The new simplest proof of Ceyley's formula and connections with Kirkwood-Salzburg equations

TL;DR

This paper presents a new, simple proof for counting labeled rooted forest-graphs, which includes Cayley's formula as a special case, providing a clearer understanding of these combinatorial structures.

Contribution

The paper introduces a novel, simplified proof for the enumeration of labeled rooted forest-graphs, connecting it with Kirkwood-Salzburg equations.

Findings

Derived a new proof for the count of labeled rooted forest-graphs

Established a connection with Kirkwood-Salzburg equations

Provided a simplified approach to Cayley's formula

Abstract

A new very simple proof of the number of labeled rooted forest-graphs with a given number of vertices is given. As a partial case of this formula we have Cayley's formula.