Bipartite graphs as polynomials and polynomials as bipartite graphs

Abstract

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial p ∈ N[x], and any directed finite bipartite graph can be considered as a polynomial p ∈ N[x, y], and vise verse. We also show that the multiplication in semirings N[x], 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 N[x], N[x, y]. Finally, we endow the set of all bipartite graphs with the Zariski topology.