Binomial edge rings of complete bipartite graphs

TL;DR

This paper introduces binomial edge rings derived from graphs, computes a SAGBI basis for complete bipartite graphs, and reveals their connection to Hibi rings and Plücker algebras, expanding algebraic graph theory.

Contribution

It defines binomial edge rings, calculates their SAGBI bases for complete bipartite graphs, and links these structures to Hibi rings and Plücker algebras, generalizing existing frameworks.

Findings

SAGBI basis for binomial edge rings of complete bipartite graphs is computed.

Initial algebra is isomorphic to a Hibi ring of a certain poset.

Framework generalizes the structure of Plücker algebras.

Abstract

We introduce a new class of algebras arising from graphs, called binomial edge rings. Given a graph $G$ on $d$ vertices with $n$ edges, the binomial edge ring of $G$ is defined to be the subalgebra of the polynomial ring with $2d$ variables generated by the binomials which correspond to $n$ edges. In this paper, we calculate a SAGBI basis for this algebra and obtain an initial algebra associated with this SAGBI basis in the case of complete bipartite graphs. It turns out that such an initial algebra is isomorphic to the Hibi ring of a certain poset. Similar phenomenon also occurs in the context of Pl\"{u}cker algebras, so the framework of binomial edge rings can be interpreted as a kind of its generalization.