Some Exceptional Cases in Mathematics: Euler Characteristic, Division Algebras, Cross Vector Product and Fano Matroid

TL;DR

This paper reviews exceptional mathematical results across graph theory, division algebras, vector cross products, and matroid theory, highlighting their unique properties and potential interconnections to motivate future research.

Contribution

It synthesizes key results in four mathematical areas and suggests exploring their links, providing a foundation for further interdisciplinary research.

Findings

Euler relation and its higher-dimensional generalization

Dimensional theorem for division algebras including Hurwitz theorem

Restrictions on vector cross product dimensions

Abstract

We review remarkable results in several mathematical scenarios, including graph theory, division algebras, cross product formalism and matroid theory. Specifically, we mention the following subjects: (1) the Euler relation in graph theory, and its higher-dimensional generalization, (2) the dimensional theorem for division algebras and in particular the Hurwitz theorem for normed division algebras, (3) the vector cross product dimensional possibilities, (4) some theorems for graphs and matroids. Our main goal is to motivate a possible research work in these four topics, putting special interest in their possible links.