From Chio Pivotal Condensation to the Matrix-Tree theorem

TL;DR

This paper introduces a determinant identity that unifies and generalizes the Chio pivotal condensation theorem and the Matrix-Tree theorem, providing a broader mathematical framework.

Contribution

The paper presents a new determinant identity that extends classical theorems, offering a unified approach to pivotal condensation and graph enumeration.

Findings

Derived a generalized determinant identity

Unified Chio pivotal condensation and Matrix-Tree theorem

Potential applications in combinatorics and matrix theory

Abstract

We show a determinant identity which generalizes both the Chio pivotal condensation theorem and the Matrix-Tree theorem.