Overview
This research focuses on the orthogonality graph associated with an Okubo algebra, specifically under the condition of an isotropic norm and defined over an arbitrary field $\mathbb{F}$. The primary objective is to characterize the structural properties of these graphs, addressing aspects such as their connected components, the diameters of these components, and the nature of shortest paths between vertices.
Research Context
The study situates itself within the domain of algebraic structures and graph theory, by examining the orthogonality graph of a specific class of non-associative algebras known as Okubo algebras. These algebras are considered with an isotropic norm, a particular property related to their norm form. The field $\mathbb{F}$ over which the algebra is defined is left arbitrary, suggesting a general inquiry into the properties irrespective of the field's specific characteristics.
Approach
The research approach involves analyzing the orthogonality graph of an Okubo algebra with isotropic norm. This analysis is performed over an arbitrary field $\mathbb{F}$. The methodology focuses on three key areas of graph-theoretic inquiry:
- Describing the connected components of the orthogonality graph.
- Computing the diameters of these identified connected components.
- Investigating the existence and uniqueness of shortest paths between any pair of vertices within the graph.
Findings
The investigation yielded several specific findings regarding the orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field $\mathbb{F}$:
- The connected components of the graph were described.
- The diameters of these connected components were computed.
- It was observed that between any two vertices in the graph, there exist at most two distinct shortest paths.
- The specific conditions under which the shortest path between any given pair of vertices is unique were determined.
Why This Matters
The findings contribute to the understanding of the combinatorial and geometric properties of algebraic structures, specifically Okubo algebras, through the lens of graph theory. Characterizing connected components, diameters, and shortest paths provides fundamental insights into the internal structure and connectivity of these mathematical objects. This work establishes specific graph-theoretic properties for a defined class of algebras, enhancing the foundational knowledge in this interdisciplinary area.