Relation graphs of the sedenion algebra

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 $4$.

References in corpus (2)

• Orthogonality graphs of real Cayley-Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons
• Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2n-ions Beyond the Sedenions

Cited by in corpus (2)

• On doubly alternative zero divisors in Cayley-Dickson algebras
• Diameter of the commutativity graph of the real sedenions