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

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.