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arXiv · 2608.26903

Relation graphs of the sedenion algebra

Alexander Guterman·Svetlana Zhilina
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Abstract

Let $\mathbb{S}$ denote the algebra of the sedenions, and $\Gamma_O(\mathbb{S})$ 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 $\mathbb{S}$ which has a convenient multiplication table. We describe explicitly the set of vertices of an arbitrary connected component of $\Gamma_O(\mathbb{S})$ and find its diameter. We then establish the bijection between the connected components of $\Gamma_O(\mathbb{S})$ and lines in the imaginary part of the octonions. Finally, we consider the commutativity graph of the sedenions and discover that all elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies between $3$ and $4$.

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Alexander Guterman, Svetlana Zhilina. 2026-08-27. Relation graphs of the sedenion algebra. https://doi.org/10.1007/s10958-021-05367-6

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