. Then we establish a bijection between the connected components of $\Gamma_O(\mathbb{S})$ and the lines in the imaginary part of the octonions. Finally, we consider the commutativity graph of the sedenions and discover that all elements whose imaginary parts are zero divisors belong to the same connected component, and its diameter lies in between $3$ and $4$.