A Bijective Proof For Forest Reciprocity Theorem

TL;DR

This paper presents a new bijective proof for the Forest Reciprocity Theorem related to a graph polynomial counting spanning rooted forests, extending Prufer coding as a special case, thus deepening understanding of graph enumeration.

Contribution

The paper introduces a novel bijective proof for the Forest Reciprocity Theorem, linking it to Prufer coding and enhancing combinatorial graph theory methods.

Findings

Established a new bijective proof for the reciprocity property

Connected the theorem to Prufer coding as a special case

Enhanced combinatorial understanding of spanning rooted forests

Abstract

In this paper, we study the graph polynomial that counts spanning rooted forests f_g of a given graph. This polynomial has a remarkable reciprocity property. We give a new bijective proof for this theorem which has Prufer coding as a special case.