# Bipartite Graphs as Polynomials, and Polynomials as Bipartite Graphs (with a view towards dividing in $\mathbb{N}[x],$ $\mathbb{N}[x,y]$)

## TL;DR

This paper establishes a correspondence between bipartite graphs and polynomials in natural number semirings, providing new insights into graph operations, polynomial division, and topological structures on graphs.

## Contribution

It introduces a novel representation of bipartite graphs as polynomials in $
[x]$ and $
[x,y]$, linking graph operations to polynomial algebra and topology.

## Key findings

- Graph-polynomial correspondence for undirected and directed bipartite graphs
- Polynomial multiplication corresponds to graph operations
- A new perspective on division in semirings $
[x]$, $
[x,y]$

## Abstract

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial $p \in \mathbb{N}[x]$, and any directed finite bipartite graph can be considered as a polynomial $p\in\mathbb{N}[x,y]$, and vise verse. We also show that the multiplication in semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$ correspondences to a operations of the corresponding graphs which looks like a ``perturbed'' products of graphs. As an application, we give a new point of view to dividing in semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$. Finally, we endow the set of all bipartite graphs with the Zariski topology.