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.

翻译：本文聚焦于行列式理论、集合论与拓扑学中的若干问题。图这一概念作为连接这些问题的纽带，其几何上的简洁性不仅有助于揭示各类看似无关问题之间的等价关系，更为本文探讨的若干开放性问题提供了解决方案。