English Translation of "\"Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre" by D\'enes K\H{o}nig

TL;DR

This paper explores the use of graphs as a unifying concept to connect and solve problems across determinant theory, set theory, and topology, demonstrating their geometric simplicity and equivalence of diverse problems.

Contribution

It introduces a graph-based approach to unify and address open problems in determinant theory, set theory, and topology, highlighting their interconnectedness.

Findings

Graphs serve as a unifying element across multiple mathematical fields.

The geometric simplicity of graphs aids in proving equivalences between problems.

Solutions to several open questions are provided using graph-theoretic methods.

Abstract

The presented work focuses on problems from determinant theory, set theory and topology. The term graph is the binding element that connects these problems. Graphs are distinguished by their geometrical simplicity, which helps in showing the equivalence between various seemingly unrelated problems, besides providing solutions to several open questions discussed here.