Hilbert Series Of Binomial Edge Ideals

TL;DR

This paper computes the Hilbert series of binomial edge ideals for various classes of graphs, including decomposable graphs, joins, and specific graph families, providing explicit formulas and decompositions.

Contribution

It introduces formulas for the Hilbert series of binomial edge ideals of decomposable graphs and their joins, extending understanding to several important graph classes.

Findings

Hilbert series of decomposable graphs expressed via indecomposable subgraphs

Hilbert series of joins of two graphs derived

Explicit Hilbert series for complete k-partite, fan, multi-fan, and wheel graphs

Abstract

Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in the polynomial ring $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the Hilbert series of binomial edge ideal of decomposable graphs in terms of Hilbert series of its indecomposable subgraphs. Also, we compute the Hilbert series of binomial edge ideal of join of two graphs and as a consequence we obtain the Hilbert series of complete $k$-partite graph, fan graph, multi-fan graph and wheel graph.