Binomial edge rings of complete bipartite graphs

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.